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+{"text":"\n\n\\subsubsection*{Acknowledgments}\nThe authors thank the ArTeCS Group from Universidad Complutense de Madrid for letting use their Xeon Phi KNL system.\n\n\n\\bibliographystyle{splncs03}\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nKNL is the second generation of Xeon Phi family and features new technologies in SIMD execution and memory access.\nIn this paper, we have evaluated a set of programming and optimization techniques for these processors taking the FW algorithm as a representative case study of graph and memory-bound applications. Among the main contributions of this research we can summarize:\n\\begin{itemize}\n\\item Blocking technique not only improved performance but also allowed us to apply a coarse-grain workload distribution in the parallel implementation.\n\n\\item SIMD exploitation was crucial to achieve top performance. In particular, the serial version run 2.9$\\times$, 6$\\times$ and 15.5$\\times$ faster using the SSE, AVX2 and AVX-512 extensions, respectively.\n\\item Aligning memory accesses and loop unrolling also showed significant speedups.\n\\item A single thread per core was enough to get maximal performance. In addition, \\emph{scatter} and \\emph{balanced} affinities provided extra performance.\n\\item Besides keeping portability, guided vectorization led to slightly better performance than the intrinsic counterpart, running upto 1.03$\\times$ faster. \n\\item MCDRAM usage demonstrated to be an efficient strategy to tolerate high-bandwidth demands with practically null programmer intervention, even when the dataset largely exceeded the MCDRAM size. In particular, it produced an average speedup of 9.8$\\times$ and a maximum speedup of 15.5$\\times$\n\\end{itemize}\n\nAs future work, we consider evaluating programming and optimization techniques in other cluster and memory modes as a way to extract more performance.\n\n\n\n\n\\section{Floyd-Warshall Algorithm}\n\\label{sec:floyd}\n\nThe FW algorithm uses a dynamic programming approach to compute the all-pairs shortest-paths\nproblem on a directed graph~\\cite{Floyd,Warshall}. \nThis algorithm takes as input a $N \\times N$ distance matrix $D$, where $D_{i,j}$ is initialized with the original distance from node \\emph{i} to node \\emph{j}. FW runs for $N$ iterations and at \\emph{k}-th iteration it evaluates all the possible paths between each pair of vertices from \\emph{i} to \\emph{j} through the intermediate vertex \\emph{k}. As a result, FW produces an updated matrix $D$, where $D_{i,j}$ now contains the shortest distance between nodes \\emph{i} and \\emph{j}. Besides, an additional matrix $P$ is generated when the reconstruction of the shortest path is required. $P_{i,j}$ contains the most recently added intermediate node between \\emph{i} and \\emph{j}. Figure~\\ref{fig:floyd_naive} exhibits the naive FW algorithm.\n\n\n\n\n\\begin{figure}[b]\n    \\centering\n    \\begin{minipage}{0.53\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{floyd_naive}\n        \\caption{\\label{fig:floyd_naive} Naive Floyd-Warshall Algorithm}\n    \\end{minipage}\\hfill\n    \\begin{minipage}{0.43\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{floyd_blocked}\n        \\caption{\\label{fig:floyd_blocked} Schematic representation of the blocked Floyd-Warshall Algorithm}\n    \\end{minipage}\n\\end{figure}\n\\section{Implementation}\n\\label{sec:implementation}\n\nIn this section, we address the optimizations performed on the Intel Xeon Phi KNL processor. First of all, we developed a serial implementation following the naive version described in Figure~\\ref{fig:floyd_naive}, as this implementation will work as baseline. Next, we optimized the serial version considering data locality and data level parallelism. Finally, we introduced thread level parallelism exploiting the OpenMP programming model to obtain a multi-threaded implementation.\n\n\\subsection{Data Locality}\n\\label{subsec:data_locality}\n\nTo improve data locality, the FW algorithm can be blocked~\\cite{floyd_blocked2}. Unfortunately, the three loops can not be interchanged in free manner due to the data dependencies from one iteration to the next in the \\emph{k}-loop (just \\emph{i} and \\emph{j} loops can be done in any order). However, under certain conditions, the \\emph{k}-loop can be put inside the \\emph{i}-loop and \\emph{j}-loop, making blocking possible. \nThe distance matrix  $D$ is partitioned into blocks of size $BS \\times BS$, so that there are $(N\/BS)^2$ blocks. The computations involve $R=N\/BS$ rounds and each round is divided into four phases based on the data dependency among the blocks:\n\\begin{enumerate}\n\\item Update the block \\emph{k,k} ($D^{k,k}$) because it is self-dependent.\n\\item Update the remaining blocks of the \\emph{k}-th row because each of these blocks depends on itself and the previously computed $D^{k,k}$.\n\\item Update the remaining blocks of the \\emph{k}-th column  because each of these blocks depends on itself and the previously computed $D^{k,k}$.\n\\item Update the rest of the matrix blocks as each of them depends on the \\emph{k}-th block of its row and the \\emph{k}-th block of its column.\n\\end{enumerate}\nIn this way, we satisfy all dependencies from this algorithm. Figure~\\ref{fig:floyd_blocked} shows a schematic representation of a round computation and the data dependences among the blocks while Figure~\\ref{fig:floyd_blocked_alg} presents the corresponding pseudo-code.\n\n\n\\begin{figure}[t]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{floyd_blocked_alg}\n\\par\\end{centering}\n\\caption{\\label{fig:floyd_blocked_alg} Blocked Floyd-Warshall algorithm.}\n\\end{figure}\n\n\\subsection{Data Level Parallelism}\n\\label{subsec:data_level_parallelism}\n\nThe innermost loop of FW\\_BLOCK code block from Figure~\\ref{fig:floyd_blocked_alg} is clearly the most computationally expensive part of the algorithm. In that sense, this loop is the best candidate for vectorization. The loop body is composed of an \\emph{if} statement that involves one addition, one comparison and (may be) two assign operations. Unfortunately, the compiler detects false dependencies in that loop and is not able to generate SIMD binary code. For that reason, we have explored two SIMD exploitation approaches: (1) guided vectorization through the usage of the OpenMP 4.0 \\emph{simd} directive  and (2) intrinsic vectorization employing the AVX-512 extensions. The guided approach simply consists of inserting the \\emph{simd} directive to the innermost loop of FW\\_BLOCK code block (line 4). On the opposite sense, the intrinsic approach consists of rewriting the entire loop body. Figures~\\ref{fig:fw_block_guided} and~\\ref{fig:fw_block_intrinsic} show the pseudo-code for FW\\_BLOCK implementation using guided and manual vectorization, respectively. In order to accelerate SIMD computation with 512-bit vectors, we have carefully managed the memory allocations so that distance and path matrices are 64-byte aligned. In the guided approach, this also requires adding the \\emph{aligned} clause to the \\emph{simd} directive. \n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{minipage}{0.43\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{FW_BLOCK_guided}\n        \\caption{\\label{fig:fw_block_guided} Pseudo-code for FW\\_BLOCK implementation using guided vectorization}\n    \\end{minipage}\\hfill\n    \\begin{minipage}{0.53\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{FW_BLOCK_intrinsic}\n        \\caption{\\label{fig:fw_block_intrinsic} Pseudo-code for FW\\_BLOCK implementation using intrinsic vectorization}\n    \\end{minipage}\n\\end{figure}\n\n\n\n\n\\subsection{Loop Unrolling}\n\\label{subsec:loop_unrolling}\n\nLoop unrolling is another optimization technique that helped us to improve the code performance. Fully unrolling the innermost loop of FW\\_BLOCK code block was found to work well. Unrolling the \\emph{i}-loop of the same code block once was also found to work well.\n\n\\subsection{Thread Level Parallelism}\n\\label{subsec:thread_level_parallelism}\n\nTo exploit parallelism across multiple cores, we have implemented a multi-threaded version of FW algorithm based on OpenMP programming model. A \\emph{parallel} construct is inserted before the loop of line 13 in Figure~\\ref{fig:floyd_blocked_alg} to create a parallel block. To respect data dependencies among the block computations, the work-sharing constructs must be carefully inserted. At each round, phase 1 must be computed before the rest. So a \\emph{single} construct is inserted to enclose line 16. Next, phases 2 and 3 must be computed before phase 4. As these blocks are independent among them, a \\emph{for} directive is inserted before the loops of lines 18 and 22. Besides, \na \\emph{nowait} clause is added to the phase 2 loop to alleviate the thread idling. Finally, another \\emph{for} construct is inserted before the loop of line 26 to distribute the remaining blocks among the threads.\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe power consumption problem represents one of the major obstacles for Exascale systems design. As a\nconsequence, the scientific community is searching for different ways to improve power efficiency of High Performance Computing (HPC) systems~\\cite{Giles2014}. One recent trend to increase compute power and, at the same time, limit power consumption of these systems lies in adding accelerators, like NVIDIA\/AMD graphic processing units (GPUs), or Intel Many Integrated Core (MIC) co-processors. These manycore devices are capable of achieving better FLOPS\/Watt ratios than traditional CPUs. For example, the number of Top500~\\cite{Top500} systems using accelerator technology grew from 54 in June 2013 to 91 in June 2017. In the same period, the number of systems based on accelerators increased from 55 to 90 on the Green500 list~\\cite{Green500}.\n\nRecently, Intel has presented the second generation of its MIC architecture (branded Xeon Phi), codenamed Knigths Landing (KNL).  Among the main differences of KNL regarding its predecessor Knights Corner (KNC), we can find the incorporation of AVX-512 extensions, a remarkable number of vector units increment, a new on-package high-bandwidth memory (HBM) and the ability to operate as a standalone processor. Even though optimizing applications on CPUs, GPUs and KNC Xeon Phi's has been largely studied in the last years, accelerating applications on KNL processors is still a pending task due to its recent commercialization. In that sense, the new features in KNL processors require the revision of programming and optimization techniques for these devices. \n\nIn this work, we selected the Floyd-Warshall (FW) algorithm as a representative case study of graph and memory-bound applications. This algorithm finds the shortest paths between all pairs of vertices in a graph and occurs in domains of communication networking~\\cite{Floyd_networking}, traffic routing~\\cite{floyd_traffic}, bioinformatics~\\cite{floyd_bioinformatics}, among others. FW is both computationally and spatially expensive since it requires $O(n^3)$ operations and $O(n^2)$ memory space, where $n$ is the number\nof vertices in a graph. Starting from the default serial version, we show how data, thread and compiler level optimizations help the parallel implementation to reach 338 GFLOPS.\n\n\n\nThe rest of the present paper is organized as follows. Section~\\ref{sec:xeon_phi} briefly introduces the Intel Xeon Phi KNL architecture while Section~\\ref{sec:floyd} presents the FW algorithm. Section~\\ref{sec:implementation} describes our implementation. In Section~\\ref{sec:results} we analyze performance results while Section~\\ref{sec:related_works} discusses related works. Finally, Section~\\ref{sec:conclusion} outlines conclusions and future lines of work.\n\n\n\n\\section{Intel Xeon Phi Knights Landing}\n\\label{sec:xeon_phi}\n\nKNL is the second generation of the Intel Xeon Phi family and the first capable of operating as a standalone processor. The KNL architecture is based on a set of \\emph{Tiles} (up to 36) interconnected by a 2D mesh. Each Tile includes 2 cores based on the out-of-order Intel's Atom micro-architecture (4 threads per core), 2 Vector Processing Units (VPUs) and a shared L2 cache of 1 MB. These VPUs not only implement the new 512-bit AVX-512 ISA but they are also compatible with prior vector ISA's such as SSE\\emph{x} and AVX\\emph{x}. \nAVX-512 provides 512-bit SIMD support, 32 logical registers, 8 new mask registers for vector predication, and gather and scatter instructions to support loading and storing sparse data. As each\nAVX-512 instruction can perform 8 double-precision (DP) operations\n(8 FLOPS) or 16 single-precision (SP)  operations (16 FLOPS), the peak performance is over 1.5 TFLOPS in DP and 3 TFLOPS in SP, more than two times higher than that of the KNC. It is also more energy efficient than its predecessor~\\cite{KNLbook}.\n\nOther significant feature of the KNL architecture is the inclusion of an in-package HBM called MCDRAM. This special memory offers 3 operating modes: \\emph{Cache}, \\emph{Flat} and \\emph{Hybrid}. In Cache mode, the MCDRAM is used like an L3 cache, caching data from the DDR4 level\nof memory. Even though application code remains unchanged, the MCDRAM can suffer lower performance rates.  In Flat mode, the MCDRAM has a physical addressable space offering the highest bandwidth\nand lowest latency. However, software modifications may be required in order to use both the DDR and the MCDRAM in the same application. Finally, in the \\emph{Hybrid mode}, HBM is divided in two parts: one part in \\emph{Cache mode} and one in \\emph{Flat mode}~\\cite{knl_best_practice_guide}.\n\n\nFrom a software perspective, KNL supports  parallel programming models used traditionally on HPC systems such as OpenMP or MPI. This fact represents a strength of this platform since it simplifies code development and improves portability over other alternatives based on accelerator specific programming languages such as CUDA or OpenCL.\n However, to achieve high performance, programmers should attend to:\n\\begin{itemize}\n\\item the efficient exploitation of the memory hierarchy, especially\nwhen handling large datasets, and\n\\item how to structure the computations to take advantage of the VPUs.\n\\end{itemize}\nAutomatic vectorization is obviously the easiest programming way to exploit VPUs. However, in most cases the compiler is unable to generate SIMD binary code since it can not detect free data dependences into loops. In that sense, SIMD instructions are supported in KNL processors through the use of guided compilation or hand-tuned codification with intrinsic instructions~\\cite{KNLbook}.  On one hand, in guided vectorization,  the programmer indicates the compiler (through the insertion of tags) which loops are independent and their memory pattern access. In this way, the compiler is able to generate SIMD binary code preserving the program portability. On the other hand, intrinsic vectorization usually involves rewriting most of the corresponding algorithm. The programmer gains in control at the cost of losing portability. Moreover, this approach also suggests the inhibition of other compiler loop-level optimizations. Nevertheless, it is the only way to exploit parallelism in some applications with no regular access patterns or loop data dependencies which can be hidden by recomputing techniques~\\cite{Culler97}. \n\n\n\n\n\n\\section{Related Works}\n\\label{sec:related_works}\n\nDespite its recent commercialization, there are some works that evaluate KNL processors. In that sense, we highlight~\\cite{Rosales2016_mcdram} that presents a study of the performance differences observed when using the three MCDRAM configurations available in combination with the three possible memory access or cluster modes. Also, Barnes et al.~\\cite{KNL_NERSC} discussed the lessons learned from optimizing a number of different high-performance applications and kernels. Besides, Haidar et al.~\\cite{Haidar2016_KNL} proposed and evaluated several optimization techniques for different matrix factorizations methods on many-core systems.\n\nObtaining high-performance in graph algorithms is usually a difficult task since they tend to suffer from irregular dependencies and large space requirements. Regarding FW algorithm, there are many works proposed to solve the all-pairs shortest paths problem on different harwdare architectures. However, to the best of the authors knowledge, there are no related works with KNL processors.\nHan and Kang~\\cite{floyd_han} demonstrated that exploiting SSE2 instructions led to 2.3$\\times$-5.2$\\times$ speedups over a blocked version. Bondhugula et al.~\\cite{floyd_fpga} proposed a tiled parallel implementation using Field Programmable Gate Arrays. In the field of GPUs, we highlight the work of Katz and Kider~\\cite{floyd_katz_gpu}, who proposed a shared memory cache efficient implementation to handle graph sizes that are inherently larger than the DRAM memory available on the device. Also, Matsumoto et al.~\\cite{floyd_matsumoto_cpu_gpu} presented a blocked algorithm for hybrid CPU-GPU systems aimed to minimize host-device communication. Finally, Hou et al.~\\cite{floyd_knc} evaluated different optimization techniques for Xeon Phi KNC coprocessor. Just as this study, they found that blocking and vectorization are key aspects in this problem to achieve high performance. Also, guided vectorization led to better results than the manual approach, but with larger performance differences. Contrary to this work, their implementation benefited from using more than one thread per core. However, as stated before, there are significant architectural differences between these platforms that support this behavior.\n\n\n\n\n\\section{Experimental Results}\n\\label{sec:results}\n\n\\subsection{Experimental Design}\nAll tests have been performed on an Intel server running CentOS 7.2 equipped with a Xeon Phi 7250 processor 68-core 1.40GHz (4 hw thread per core and 16GB MCDRAM memory) and 48GB main memory. The processor was run in \\emph{Flat} memory mode and \\emph{Quadrant} cluster mode.\n\nWe have used Intel's ICC compiler (version 17.0.1.132) with the \\emph{-O3} optimization level. To generate explicit AVX2 and AVX-512 instructions, we employed the \\emph{-xAVX2} and \\emph{-xMIC-AVX512} flags, respectively. Also, we used the \\emph{numactl} utility to exploit MCDRAM memory (no source code modification is required). Besides, different workloads were tested: \\emph{N} = \\{4096, 8192, 16384, 32768, 65536\\}. \n\n\\subsection{Performance Results}\n\\label{sec:perf-knl}\n\nFirst, we evaluated the performance improvements of the different optimization techniques applied to the naive serial version, such as blocking (\\emph{blocked}), data level parallelism (\\emph{simd}, \\emph{simd (AVX2)} and \\emph{simd (AVX-512)}), aligned access (\\emph{aligned}) and loop unrolling (\\emph{unrolled}). Table~\\ref{tab:performance_serial} shows the execution time (in seconds) of the different serial versions when \\emph{N}=4096. As it can be observed, blocking optimization reduces execution time by 5\\%. \nRegarding the block size, 256 $\\times$ 256 was found to work best. In the most memory demanding case of each round (phase 4), four blocks are loaded into the cache (3 distance blocks and 1 path block). The four blocks requires 4 $\\times$ 256 $\\times$ 256 $\\times$ 4 bytes = 1024 KB = 1MB, which is exactly the L2 cache size.\n\nAs stated in Section~\\ref{subsec:data_level_parallelism}, the compiler is not able to generate SIMD binary code by itself in the blocked version. Adding the corresponding \\emph{simd} constructs to the blocked version reduced the execution time from 572.66 to 204.52 seconds, which represents a speedup of 2.8$\\times$. However, AVX-512 instructions can perform 16 SP operations at the same time. After inspecting the code at assembly level, we realized that the compiler generates SSE\\emph{x} instructions by default. As SSE\\emph{x} can perform 4 SP operations at the same time, the 2.8$\\times$ speedup has more sense since not all the code can be vectorized. Next, we re-compiled the code including the \\emph{-xAVX2} and \\emph{-xMIC-AVX512} flags to force the compiler to generate AVX2 and AVX-512 SIMD instructions, respectively. AVX2 extensions accelerated the blocked version by a factor of 5.8$\\times$ while AVX-512 instructions achieved an speedup of 15.5$\\times$. So, it is clear that this application benefits from larger SIMD width. In relation to the other optimization techniques employed, we have found that the \\emph{simd (AVX-512)} implementation runs 1.11$\\times$ faster when aligning memory accesses in AVX-512 computations (\\emph{aligned}). Additionally, applying the loop unrolling optimization to the \\emph{aligned} version led to higher performance, gaining a 1.45$\\times$ speedup. In summary, we achieve a 26.3$\\times$ speedup over the naive serial version through the combination of the different optimizations described.\n\n\\begin{table} [t!]\n\\centering\n\\caption{\\label{tab:performance_serial} Execution time (in seconds) of the different optimization techniques applied to the naive serial version when \\emph{N}=4096.}\n\\begin{tabular*}{12cm}{@{\\extracolsep{\\fill}}>{\\centering}p{1.2cm}>{\\centering}m{1.2cm}>{\\centering}m{1.4cm}>{\\centering}p{2.3cm}>{\\centering}m{2.7cm}>{\\centering}m{1.3cm}>{\\centering}m{1.4cm}}\n\\hline \n\\noalign{\\vskip0.2cm}\n\\emph{naive} & \\emph{blocked} & \\emph{simd} & \\emph{simd (AVX2)} & \\emph{simd (AVX-512)} & \\emph{aligned} & \\emph{unrolled}\\tabularnewline\n\\hline \n\\noalign{\\vskip0.2cm}\n602.8 & 572.66 & 204.52 & 100.47 & 36.95 & 33.28 & 22.95\\tabularnewline\n\\hline \n\\end{tabular*}\n\\end{table}\n\nTaking the optimized serial version, we developed a multi-threaded implementation as described in Section~\\ref{subsec:thread_level_parallelism}. Figure~\\ref{fig:performance_threads_affinity} shows the performance (in terms of GFLOPS) for the different affinity types used varying the number of threads when \\emph{N}=8192. As expected, \\emph{compact} affinity produced the worst results since it favours using all threads on a core before using other cores. \\emph{Scatter} and \\emph{balanced} affinities presented similar performances improving the \\emph{none} counterpart. As the KNL processor used in this study has all its cores in the same package, \\emph{scatter} and \\emph{balanced} affinities distribute the threads in the same manner when one thread per core is assigned.\nRegarding the number of threads, using a single thread per core is enough to get maximal performance (except in \\emph{compact} affinity). This behavior is opposed to the KNC generation where two or more threads per core where required to achieve high performance. However, it should not be a surprise since the KNL cores were designed to optimize single thread performance including out-of-order pipelines and two VPUs per core.\n\nIt is important to remark that, unlike the optimized serial version, the parallel implementation used a smaller block size  since it delivered higher performance. A smaller block size allowed a finer-grain workload distribution and decreased thread idling, especially when the number of threads was larger than the number of blocks in phases 2 and 3. Another reason to decrease block size was that the L2 available space is now shared between the threads in a tile, contrary to the single threaded case. In particular, \\emph{BS}=64 was found to work best.\n\n\\begin{figure}[t!]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{performance_threads_affinity}\n\\par\\end{centering}\n\\caption{\\label{fig:performance_threads_affinity}Performance for the different affinity types used varying the number of threads  when \\emph{N}=8192.}\n\\end{figure}\n\nFigure~\\ref{fig:performance_n_mcdram} illustrates performance evolution varying workload and MCDRAM exploitation for the different vectorization approaches. For small workloads (\\emph{N} = 8192), the performance improvement is little ($\\sim$1.1$\\times$). However, MCDRAM memory presents remarkable speedups for greater workloads, even when the dataset largely exceeds the MCDRAM size (\\emph{N} = 655536). In particular, MCDRAM exploitation achieves an average speedup of 9.8$\\times$ and a maximum speedup of 15.5$\\times$. In this way, we can see how MCDRAM usage is an efficient strategy for bandwidth-sensitive applications.\n\nIn relation to the vectorization approach, we can appreciate that guided vectorization leads to slightly better performance than the intrinsic counterpart, running upto 1.03$\\times$ faster. The best performances are 330 and 338 GFLOPS for the intrinsic and guided versions, respectively. After analyzing the assembly code, we realized that this difference is caused by the prefetching instructions introduced by the compiler when\nguided vectorization is used. Unfortunately, the compiler disables automatic prefetching when code is manually vectorized.\n\n\\begin{figure}[t]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{performance_n_mcdram}\n\\par\\end{centering}\n\\caption{\\label{fig:performance_n_mcdram}Performance evolution varying workload and the MCDRAM exploitation.}\n\\end{figure}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{SectionIntroduction}\n\nIn this paper we start the program of developing a general theory of rough PDEs aiming at extending classical PDE tools such as weak solutions, a priori estimates, compactness results, duality. This is a quite unexplored territory where few tools are available, so as a start, we will content ourselves in this work with the study of linear symmetric hyperbolic systems of the form\n\\begin{equation}\n\\label{eq:basic}\n\\partial_t f + a\\nabla f = 0,\n\\end{equation}\nwhere $f$ is an $\\RR^N$-valued space-time distribution on $\\RR_+\\times\\RR^d$, and $a :\\RR_+\\times \\RR^d \\to  \\textrm{L}(\\RR^d, \\RR^{N\\times N})$ is a $N\\times N$ matrix-valued family of time--dependent vector fields in $\\RR^d$. This setting includes as a particular case scalar transport equations. Moreover we restrict our attention to the case where the matrix-valued vector field $a$ is only a distribution in the time variable, rather than a regular bounded function. We however retain some smoothness assumption in the space variable, as expected from the fact that general transport equations do not possess the regularisation properties needed to drive them with space-time irregular signals. Even in the classical setting it is known that non-regular coefficients  can give rise to non-uniqueness of weak solutions~\\cite{FlandoliMilano}.\n\n\\smallskip\n\nWhen $a$ is only a distribution in the time variable the above weak formulation is not available since in the classical setting solutions are considered in spaces like $C([0,T],\\LL^2(\\RR^d))$ for general symmetric systems  or $L^\\infty(\\RR\\times \\RR^d)$ for scalar transport equation. In this case, the product $a\\nabla f$ is not well-defined, not even in a distributional setting. Rough paths have their origin in the need to handle such difficulties in the case of ordinary differential equations driven by distribution valued signals \\cite{Lyons98,Lyons2002,Lyons2007,Friz2010}. Controlled rough paths have been introduced in \\cite{Gubinelli04} as a setting for considering more general problems; they were used successfully in the study of some stochastic partial differential equations~\\cite{Bessaih2005,GT,Deya2012,Hairer2011,Gubinelli2012a,Hairer2012,Hairer2013},  including the remarkable solution by Hairer of the KPZ equation \\cite{HairerKPZ}. \n\n\\smallskip\n\nThese developments have ultimately lead to the notion of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski~\\cite{GIP} and to Hairer's general theory of regularity structures~\\cite{HairerRegularity},  providing a  framework for the analysis of non-linear operations on distributions. Despite their successes, these new tools and methods are somehow designed to deal with a prescribed class of singular PDEs which is far from exhausting the set of all interesting singular PDEs. \n\n\\smallskip\n\nPDEs with irregular signals have been studied using directly rough path methods also by Friz and co-authors~\\cite{Caruana2009, CaruanaFrizOberhauser, FrizOberhauser1, FrizOberhauser2, FrizGess}. They have developed an approach to some  fully non-linear PDEs and conservation laws driven by rough signals by interpreting these equations  as transformations of classical PDEs, generalising the method of characteristics. Subsequently a combination of rough path stability and PDE stability allows to go from smooth signal to the wider class of signals described by rough paths. Entropy solutions to scalar conservation laws with rough drivers have also been analysed also by P.-L. Lions, Souganidis and coauthors~\\cite{LionsPerthameSouganidis1,LionsPerthameSouganidis2, GessSouganidis}. A major drawback of this otherwise effective approach is that there is no intrinsic notion of solution to the PDE and that the study of the properties of the PDE has to be done on a global level.\n\n\\smallskip\n\nIn recent works intrinsic notions of weak solution to rough PDEs have been proposed by Tindel, Gubinelli and Torecilla~\\cite{GTT} for viscosity solutions,  Catellier~\\cite{CatellierTransport} for weak solutions to linear transport equations (see also Hu and Le~\\cite{HuLe} for classical solutions to transport equations) and more recently by Diehl, Friz and Stannat~\\cite{DiehlFrizStannat} for a general class of parabolic equations. All these notions are based on a weak formulation of the equation where the irregularity of some data is taken into account via the framework of controlled paths introduced in~\\cite{Gubinelli04}. However in all these papers explicit formulas involving the flow of rough characteristics play an important role, and this sets apart the study of rough PDEs with respect to the study of weak solutions to more regular PDEs. One of the main motivations of our investigations is an effort to understanding what kind of robust arguments can be put forward in the study of the a priori regularity of distributions satisfying certain rough PDEs formulated in the language of controlled paths. Extensions to regularity structures or paracontrolled distributions will be considered in forthcoming work. \n\n\\bigskip\n\nWe study equation \\eqref{eq:basic} by working in the technically easier setting of controlled paths. To motivate our formalism, note that a formal integration of the weak formulation \\eqref{eq:basic} over any time interval $[s,t]$, gives an equation of the form\n$$\nf_t = f_s + \\int_s^t V_r f_r dr, \n$$ \nwhere $V_r = a_r\\nabla$ is a matrix-valued vector-field and $f_r(x) = f(r,x)$, is a convenient notation of the distribution $f$ evaluated at time $r$, assuming this make sense. An expansion for the time evolution of $f$ is obtained by iterating the above equation, and reads\n\\begin{equation}\n\\label{eq:basic-increments}\nf_t = f_s + A^1_{ts} f_s + A^2_{ts} f_s + R_{ts},\n\\end{equation}\nwhere\n$$\nA^1_{ts} = \\int_s^t V_r  dr,\\qquad\\textrm{ and }\\qquad A^2_{ts} = \\int_s^t \\int_s^r V_r V_{r'} dr' dr,\n$$\nare respectively a first order differential operator (that is a vector field) and a second order differential operator, for each $s\\leq t$. As a function of $(s,t)$, they satisfy formally \\textit{Chen's relation}\n\\begin{equation}\n\\label{eq:operator-chen}\nA^2_{ts} =A^2_{tu}+ A^2_{us} + A^{1}_{tu}A^{1}_{us}\n\\end{equation}\nfor all $0\\leq s \\leq u \\leq t$. It is a key observation of rough path theory that equation \\eqref{eq:basic-increments} can be used as a replacement for the differential or integral formulation \nof equation \\eqref{eq:basic} if the remainder term can be shown to be sufficiently small as $t-s$ goes to $0$, in a suitable sense.\n\n\\medskip\n\nWe shall call a \\emph{rough driver} an operator-valued $2$-index maps ${\\bfA}_{ts}=\\big(A^1_{ts},A^2_{ts}\\big)$ satisfying the operator Chen relation \\eqref{eq:operator-chen} and some regularity assumptions. Building on the above picture, a path with values in some Banach space where the operators $A^1_{ts}$ and $A^2_{ts}$ act, will be said to solve the rough linear equation\n$$\ndf_s = {\\bfA}(ds)\\,f_s.\n$$\nif the Taylor expansion \\eqref{eq:basic-increments} holds.\n\n\\smallskip\n\nThere is a complete theory of such equations in the case where the equation is set in a Banach algebra and the operators $A^1_{ts}$ and $A^2_{ts}$ are given by left multiplication by some elements of the algebra. It is however natural, in the present PDE setting, to consider also unbounded operators $A^1, A^2$, which makes the use of rough paths ideas non-trivial, unless we work in the analytic category or in similar topologies. \n\n\\medskip\n \nWe lay out in this work a theory of such rough linear equation driven by unbounded drivers $\\bfA$, and obtain some a priori estimates that are used to study the well--posedness of some classes of linear symmetric systems in $\\LL^2$ and of the rough transport equation in $\\LL^\\infty$. The major difficulty which has to be overcome is the lack of a Gronwall lemma for rough equations and the main contribution to this paper is to develop suitable a priori estimates on weak controlled solutions that replace the use of Gronwall lemma in classical proofs. Along the way we refine the standard theory of controlled path by introducing weighed norms compatible with the sewing map and by revisiting the theory of linear rough differential equations in the context of bounded drivers. \n\n\\medskip\n\nAs a guide for the reader, here is how we have organised our work. Section~\\ref{SectionWeightedNorms} provides a refined version of the sewing lemma that allows to keep track of the growth in time of the additive function associated with an almost-additive $2$-index map. This result is used in Section~\\ref{SectionBoundedDrivers} in the proof of the well-posed character of linear differential equations driven by the bounded rough drivers defined there. \\emph{Unbounded rough drivers} are introduced in Section~\\ref{SectionUnboundedRoughDrivers}, where some fundamental a priori estimate is proved. An $\\LL^2$ theory of rough linear equations is developed for a class of unbounded drivers, that contains as a particular example the rough linear transport equation. Our main workhorse here is a novel renormalisation lemma obtained from a tensorization argument in the line of the \"doubling of variables\" method commonly used in the setting of transport equations or conservation laws. A complete $\\LL^\\infty$ theory of rough transport equations is given in Section~\\ref{SectionLInftyTheory}.\n\n\\bigskip\n\n\\noindent \\textbf{Acknowledgments} -- The authors would like to express their gratitude to Martina Hofmanova and Mario Maurelli who discovered a major error in the first version of the paper and hinted to a possible strategy to overcome it.\n\\bigskip\n\n\\noindent \\textbf{Notations} -- We gather here for reference a number of notations that will be used throughout the text.\n\\begin{itemize}\n   \\item We shall denote by E a generic Banach space. Given two Banach spaces E and F, we denote by $\\textrm{L}(\\textrm{E},\\textrm{F})$ the set of continuous linear maps from E to F. \\vspace{0.1cm}\n   \\item We shall denote by $c$ a constant whose value may change from place to place. \\vspace{0.1cm}\n   \\item Given two positive real numbers $\\alpha,\\beta$, we shall write $\\alpha \\lesssim \\beta$ to say that $\\alpha\\leq c\\beta$, for some positive constant $c$. To indicate that this constant $c$ depends on a parameter $\\lambda$, we write $\\alpha \\lesssim_\\lambda \\beta$. \\vspace{0.1cm}\n   \\item Denote by $\\|\\cdot\\|_{\\alpha\\,;\\,k}$ the $\\alpha$-H\\\"older norm of an $E_k$-valued path, for $k\\in\\ZZ$, and by $\\|\\cdot\\|_{\\alpha\\,;\\,\\textrm{E}}$ the $\\alpha$-H\\\"older norm of an E-valued path.\n\\end{itemize}\n\n\n\\bigskip\n\n\\section{Weighted norms}\n\\label{SectionWeightedNorms}\n\nWe introduce  some weighted norms that will be useful in getting a priori estimates on the growth of solutions to the linear differential equations studied in Section~\\ref{SectionBoundedDrivers}. These norms are modelled on Picard's well-known norms\n$$\n\\llparenthesis\\, f \\,\\rrparenthesis := \\underset{t\\geq 0}{\\sup}\\,e^{-\\lambda^{-1} t}\\big| f(t) \\big|,\n$$\nintroduced in the study of ordinary differential equations in order to provide a functional setting where to get well--posedness results on the whole time interval $[0,\\infty)$, as a direct consequence of the Banach fixed point theorem, and to get as a consequence a control on the growth of the size of solutions. \n\n\\medskip\n\nLet $T$ be a possibly infinite positive time horizon. As is common in rough paths theory, we shall work with Banach space-valued multi-index maps, mainly $2$ and $3$-index maps, defined on the simplexes\n$$\n\\Big\\{(s,t)\\in [0,T)^2\\,;\\,s\\leq t\\Big\\}\\quad \\textrm{ and } \\quad \\Big\\{(s,u,t)\\in [0,T)^3\\,;\\,s\\leq u\\leq t\\Big\\}.\n$$\nWith Picard's norm in mind, we introduce a norm on the set of $2$ and $3$-index maps which captures both their H\\\"older size and their growth at infinity. \nGiven $\\lambda >0$, an increasing non-negative function $g$ defined on $\\RR_+$ and a non-negative H\\\"older exponent $\\gamma$, we define the $(\\gamma,g)$-norm of a $2$-index map $a$, and a $3$-index map $b$, by the formulae\n$$\n\\llparenthesis\\, a \\,\\rrparenthesis_{\\gamma,g} := \\underset{\\stackrel{0\\leq s<t<T}{|t-s|\\le \\lambda}}{\\sup}\\;\\,\\frac{\\big| a_{ts}\\big|}{g\\big(\\lambda^{-1} t\\big)\\,|t-s|^\\gamma},\n$$\nand \n$$\n\\llparenthesis\\, b \\,\\rrparenthesis_{\\gamma,g} := \\underset{\\stackrel{0\\leq s< u<t<T}{{|t-s|\\le \\lambda}}}{\\sup}\\;\\,\\frac{\\big| b_{tus}\\big|}{g\\big(\\lambda^{-1} t\\big)\\,|t-s|^\\gamma};\n$$\nnote the following comparison: For $0\\leq \\gamma'\\leq \\gamma$, we have\n\\begin{equation}\n\\label{EqComparisonNorms}\n\\llparenthesis\\, \\cdot \\,\\rrparenthesis_{\\gamma',g} \\leq \\lambda^{\\gamma-\\gamma'} \\llparenthesis\\, \\cdot \\,\\rrparenthesis_{\\gamma,g}.\n\\end{equation}\n\n\\smallskip\n\nRecall that given an E-valued $2$-index map $a$, the \\textit{sewing lemma} \\cite{Gubinelli04, FdlP} asserts that if the inequality\n\\begin{equation}\n\\label{EqAlmostAdditivity}\n\\big| a_{ts} - (a_{tu}+a_{us})\\big| \\leq c|t-s|^\\zeta\n\\end{equation}\nholds for all $0\\leq s\\leq u\\leq t<T$, with $t-s\\leq 1$ say, for some exponent $\\zeta>1$ and some positive constant $c$, then there exists a unique map $A : [0,T)\\rightarrow \\textrm{E}$ whose increments $\\delta A_{ts} := A_t-A_s$ are well-approximated by $a_{ts}$, in the sense that \n$$\n\\big|\\delta A_{ts} - a_{ts} \\big| \\lesssim |t-s|^\\zeta,\n$$\nfor all $t-s\\leq 1$ say. Moreover, it $t_i$ denotes the times of a finite partition $\\pi_{ts}$ of an interval $(s,t)$, with mesh $\\big|\\pi_{ts}\\big|$, we have\n\\begin{equation}\n\\label{EqEstimateAlmostAdditiveFunctional}\n\\big|\\delta A_{ts} - \\sum a_{t_{i+1}t_i}\\big| \\lesssim |t-s|\\big|\\pi_{ts}\\big|^{\\zeta-1}.\n\\end{equation}\n\n\\smallskip\n\nThe \\textbf{sewing map} associates to the above $2$-index map $a$ the $2$-index map\n$$\n\\Lambda(a)_{ts} := \\delta A_{ts} - a_{ts}.\n$$\nFor a given function $g:\\RR\\to\\RR$ let \n$$\nG(t) = g(t) + \\int_0^t g(r)\\,dr,\n$$\nand write as usual  $\\delta a_{tus}$ for $a_{ts} - (a_{tu}+a_{us})$, for any $0\\leq s\\leq u\\leq t<T$. The following lemma provides an estimate of the weighted norm of $\\Lambda(a)$ in terms of the weighted norm of $\\delta a$; an estimate for the growth of $A$ follows as a consequence. \n\n\\begin{lem}\n\\label{LemmaSewingMap}\nThere exists a positive constant $c_\\zeta$, depending only on $\\zeta$, such that \n$$\n\\llparenthesis\\,\\Lambda(a)\\,\\rrparenthesis_{\\zeta,g} \\leq c_\\zeta \\, \\llparenthesis\\,\\delta a\\,\\rrparenthesis_{\\zeta,g}.\n$$\n\\end{lem}\n\n\\medskip\n\n\\begin{Dem}\nGiven $0\\leq s<t$ and $n\\geq 1$, set $t_i = s+i2^{-n}(t-s)$, and note that the telescopic sum\n\\begin{equation*}\n(\\star) := \\sum_{i=0}^{2^n-1}a_{t_{i+1}t_i}\\; - a_{ts} = \\sum_{k=0}^{n-1}\\;\\sum_{\\ell=0}^{2^{n-(k+1)}} \\Big(a_{(\\ell+2)2^k\\,(\\ell+1)2^k}+a_{(\\ell+1)2^k\\,\\ell2^k} - a_{(\\ell+2)2^k\\,\\ell2^k} \\Big)\n\\end{equation*}\nprovides a control of the quantity $(\\star)$ in terms of $\\delta a$ only. Sending $n$ to infinity, we see by identity \\eqref{EqEstimateAlmostAdditiveFunctional} that there exists a constant $c_\\zeta$ depending only on $\\zeta$, such that \n$$\n\\big| \\Lambda(a)_{ts} \\big| \\leq c_\\zeta \\, |t-s|^\\zeta \\,\\sup\\left\\{\\frac{\\big|\\delta a_{t'u's'}\\big|}{|t'-s'|^\\zeta}\\,;\\,0\\leq s<s'<u'<t'\\leq t\\right\\}.\n$$\nIf $|t-s| \\leq \\lambda$ the above inequality implies clearly that we have\n\\begin{equation}\n\\label{EqEstimateLambdaA}\n\\frac{\\big| \\Lambda(a)_{ts} \\big|}{g(\\lambda^{-1} t)\\,|t-s|^\\zeta} \\leq c_\\zeta \\, \\big|\\delta a\\big|_{\\zeta,g}.\n\\end{equation}\n\\end{Dem}\n\n\\medskip\n\nWe will consider only the particular choice of function $g(t) = e^{t}$ and we shall set for a path $f : [0,T)\\rightarrow \\textrm{E}$\n\\begin{equation}\n\\label{EqDefnNormFunction}\n\\llparenthesis\\, f_\\bullet \\,\\rrparenthesis :=  \\underset{t\\geq 0}{\\sup}\\,e^{-\\lambda^{-1} t}\\,\\big| f(t) \\big|,\n\\end{equation}\nand for a $2$-index map $a$, and a  $3$-index map $b$,\n\\begin{equation}\n\\label{EqDefnNormMultiIndexMaps}\n\\llparenthesis\\, a \\,\\rrparenthesis_\\gamma := \\underset{\\stackrel{0\\leq s<t<T}{|t-s|\\le \\lambda}}{\\sup} \\;\\, e^{-\\lambda^{-1} t} \\,\\frac{\\big| a_{ts}\\big|}{|t-s|^\\gamma}\\,, \\quad \\textrm{ and }\\quad \n\\llparenthesis\\, b \\,\\rrparenthesis_\\gamma := \\underset{\\stackrel{0\\leq s< u<t<T}{{|t-s|\\le \\lambda}}}{\\sup} \\;\\, e^{-\\lambda^{-1} t} \\,\\frac{\\big| b_{tus}\\big|}{|t-s|^\\gamma}\\,.\n\\end{equation}\nNote that these norms depend on a choice of parameter $\\lambda$, which may be tuned on demand. This will be particularly useful in the statement and proof of Theorem~\\ref{ThmIntegrationLinearBounded} below, giving the well-posed character of some linear rough differential equation. Note also that we can compare $\\llparenthesis\\, f_\\bullet \\,\\rrparenthesis$ and $\\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma$, as the inequality\n$$\n\\llparenthesis\\, f_\\bullet \\,\\rrparenthesis \\leq \\big|f_0\\big| + e \\lambda^\\gamma\\,\\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma\n$$\nholds for all $\\lambda>0$. This will be our starting point in the proof of the a priori estimate \\eqref{EqLinearBound} in Theorem~\\ref{ThmIntegrationLinearBounded} below. The proof is easy: for fixed $t$ let $\\{0=t_0< t_1 < \\cdots < t_n = t\\}$ be a partition of $[0,t]$ with intervals of at most size $\\lambda$. Then \n$$\n|f(t)| \\le |f(0)| + \\sum_{k=0}^{n-1}|f(t_k)-f(t_{k+1})| \\le  |f(0)| +  \\lambda^\\gamma  \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma \\sum_{k=0}^{n-1} \\exp(t_{k+1}\/\\lambda).\n$$\nBut now\n$$\n\\sum_{k=0}^{n-1} \\exp(t_{k+1}\/\\lambda)\n\\le   \\lambda^{-1}\\int_{t_{k}+\\tau}^{t_{k+1}+\\lambda} \\exp(s\/\\lambda) d s \\le  \\lambda^{-1} \\int_0^{t+\\lambda} \\exp(s\/\\lambda) d s \\le \\exp(t\/\\lambda + 1).\n$$\n\n\\smallskip\n\nLast, a $2$-index map $a$ such that $\\sup_{t-s\\leq 1}\\,\\frac{| a_{ts} |}{|t-s|^\\gamma}$ is finite will be called a $\\gamma$\\textbf{-H\\\"older map.}\n\n\n\\section{Linear differential equations with bounded rough drivers}\n\\label{SectionBoundedDrivers}\n\nLet $\\big(\\mcA,|\\cdot|\\big)$ be a Banach algebra with unit ${\\bf 1}_{\\mathcal{A}}$; one may think for instance to the space of continuous linear maps from some Hilbert space to itself, or to the truncated tensor algebra over some Banach space, equipped with a tensor norm and completed for that norm. We introduce in this section a notion of bounded rough driver in the Banach algebra $\\mcA$, and show that they generate some flows on the algebra. \n\n\\begin{defn}\nLet $\\frac{1}{3}<\\gamma\\leq \\frac{1}{2}$. A \\textbf{\\emph{bounded $\\gamma$-rough driver in $\\mcA$}} is a pair ${\\bf A}= \\big(A^1,A^2\\big)$ of $\\mcA$-valued $2$-index maps satisfying \\emph{Chen's relations}\n\\begin{equation}\n\\label{EqChenRelations}\n\\delta A^1=0, \\quad \\textrm{ and }\\quad \\delta A^2_{tus} = A^1_{tu}A^1_{us},\n\\end{equation}\nand such that $A^1$ is $\\gamma$-H\\\"older and $A^2$ is $2\\gamma$-H\\\"older. The norm of $\\bf A$ is defined by the formula\n$$\n\\| {\\bf A} \\| := \\sup_{0\\leq s<t<T\\,;\\,t-s\\leq 1}\\;\\frac{\\big|A^1_{ts}\\big|}{|t-s|^\\gamma}\\vee\\frac{\\big|A^2_{ts}\\big|}{|t-s|^{2\\gamma}}.\n$$\n\\end{defn}\n\n\\medskip\n\nAs an elementary example, think of $\\mcA$ as the truncated tensor algebra $\\bigoplus_{i=0}^N(\\RR^\\ell)^{\\otimes i}$ over $\\RR^\\ell$, for $N\\geq 2$, and consider a weak geometric $\\gamma$-H\\\"older rough path ${\\bfX}_{ts} = 1\\oplus X_{ts}\\oplus \\bbX_{ts}\\in\\bigoplus_{i=0}^N(\\RR^\\ell)^{\\otimes i}$, with $2\\leq \\gamma<3$. Left multiplication by $X_{ts}$ and $\\bbX_{ts}$ define operators $A^1$ and $A^2$ that are the components of a rough driver.\n\n\\medskip\n\nRecall the weighted norms $\\llparenthesis\\,\\cdot\\,\\rrparenthesis$ and $\\llparenthesis\\,\\cdot\\,\\rrparenthesis_\\gamma$ defined by identities \\eqref{EqDefnNormFunction} and \\eqref{EqDefnNormMultiIndexMaps}, respectively, depend on some parameter $\\lambda$. The proof of the following well--posedness result for linear rough differential equations driven by bounded $\\gamma$-rough drivers shows the interest of being flexible on the tuning of $\\lambda$.\n\n\\begin{thm}[Integration of bounded rough drivers]\n\\label{ThmIntegrationLinearBounded}\nGiven any initial condition $J_0\\in\\mcA$, there  exists a unique $\\gamma$-H\\\"older path $f_\\bullet$ starting from $f_0$, and such that the formula \n  \\begin{equation*}\n    \\delta f_{t, s} - \\Big(A^1_{t, s} +A^2_{t, s}\\Big) f_s \n  \\end{equation*}\ndefines a $3\\gamma$-H\\\"older $2$-index map $f^\\sharp$. Moreover, the following estimate holds\n  \\begin{equation}\n  \\label{EqLinearBound}\n    \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis \\leqslant 2\\,\\big| f_0 \\big|\n  \\end{equation}\nfor all $\\lambda$ greater than some $\\lambda_0$ depending only on $\\| \\bf A \\|$ and $\\gamma$. When $f_0  ={\\bf 1}_{\\mathcal{A}}$, we will use the notation $ f_t =e^{\\bf A}_{t,0}$; the flow property\n$$\ne^{\\bf A}_{t, s} = e^{\\bf A}_{t, u} e^{\\bf A}_{u, s},\n$$\nholds for all $0\\leq s\\leq u\\leq t<T$.\n\\end{thm}\n\n\\medskip\n\nApplied to the above example of rough driver in the truncated tensor product space, this well--posedness result provides a proof of Lyons' extension theorem \\cite{Lyons98}.\n\n\\medskip\n\n\\begin{Dem}\n{\\bf a) A priori estimate --} Let us prove the a priori bound \\eqref{EqLinearBound} first; the uniqueness claim in the theorem follows from this bound and the linear character of the problem. As mentioned above, we start from the inequality\n$$\n\\llparenthesis\\,f_\\bullet \\,\\rrparenthesis \\leq | f_0 | + c \\lambda^\\gamma \\llparenthesis\\, \\delta f \\,\\rrparenthesis _\\gamma,\n$$\nand try and write $\\llparenthesis\\, \\delta f \\,\\rrparenthesis _\\gamma$ in terms of $\\llparenthesis\\,f_\\bullet \\,\\rrparenthesis$; this can be done as follows.\n\nBy using Chen's relation \\eqref{EqChenRelations} and the definition of the remainder $f^\\sharp$, the identity\n$$\n- \\delta f^\\sharp_{t, u, s} = A^1_{t, u} \\Big(\\delta f_{u, s}  -A^1_{u, s} f_s\\Big) +A^2_{t, u} \\delta f_{u, s} = A^1_{t, u} \\Big(A^2_{t, s} f_s + f^{\\sharp}_{s, t}\\Big) + A^2_{t, u} \\delta f_{u, s}, \n$$\ngives us the estimate\n$$\n\\llparenthesis\\, \\delta f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma} \\leq \\| {\\bf A} \\| ^2 \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis + \\| {\\bf A} \\| \\llparenthesis\\, f^{\\sharp} \\,\\rrparenthesis_{2 \\gamma} + \\| {\\bf A} \\| \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma.\n$$ \nBut since $3 \\gamma > 1$, we have \n$$\nf^{\\sharp} = \\Lambda \\delta f^{\\sharp},\n$$ \nso the inequality\n$$\n\\llparenthesis\\, f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_\\gamma \\| {\\bf A} \\|^2 \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis + \\| {\\bf A} \\| \\llparenthesis\\, f^{\\sharp} \\,\\rrparenthesis_{2 \\gamma} + \\| {\\bf A} \\| \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma\n$$\nfollows from Lemma~\\ref{LemmaSewingMap}. Using the inequality $\\llparenthesis\\,  f^{\\sharp} \\,\\rrparenthesis_{2 \\gamma} \\leq \\lambda^\\gamma \\llparenthesis\\,  f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma}$, emphasized in \\eqref{EqComparisonNorms}, the above equation gives \n$$\n\\llparenthesis\\,  f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_\\gamma \\| {\\bf A} \\|^2\\, \\llparenthesis\\,  f_\\bullet \\,\\rrparenthesis  + \\lambda^\\gamma \\| {\\bf A} \\|\\,\\llparenthesis\\,  f^{\\sharp}\\,\\rrparenthesis_{3 \\gamma} + \\| {\\bf A} \\|\\, \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma.\n$$\nFor $\\lambda$ small enough so that $\\lambda^\\gamma \\| {\\bf A} \\| \\leq \\frac{1}{2}$, we obtain\n$$\n\\llparenthesis\\,  f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_{\\gamma} \\| {\\bf A} \\|^2\\, \\llparenthesis\\,  f_\\bullet \\,\\rrparenthesis  + \\| {\\bf A} \\|\\, \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma,\n$$\nso, using again the definition of the remainder $f^\\sharp$, and the observation that \n$$\n\\llparenthesis\\, A^2 f\\,\\rrparenthesis_{\\gamma} \\lesssim_{\\gamma}  \\lambda^\\gamma \\llparenthesis\\, A^2 f\\,\\rrparenthesis_{2 \\gamma} \\lesssim_{\\gamma} \\lambda^\\gamma \\| {\\bf A} \\|\\, \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis,\n$$\nwe obtain the estimate\n\\begin{equation*}\n\\begin{split}\n\\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma &\\lesssim_{\\gamma} \\| {\\bf A} \\|\\llparenthesis\\,f_\\bullet \\,\\rrparenthesis + \\llparenthesis\\, A^2 f\\,\\rrparenthesis_\\gamma + \\llparenthesis\\, f^{\\sharp}\\,\\rrparenthesis_\\gamma \\\\\n                            &\\lesssim_{\\gamma} \\Big\\{\\| {\\bf A} \\| \\big(1 + \\lambda^{2 \\gamma}\\| {\\bf A} \\|\\big) + \\lambda^\\gamma \\| {\\bf A} \\|\\Big\\} \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis + \\lambda^{2 \\gamma}\\| {\\bf A} \\|\\,\\llparenthesis\\, \\delta f \\,\\rrparenthesis_{\\gamma}.\n\\end{split}\n\\end{equation*}\nTaking $\\lambda$ small enough, depending only on $\\| \\bf A \\|$, we eventually see that \n$$ \n\\llparenthesis\\,\\delta f \\,\\rrparenthesis_\\gamma \\lesssim_{\\gamma}\\Big\\{\\| {\\bf A} \\| \\big(1 +  \\lambda^{2 \\gamma} \\| {\\bf A} \\|\\big) + \\lambda^\\gamma  \\| {\\bf A} \\|\\Big\\} \\llparenthesis\\, f_\\bullet\\,\\rrparenthesis.\n$$\nThe a priori estimate $\\llparenthesis\\, f _\\bullet\\,\\rrparenthesis \\leqslant 2 | f_0 |$, follows now from a choice of sufficiently small  parameter $\\lambda$, since $\\llparenthesis\\,f_\\bullet \\,\\rrparenthesis \\leq | f_0 | + c \\lambda^\\gamma \\llparenthesis\\, \\delta f \\,\\rrparenthesis _\\gamma$.\n\n\\bigskip\n\n{\\bf b) Existence --} We can run a Picard iteration to prove the existence of a path satisfying the conditions of the theorem. Set first\n$$\nf^0_t = f_0,\\quad \\textrm{ and } \\quad f^1_t =A^1_{t0} f_0\n$$ \nfor all $t \\in \\RR$. Given the paths $f^{n-1}_\\bullet, f^n_\\bullet$, the $2$-index map \n$$\na^n_{ts} := A^1_{ts} f^n_s +A^2_{ts} f_s^{n - 1}\n$$\nsatisfies the almost-additivity condition \\eqref{EqAlmostAdditivity} with $\\zeta=3\\gamma>1$ here, so there is, by the sewing lemma, a unique $\\gamma$-H\\\"older path $f^{n + 1}_\\bullet$ for which the formula\n$$ \n\\delta f^{n + 1}_{ts} - \\Big(A^1_{ts} f^n_s +A^2_{ts} f_s^{n - 1}\\Big) \n$$\ndefines $2$-index $3\\gamma$-H\\\"older map $f^{n + 1, \\sharp}$. Setting $g^0_\\bullet := 0$ and\n$$\ng^{n + 1}_\\bullet := f^{n + 1}_\\bullet - f^n_\\bullet, \\quad  g^{n + 1, \\sharp} := f^{n + 1, \\sharp} - f^{n, \\sharp},\n$$ \nfor all $n \\geq 0$, we have\n$$\n\\delta g^{n + 1}_{ts} =A^1_{ts} g^n_s +A^2_{ts} g_s^{n - 1} + g^{n + 1, \\sharp}_{ts}. \n$$\nNote moreover that we have the identity\n\\begin{equation*}\n\\begin{split}\n- \\delta g^{n + 1, \\sharp}_{t, u, s} &= A^1_{t, u} \\Big(\\delta  g^n_{u, s} -A^1_{u, s} g^{n - 1}_s\\Big) +A^2_{t, u} \\delta g^{n - 1}_{u, s} \\\\\n&= A^1_{t, u} \\Big(A^2_{t, s} g^{n - 2}_s + g^{n, \\sharp}_{s, t}\\Big) +A^2_{t, u} \\delta g^{n - 1}_{u, s},\n\\end{split}\n\\end{equation*}\nso, proceeding as in the proof of the a priori bound, we see that the inequality\n$$\n\\llparenthesis\\, g^{n - 1}_\\bullet\\,\\rrparenthesis + \\llparenthesis\\, \\delta g^n \\,\\rrparenthesis_\\gamma + \\llparenthesis\\, g^{n + 1, \\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_{\\gamma, \\| {\\bf A} \\|} \\lambda^\\gamma \\Big\\{\\llparenthesis\\, g^{n - 2}_\\bullet \\,\\rrparenthesis + \\llparenthesis\\, \\delta g^{n - 1} \\,\\rrparenthesis_\\gamma + \\llparenthesis\\, g^{n, \\sharp} \\,\\rrparenthesis_{3 \\gamma}\\Big\\}\n$$  \nholds, by choosing $\\lambda$ small enough. The estimate\n$$\n\\llparenthesis\\, g^{n - 1}_\\bullet\\,\\rrparenthesis + \\llparenthesis\\, \\delta g^n \\,\\rrparenthesis_\\gamma + \\llparenthesis\\, g^{n + 1, \\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_{\\gamma, \\| {\\bf A} \\|} \\lambda^{\\gamma n}\n$$\nfollows as a consequence, so the series $f^n = f^0 + \\sum_{n \\geqslant 1} g^n$ converges in the complete space of $\\mcA$-valued $\\gamma$-H\\\"older paths, and defines a path in $\\mathcal{A}$ satisfying the conditions of the theorem.\n\\end{Dem}\n\n\\smallskip\n\n\\begin{rems}\n\\begin{enumerate}\n   \\item Note that the proof given above gives back the known sharp growth rate $\\exp\\big((2\\| {\\bf A}\\|)^\\gamma\\,t\\big)$ for $\\big|f_t\\big|$; see \\cite{FrizOberhauser}. Bounded rough drivers can also be integrated by defining recursively the $(n\\gamma)$-H\\\"older $\\mcA$-valued $2$-index map $A^n$ using the formula\n$$\n\\delta A^n_{t, u, s} = \\sum_{k = 1}^{n - 1} A^{n - k}_{t, u} A^k_{u, s},\n$$\nand setting \n$$\ne^{\\bf A}_{t, s} = \\sum_{n = 0}^{\\infty} A^n_{t, s}.\n$$ \nStandard estimates on the sewing map \\cite{Gubinelli04} show that $\\delta A^n$ has $n \\gamma$-H\\\"older norm  no greater than $(n!)^{- \\gamma}$, so the above series converges in $\\mcA$ for all $0 \\leq s \\leq t$. The flow property is obtained by a direct calculation, and setting $f_t := e^{\\bf A}_{t, 0} f_0$, we see that the path $f_\\bullet$ solves the problem.   \\vspace{0.2cm}\n\n   \\item {\\bf Linear rough differential equations with a linear drift -- } The above theory extends easily to rough equations of the form\n\\begin{equation}\n\\label{EqRoughLinearDrift}\n\\delta f_{t, s} = \\int_s^t B_r f_r\\,dr + A^1_{t, s} f_s + A^2_{t, s} f_s + f^{\\natural}_{t, s}\n\\end{equation}\nwhere $B \\in \\LL^{\\infty} (\\RR; \\mcA)$ is a bounded measurable family of bounded operators. This equation is the rigorous meaning to give to solutions of the differential equation\n$$\n\\frac{d}{dt} f = (B_r + \\dot{A}_r) f_r\n$$\nwhere $\\dot{A}_t = \\partial_t A_{t0}^1$. In case $\\dot{A}_t \\in \\LL^{\\infty} (\\RR; \\mcA)$, the two formulations are equivalent provided ${\\bf A} = \\big(A^1, A^2\\big)$ is defined by the formula\n$$ \nA^1_{t, s} = \\int_s^t \\dot{A}_r \\,dr, \\quad A^2_{t, s} = \\int_s^t \\int_s^r \\dot{A}_u \\dot{A}_r \\,du dr.\n$$\nThe proof of Theorem~\\ref{ThmIntegrationLinearBounded} can be easily adapted in the present setting, and the final lower bound on $\\lambda$ gets an additional dependence on $\\| B \\|_\\infty$. It yields moreover the following Duhamel formula\n$$\nf_t = e^{\\bf A}_{t, 0} f_0 + \\int_0^t e^{\\bf A}_{t, r} B_r f_r\\,dr.\n$$\nIndeed, let $f_\\bullet$ be a function satifsying the above identity. If one computes the increment of the right hand side in the above equation, we get\n$$\n\\delta f_{t, s} = \\int_s^t e^{\\bf A}_{t, r} B_r f_r \\,dr - \\Big(e^{\\bf A}_{t, s} -\\textrm{\\emph{Id}}\\Big) f_s = \\int_s^t B_r f_r \\,dr + A^1_{t, s} f_s + A^2_{t, s} f_s + f^\\sharp_{t, s}\n$$\nwhere\n$$\nf^\\sharp_{s, t} = \\int_s^t \\Big(e^{\\bf A}_{t, r} -\\textrm{\\emph{Id}}\\Big) B_r f_r \\,dr + \\Big(e^{\\bf A}_{t, s} -\\textrm{\\emph{Id}} - A^1_{t, s} - A^2_{t, s}\\Big) f_s.\n$$\nUsing the bounds\n$$\n\\Big| e^{\\bf A}_{t, r} -\\textrm{\\emph{Id}}\\Big| \\lesssim_{\\|{\\bf A}\\|} | t -  r |^{\\gamma}, \\quad \\textrm{ and }\\quad  \\Big| e^{\\bf A}_{t, s}  -\\textrm{\\emph{Id}} - A^1_{t, s} - A^2_{t, s} \\Big| \\lesssim_{\\| {\\bf A} \\|} | t - s |^{3 \\gamma},\n$$\nthis allows to conclude that $\\big|f^{\\sharp}_{t, s}\\big| \\lesssim | t - s |^{3\\gamma}$, and that the path $f_\\bullet$ is indeed the unique solution to equation \\eqref{EqRoughLinearDrift}. \\vspace{0.2cm}\n\n   \\item Bounded rough drivers have also been introduced previously in the work \\cite{CoutinLejayLinear} of Coutin and Lejay, and studied in relation with the Magnus formula for what is called there the \\emph{resolvent operator} $e^{\\bf A}$. The above short proof of Theorem~\\ref{ThmIntegrationLinearBounded} can be considered an alternative proof of the main result of section 3 in \\cite{CoutinLejayLinear}. They also consider perturbed linear equations, with an a priori given drift of the more general form $C_{ts}$, rather than $\\int_s^t B_rf_r\\,dr$, with $C$ satisfying some regularity conditions. The pioneering work \\cite{FdlPM} of Feyel-de la Pradelle-Mokobodzki is also closely related to these questions.\n\\end{enumerate}\n\\end{rems}\n\n\n\\medskip\n\n\n\n\\section{Unbounded rough drivers and rough linear equations}\n\\label{SectionUnboundedRoughDrivers}\n\nThe above results apply in the particular case where $\\mcA$ is the Banach algebra of bounded operators on an Hilbert space $H$. We shall study in the remaining sections the integration problem \n\\begin{equation}\n\\label{EqLinearEquationHilbert}\n\\delta f_{ts} = \\Big(A^1_{ts} + A^2_{ts}\\Big) f_s +  f^\\sharp_{ts},\n\\end{equation}\nfor a particular class of drivers $\\bf A$ associated to a class of \\textit{unbounded} operators on $H$, or other Banach spaces, with in mind the model case of the \\textbf{rough transport equation}\n$$\n\\delta f(\\varphi)_{ts} = X_{ts}\\,f_s\\big(V^*\\varphi\\big) +  \\bbX_{ts}\\,f_s\\big(V^*V^*\\varphi\\big) + f^\\sharp_{ts}(\\varphi),\n$$\nwhere ${\\bfX} = (X,\\bbX)$ is an $\\ell$-dimensional $\\gamma$-H\\\"older rough path and $V = \\big(V_1,\\dots,V_\\ell\\big)$ is a collection of $\\ell$ vector fields on $\\RR^d$.\n\n\n\n\\subsection{Rough drivers}\n\\label{SubsectionRoughDrivers}\n\nTo make sense of this equation we need to complete the functional setting by the datum of a scale of Banach spaces $\\big(E_n,|\\cdot|_n\\big)_{n\\geq 0}$, with $E_{n+1}$ continuously embedded in $E_n$. For $n\\geq 0$, we shall denote by $E_{-n}=E_n^*$ the dual space of $E_n$, equipped with its natural norm, \n$$\n|e|_{-n} := \\underset{\\varphi\\in E_n,\\,|\\varphi|_n\\leq 1}{\\sup}\\; (\\varphi,e), \\quad\\quad e\\in E_{-n}.\n$$\nWe require that the following continuous inclusions \n\\begin{equation*}\nE_n\\subset \\cdots\\subset E_2\\subset E_1\\subset E_0 \\\\ \n\\end{equation*}\nhold for all $n\\geq 2$. One can think of $n$ as quantifying the 'regularity' of elements of some test functions, with the elements of $E_n$ being more regular with $n$ increasing. Denote by $\\|\\cdot\\|_{(b,a)}$ for the norm of a linear operator form $E_a$ to $E_b$. (Note that we use $(b,a)$ and not $(a,b)$ in the lower index for the norm.) We also assume the existence of a family $\\big(J^\\varepsilon\\big)_{0<\\varepsilon\\leq 1}$ of operators from $E_0$ to itself such that the estimates\n\\begin{equation}\n\\label{EqApproximationProperties}\n\\big\\|J^\\varepsilon - \\textrm{Id} \\big\\|_{(n+k,n)} \\leq c\\,\\varepsilon^k, \\quad \\big\\| J^\\varepsilon \\big\\|_{(n,n + k)} \\leq c\\,\\varepsilon^{- k}\n\\end{equation}\nhold for all $n, k \\geq 0$, for some positive constant $c$ independent of $\\varepsilon$. For $\\varphi\\in E_0$, the elements $\\varphi_\\varepsilon := J^\\varepsilon\\varphi$ are in particular 'smooth', that is in the intersection of all the spaces $E_n$, for $n\\geq 0$. \n\nWhenever we will work with Sobolev--like scales $E_n = W^{n,p}(\\RR^d)$ ($p\\ge 1$) we will take the operators $J^\\epsilon = \\big(\\textrm{I}-\\epsilon\\triangle\\big)^{-j_0}$, for $j_0$ big enough.\n\n\n\\smallskip\n\n\\begin{defn}\n\\label{DefnUnboundedRoughDriver}\nLet $\\frac{1}{3}<\\gamma\\leq \\frac{1}{2}$ be given. An \\textbf{\\emph{unbounded $\\gamma$-rough driver on the scales}} $\\big(E_n,|\\cdot|_n\\big)_{n\\geq 0}$, is a pair ${\\bf A} = \\big(A^1,A^2\\big)$ of $2$-index maps, with \n\\begin{equation}\n\\label{EqBoundednessAssumptionRoughDriver}\n\\begin{split}\n&A^1_{ts}\\in \\textrm{\\emph{L}}\\big(E_n,E_{n-1}\\big), \\textrm{ for } n\\in\\{-0,-2\\}, \\\\\n&A^2_{ts}\\in\\textrm{\\emph{L}}\\big(E_n,E_{n-2}\\big), \\textrm{ for } n\\in\\{-0,-1\\},\n\\end{split}\n\\end{equation}\nfor all $0\\leq s\\leq t<T$, which satisfies Chen's relations \\eqref{EqChenRelations}, and such that $A^1$ is $\\gamma$-H\\\"older and $A^2$ is $2\\gamma$-H\\\"older. \n\\end{defn}\n\n\\medskip\n\nEquation \\eqref{EqDefnSolutionRDE} below will make it clear that unbounded rough drivers can be thought of a some kind of (dual objects to some) multi-scales velocity fields, with two time scales given by $A^1$ and $A^2$. This is particularly clear on the following elementary example, where\n$$\nA^1_{ts} = \\sum_{i=1}^\\ell X^i_{ts}V_i, \\quad \\textrm{and } \\quad A^2_{ts} = \\sum_{j,k=1}^\\ell \\bbX^{jk}_{ts}V_jV_k,\n$$\nwhere ${\\bfX} = (X,\\bbX)$ is a weak geometric $\\gamma$-H\\\"older rough path over $\\RR^\\ell$, with $\\frac{1}{3}<\\gamma\\leq \\frac{1}{2}$, and $V_1,\\dots, V_\\ell$ are regular enough vector fields on $\\RR^d$ with values in $N\\times N$ matrices, understood here as first order differential operators. On can also take $V_i^* = -V_i - \\textrm{div}V_i$ in the above definition of a rough driver.\n\n\\smallskip\n\nIn a probabilistic setting, the above rough path $\\bfX$ could be the (Stratonovich) Brownian rough path. More generally, one could take as first level $A^1_{ts}$ a semimartingale velocity field of Le Jan-Watanabe-Kunita type (or its dual), where noise (the above $X_{ts}$) and dynamics (the above $V_i$) cannot be separated from one another. It is shown in the forthcoming work \\cite{BailleulRiedel} that these velocity fields can be lifted into a(n object very similar to a) rough driver under some mild conditions on the semimartingale structure. In the same spirit, Catellier has shown in \\cite{Catellierregularisation, CatellierTransport} the interest for the study of rough transport equations with irregular drift, of considering velocity fields in $\\RR^d$ given by the regularisation of some field $V$ (that may even be a distribution) along some irregular path $w$, such as a typical trajectory of a fractional Brownian motion with any Hurst index. One deals in this particular setting with the 'Young analogue' of the above $\\gamma$-rough drivers, corresponding to $\\frac{1}{2}<\\gamma\\leq 1$, with only one level, and given in this example by the formula\n$$\nA^1_{ts}(x) = \\int_s^t V(x+w_u)\\,du.\n$$\n\n\\medskip\n\nWe denote by $\\epsilon(e)$ the duality pairing between an element $\\epsilon$ of $E_{-n}$ and an element $e$ of $E_n$, for any $n\\in\\ZZ$. The dual of a continuous operator $A$ from $E_{-a}$ to $E_{-b}$ is denoted by $A^*$; this is a continuous operator from $E_b$ to $E_a$. We can now make sense of equation \\eqref{EqLinearEquationHilbert}. \n\n\\begin{defn}\n\\label{DefnSolutionEquation}\nAn $E_{-0}$-valued \\emph{\\textbf{path}} $f_\\bullet$ is said to \\emph{\\textbf{solve the linear rough differential equation}}\n\\begin{equation}\n\\label{EqRDE}\ndf_s = {\\bf A}(ds)f_s\n\\end{equation}\nif there exists an $E_{-2}$-valued $2$-index map $f^\\sharp$ such that one has\n\\begin{equation}\n\\label{EqDefnSolutionRDE}\n\\delta f_{ts}(\\varphi) = f_s\\Big(\\big\\{A^{1,*}_{ts}+A^{2,*}_{ts}\\big\\}\\varphi\\Big) + f^\\sharp_{ts}(\\varphi)\n\\end{equation}\nfor all $0\\leq s\\leq t<T$ and all $\\varphi\\in E_2$, and the map $f^\\sharp_{ts}(\\varphi)$ is $3\\gamma$-H\\\"older, for each $\\varphi\\in E_2$.\n\\end{defn}\n\n\\smallskip\n\nLet us insist on the fact that this notion of solution depends on the scale $(E_n)_{n\\geq 0}$. We define, for all $0\\leq s\\leq t<T$, an  $E_{-1}$-valued $2$-index map $f^\\flat$ setting, for $\\varphi\\in E_1$,\n\\begin{equation*}\nf^\\flat_{ts}(\\varphi) := \\delta f_{ts}(\\varphi) - f_s\\big(A_{ts}^{1,*} \\varphi\\big);\n\\end{equation*}\nfor $\\varphi\\in E_2$, one also has another expression for that quantity\n$$\nf^\\flat_{ts}(\\varphi) = f_s\\big(A^{2,*}_{ts}\\varphi\\big) + f^\\sharp_{ts}(\\varphi).\n$$\n\n\n\\bigskip\n\n\n\\subsection{A priori estimates}\n\\label{SubsectionAPrioriEstimates}\n\nWe show in this section that solutions $f_\\bullet$ of equation \\eqref{EqRDE} satisfy some a priori bounds that depend only on certains norms on the rough driver $A$ and on the uniform norm of $f$, when seen as a continuous path with values in some spaces of distributions. \n\n\\smallskip\n\nThe next lemma shows that the map $f^\\flat$ is actually $2\\gamma$-H\\\"older with values in a  space of distributions \"of order 2\" in a certain sense while $f^\\sharp$ is only expected to be $\\gamma$-H\\\"older from its definition. This \\textit{gain of time regularity, traded against a loss of 'space regularity'} may well be one of our main contribution, despite its elementary nature. It leads to a similar (and even better) result for $f^\\sharp$, as expressed in Theorem~\\ref{ThmGeneralRegularityGain}, that is the key technical result of this paper and which opens the road  to a quite complete theory of linear rough equation.\n\nRecall the notations $\\llparenthesis \\cdot\\rrparenthesis$ and $\\llparenthesis \\cdot\\rrparenthesis_\\gamma$ for the norms introduced in equations \\eqref{EqDefnNormFunction} and \\eqref{EqDefnNormMultiIndexMaps}. They depend implicitly on some positive parameter $\\lambda$ that we shall tune as needed along the way. Given $\\varphi\\in E_0$, set for all $n\\ge 0$\n\\begin{equation*}\n\\begin{split}\nK_n(\\varphi) := \\sup_{0 < \\varepsilon \\leq 1} \\inf_{\\varphi_1+\\varphi_2=\\varphi} \\Big\\{&   2\n (\\lambda^\\gamma \\varepsilon)^{0-n} \\, \\llparenthesis\\, f\\big(\\varphi_x\\big) \\,\\rrparenthesis\n  +  (\\lambda^\\gamma \\varepsilon)^{1-n} \\llparenthesis\\, f\\big(A^{1,*}\\varphi_x \\big) \\,\\rrparenthesis_\\gamma  \n \\\\ &\n + (\\lambda^\\gamma \\varepsilon)^{2-n}\\llparenthesis\\, f\\big(A^{2,*} \\varphi_x) \\,\\rrparenthesis_{2\\gamma} \n  +  (\\lambda^\\gamma \\varepsilon)^{3-n}\\,\\llparenthesis\\, f^\\sharp \\big(  \\varphi_x \\,) \\,\\rrparenthesis_{3\\gamma} \n  \\Big\\}\n\\end{split}\n\\end{equation*}\nwhere $\\varphi_x$ is $\\varphi_1$ if corresponds to a term of order $<n\\gamma$ or $\\varphi_2$ otherwise. \n\\smallskip\n\n\\begin{lem}\n\\label{LemmaGeneralImproveEstimates}\nWe have $\\llparenthesis\\, \\delta f(\\varphi) \\,\\rrparenthesis_\\gamma \\lesssim\\; K_1(\\varphi)$, and $\\llparenthesis\\,  f^\\flat(\\varphi) \\,\\rrparenthesis_{2\\gamma} \\lesssim\\; K_2(\\varphi)$.\n\\end{lem}\n\n\\medskip\n\n\\begin{Dem}\nWe start by an arbitrary decomposition of $\\varphi\\in E_1$ as $\\varphi = \\varphi_1 + \\varphi_2$ giving\n$$\n|\\delta f_{ts}(\\varphi)| \\leq |\\delta f_{ts}(\\varphi_1)| + |\\delta f_{ts}(\\varphi_2)|.\n$$\nTake $t,s$ such that ${|t-s|\\le \\lambda}$. The second term is bounded above by \n$$\ne^{-\\lambda^{-1}t} \\frac{|\\delta f_{ts}(\\varphi_2)|}{|t-s|^\\gamma} \\leq  2 \\lambda^{-\\gamma}\\varepsilon^{-1}\\llparenthesis\\, \\varphi_2 \\rrparenthesis ,\n$$ \nwhere  $\\varepsilon = \\lambda^{-\\gamma} | t - s |^\\gamma \\in(0,1]$ if $|t-s|\\le \\lambda$.\nThe first term can be estimated using properties \\eqref{EqApproximationProperties} and the defining identity \\eqref{EqDefnSolutionRDE}:\n$$\ne^{-\\lambda^{-1}t} \\frac{|\\delta f_{ts}(\\varphi_1)|}{|t-s|^\\gamma} \\leq\n\\llparenthesis  f(A^{1,*} \\varphi_1) \\rrparenthesis_\\gamma\n+ \\lambda^{\\gamma} \\varepsilon  \\llparenthesis f( A^{2,*} \\varphi_1) \\rrparenthesis_{2\\gamma}  \n+ \\lambda^{2\\gamma}\\varepsilon^2 \\llparenthesis f^\\sharp (  \\varphi_1) \\rrparenthesis_{3\\gamma}.\n$$ \n\n\\medskip\n\n$\\bullet$  This gives us an upper bound for $\\big|\\delta f_{ts}(\\varphi)\\big|$ depending on $\\varepsilon$ and $\\varphi_1, \\varphi_2$ over which we can optimise to obtain\n\\begin{equation*}\n\\llparenthesis \\delta f(\\varphi) \\rrparenthesis_\\gamma \\leq  K_1(\\varphi) .\n\\end{equation*}\n\n\\smallskip\n\n$\\bullet$ One can proceed in a similar way to estimate $\\big|f^\\flat_{ts}(\\varphi)\\big|$, starting from the inequality\n\\begin{equation*}\n\\begin{split}\n| f^\\flat_{ts}(\\varphi)| &\\leq | \\delta f_{ts}(\\varphi_1)| + | f_s(A^{1,*}_{ts}\\varphi_1)| + | f_s(A^{2,*}_{ts}\\varphi_2)| + |f^\\sharp_{ts}(\\varphi_2)|\n\\end{split}\n\\end{equation*}\nand optimizing as above we get the upper bound on $\\big|f^\\flat_{ts}(\\varphi)\\big|$.\n\\end{Dem}\n\n\\medskip\n\nThe estimates proved in Lemma~\\ref{LemmaGeneralImproveEstimates} are all we need to give an upper bound on the $3\\gamma$-H\\\"older norm of $f^\\sharp$ in terms of $f$ itself. As this result will be our key tool for proving a number of results in different situations, we formulate it here in some generality. Given a Banach space B and a B-valued path $m_\\bullet$, we set\n$$\n\\llparenthesis\\,m\\,\\rrparenthesis_{\\gamma\\,;\\,\\textrm{B}} := \\sup_{\\stackrel{0\\leq s<t\\leq T}{{|t-s|\\le \\lambda}}}e^{-\\lambda^{-1}t}\\,\\frac{\\big| \\delta m_{ts} \\big|_\\textrm{B}}{|t-s|^\\gamma},\n$$\nfor $\\gamma\\geq 0$. Let\n$$\nN({\\bfA}) := \\sup_\\varphi \\frac{\\|J_1(A^{1,*}\\varphi)\\|_\\gamma + \\|J_2(A^{2,*}\\varphi)\\|_{2\\gamma}}{|\\varphi|_F}\n$$ \nwith\n\\begin{equation*}\n\\begin{split}\nJ_n(\\varphi) :=\n\\sup_{0 < \\varepsilon \\leq 1} \\inf_{\\varphi_1+\\varphi_2=\\varphi} \\Big\\{&   2\n ( \\varepsilon)^{0-n} \\, | \\varphi_x|_E\n  +  (\\varepsilon)^{1-n} \\| A^{1,*}\\varphi_x \\|_{\\gamma;E}  \n + ( \\varepsilon)^{2-n}\\|A^{2,*} \\varphi_x\\|_{2\\gamma;E} \n \\\\ &\n  +  ( \\varepsilon)^{3-n}| \\varphi_x |_F \n  \\Big\\} .\n\\end{split}\n\\end{equation*}\n\n\\smallskip\n\n\\begin{thm}\n\\label{ThmGeneralRegularityGain} \nLet $\\textrm{\\emph{E}}\\subset E_0$ and $\\textrm{\\emph{F}}\\subset E_3$ be two Banach spaces of distributions such that the above quantity $N({\\bfA})$ is finite. Let  $\\lambda \\leq 1$ be sufficiently small so that $2\\lambda^{\\gamma} N({\\bfA}) \\leq 1$. Then any solution of the rough linear equation \\eqref{EqRDE} satisfies the a priori bound \n$$\n\\llparenthesis\\, f^\\sharp\\,\\rrparenthesis_{3\\gamma\\,;\\,F^*} \\leq 8 \\lambda^{-2\\gamma} N({\\bfA})\\,\\llparenthesis\\,f\\,\\rrparenthesis_{E^*}. \n$$\n\\end{thm}\n\n\\smallskip\n\n\\begin{Dem}\nGiven $\\varphi\\in E_3$, note that\n$$\n\\delta f_{tus} = f^\\flat_{us}\\big(A^{1,*}_{tu}\\varphi\\big) + \\delta f_{us}\\big(A^{2,*}_{tu}\\varphi\\big),\n$$\nis $3\\gamma$-H\\\"older by Lemma~\\ref{LemmaGeneralImproveEstimates}, so the sewing Lemma~\\ref{LemmaSewingMap} can be used, giving\n\\begin{equation*}\n\\begin{split}\n\\llparenthesis\\, f^\\sharp(\\varphi)\\,\\rrparenthesis_{3\\gamma} \n&\\lesssim \\llparenthesis\\,  f^\\flat \\big(A^{1,*}\\varphi\\big) \\,\\rrparenthesis_{3\\gamma} +  \\llparenthesis\\,  \\delta f (A^{2,*}\\varphi) \\,\\rrparenthesis_{3\\gamma}\\\\\n&\\lesssim  \\|  K_2\\big(A^{1,*}\\varphi\\big) \\|_\\gamma +  \\|  K_1\\big(A^{2,*}\\varphi\\big)\\|_{2\\gamma}\\\\\n\\end{split}\n\\end{equation*}\nNow\n\\begin{equation*}\n\\begin{split}\nK_n(\\varphi) \\leq \\lambda^{-n\\gamma}[(1+\\lambda^\\gamma+\\lambda^{2\\gamma})\\llparenthesis\\, f\\,\\rrparenthesis_{\\textrm{E}^*} + \\lambda^{3\\gamma} \\llparenthesis\\, f^\\sharp\\,\\rrparenthesis_{3\\gamma;\\textrm{F}^*}] J_n(\\varphi)\n\\end{split}\n\\end{equation*}\nso\n\\begin{equation*}\n\\begin{split}\n \\llparenthesis\\, f^\\sharp\\,\\rrparenthesis_{3\\gamma;\\textrm{F}^*}\n&\\leq  [4 \\lambda^{-2\\gamma}\\llparenthesis\\, f\\,\\rrparenthesis_{\\textrm{E}^*} + \\lambda^{\\gamma} \\llparenthesis\\, f^\\sharp\\,\\rrparenthesis_{3\\gamma;\\textrm{F}^*}] N({\\bfA})\n\\end{split}\n\\end{equation*}\nwhere we assumed  $\\lambda \\leq 1$, with no loss of generality, to get a simpler expression for the formula. \nTaking $\\lambda\\leq 1$ small enough so that\n$\n\\lambda^{\\gamma} N({\\bfA}) \\leq 1\/2,\n$\nwe obtain the  inequality\n$$\n\\llparenthesis\\, f^\\sharp \\,\\rrparenthesis_{3\\gamma\\,;\\,\\textrm{F}^*} \\leq  8 \\lambda^{-2\\gamma} \n\\llparenthesis\\, f\\,\\rrparenthesis_{\\textrm{E}^*} N({\\bfA}).\n$$\n\\end{Dem}\n\n\\bigskip\n\nTaking\n$$\n\\textrm{E} = E_0, \\quad \\textrm{ and }\\quad \\textrm{F} = E_3,\n$$ \nand assuming that \n$$\nA^{1,*}_{ts} \\in \\textrm{L}\\big(E_1,E_0\\big) \\cap \\textrm{L}\\big(E_3,E_2\\big)\n,\\quad\n\\text{and}\\quad \nA^{2,*}_{ts} \\in \\textrm{L}\\big(E_3 , E_1\\big) \\cap \\textrm{L}\\big(E_2,E_0\\big)\n$$ \nfor all $0\\leq s\\leq t\\leq T$, then \n$\nN({\\bfA}) \\lesssim C_0^2\n$ \nwhere\n\\begin{equation}\n\\label{EqDefnC0}\nC_0 := 1+\\big\\|A^1\\big\\|_{\\gamma\\,;\\,(-0,-1)}  + \\big\\|A^2\\big\\|_{2\\gamma\\,;\\,(-0,-2)} +  \\big\\|A^1\\big\\|_{\\gamma\\,;\\,(-2,-3)}  + \\big\\|A^2\\big\\|_{2\\gamma\\,;\\,(-1,-3)}  < \\infty .\n\\end{equation}\nIndeed in this case we can take the decomposition $\\varphi_2 = J^\\varepsilon \\varphi$ and $\\varphi_1 = (\\mathrm{Id} - J^\\varepsilon) \\varphi$ in the definition of $J_n(\\varphi)$ and obtain, for example,\n \\begin{equation*}\n\\begin{split}\nJ_1(\\varphi) \\leq\n\\sup_{0 < \\varepsilon \\leq 1} \\Big\\{&   2\n ( \\varepsilon)^{-1} \\, | (\\mathrm{Id} - J^\\varepsilon) \\varphi|_0\n  +   \\| A^{1,*}J^\\varepsilon \\varphi\\|_{\\gamma;0}  \n + ( \\varepsilon)^{1}\\|A^{2,*} J^\\varepsilon \\varphi\\|_{2\\gamma;0} \n \\\\ &\n  +  ( \\varepsilon)^{2}|J^\\varepsilon \\varphi|_3 \n  \\Big\\} \\lesssim  C_0 |\\varphi|_1\n\\end{split}\n\\end{equation*}\nand similarly $J_2(\\varphi) \\lesssim C_0 |\\varphi|_2$ from which is easy to conclude the bound $N({\\bfA}) \\lesssim C_0^2$.\n\n\n\\smallskip\n\nNote that it follows from Banach uniform boundedness principle that for a solution path $\\big(f,f^\\sharp\\big)$, we have\n$$\n\\big\\| f^\\sharp\\big\\|_{3\\gamma\\,;\\,-3} \\leq \\big\\| f^\\sharp\\big\\|_{3\\gamma\\,;\\,-2} < \\infty.\n$$\nTheorem~\\ref{ThmGeneralRegularityGain} shows that one has an explicit upper bound for $\\big\\| f^\\sharp_{ts}\\big\\|_{3\\gamma\\,;\\,-3}$ in terms of $ \\llparenthesis f\\rrparenthesis_{-0}$ and $C_0$ only. This fact is recorded in the following :\n\n\\begin{thm}\n\\label{ThmRegularityGain} \nFor any $\\lambda \\leq 1$ such that $\\lambda^\\gamma C_0^2 \\lesssim 1$ we have the a priori bounds\n\\begin{equation}\n\\label{EqFundamentalAPrioriEstimate}\n\\max\\Big\\{  \\llparenthesis\\, \\delta f \\,\\rrparenthesis_{\\gamma\\,;\\, - 1} , \\llparenthesis\\, f^\\flat \\,\\rrparenthesis_{2 \\gamma\\,;\\, - 2}, \\llparenthesis\\, f^\\sharp \\,\\rrparenthesis_{3 \\gamma\\,;\\, - 3} \\Big\\} \\lesssim_{\\gamma} C_0^2\\,\\lambda^{-2\\gamma}\\,\\llparenthesis\\, f\\,\\rrparenthesis_{-0} .\n\\end{equation}\n\\end{thm}\n\n\\medskip\n\n\n\\section{Integration of unbounded rough drivers in Hilbert spaces}\n\\label{SectionHilbertSpaceTheory}\n\nWe develop in this section the theory of integration of unbounded rough drivers in the Hilbert space $\\LL^2(\\RR^d)=E_0=E_{-0}$. We are able to give a rather complete theory a class of drivers that enjoys the same algebraic properties as the rough drivers ${\\bfA} = \\big(X\\,V,\\bbX\\,V\\,V\\big)$ involved in the rough transport equation, when the vector fields $V=\\big(V_1,\\dots,V_\\ell\\big)$ are divergence free. These drivers are called conservative. A general existence result for the rough linear equation $df_s = {\\bfA}(ds)f_s$, driven by conservative drivers is given in Section~\\ref{SubsectionIntegrationConservative}. To prove uniqueness of solutions to such equations, we develop in section \\eqref{SubsectionTensorization} a robust tensorization argument for a larger class of unbounded rough drivers that is the key to obtain some a priori bounds. These bounds imply uniqueness for rough linear equations driven by conservative drivers under a mild additional assumption, but they also lead to a complete $\\LL^2$-theory of rough transport equations, as illustrated in section \\eqref{SubsectionL2Transport}.\n\n\\smallskip\n\nNote that working in a Hilbert space setting, we have the continuous inclusions\n\\begin{equation}\n\\label{EqDoubleInclusion}\nE_n\\subset \\cdots \\subset E_1\\subset E_0 = E_{-0}\\subset E_{-1}\\subset \\cdots\\subset E_{-n}.\n\\end{equation}\n\n\\medskip\n\n\\subsection{Conservative drivers}\n\\label{SubsectionIntegrationConservative}\n\nWe start with the simple situation where the driver is \\emph{conservative} according to the following definition.\n\n\\begin{defn}\nA rough driver is said to be \\textbf{\\emph{conservative}} if we have \n\\begin{equation*}\n\\begin{split}\n&{\\bf (i)}\\quad A^{1,*}_{ts} +A^1_{ts} = 0, \\quad \\textrm{ on } E_1, \\\\\n&{\\bf (ii)}\\quad A^{2,*}_{ts} + A^2_{ts} + A_{ts}^{1,*}  A^1_{ts} = 0, \\quad \\textrm{ on } E_2,\n\\end{split}\n\\end{equation*}\nfor all $0\\leq s\\leq t<T$.\n\\end{defn}\nNotice that the conservative conditions make sense because of the above continuous inclusions \\eqref{EqDoubleInclusion}. As an elementary example of conservative unbounded driver, take as Banach spaces $E_n$ the Sobolev spaces $W^{n,2}(\\RR^d)$, with norm $|\\varphi|_n := \\sum_{k=0}^n \\big|\\nabla^k\\varphi\\big|_{\\LL^2}$, and consider the unbounded rough driver $\\bf A$  given by the formula\n\\begin{equation}\n\\label{EqRoughDriverRoughPath}\nA^1_{ts} = X^i_{ts}V_i,  \\quad\\quad A^2_{ts} = \\bbX^{jk}_{ts}V_jV_k,\n\\end{equation}\nfor some $\\ell$-dimensional geometric $\\gamma$-rough path, with $\\frac{1}{3}<\\gamma\\leq 1$, and some \\emph{divergence-free} vector fields $(V_i)_{i=1..\\ell}$ on $\\RR^d$, with the latter understood as first order differential operators. Condition {\\bf (ii)} partly plays the role in our setting that the notion of weak geometric rough path plays in a rough paths setting. Also, the antisymmetry condition {\\bf (i)} holds due to the fact that $V_i$ have null divergence, and condition {\\bf (ii)} holds as a consequence of the weak geometric character of $\\bfX$. Indeed, in this setting we have\n\\begin{equation*}\n\\begin{split}\n&A^{1,*}_{ts} = X^i_{ts}(V_i)^* = - X^i_{ts} V_i = -A^1_{ts}, \\\\\n&A^{2,*}_{ts} + A^2_{ts} =  \\bbX^{jk}_{ts}\\big(V_k^*V_j^*+V_j V_k\\big) = \\frac1 2  X^j_{ts}X^k_{ts} \\big(V_kV_j +V_j V_k\\big) = - A^{1,*}_{ts} A^1_{ts}\n\\end{split}\n\\end{equation*}\non $E_1$ and $E_2$ respectively. The boundedness assumptions \\eqref{EqBoundednessAssumptionRoughDriver} that $A^1_{ts}$ and $A^2_{ts}$ need to satisfy hold if, for instance, the vector fields $V_i$ are $\\mcC^2_b$. (Note that our setting is by no means restricted to working with vector fields. Working in the spaces $E_n = W^{kn,2}(\\RR^d)$, we may take $V_i$ in the above formula for $\\bfA$ the operators $V_i=W_i^k$, for some divergence-free vector fields $W_i$. Less trivial operators appear in a number of examples.)\n\n\\smallskip\n\nA general existence result holds for equations driven by conservative rough drivers under very mild conditions on the functional setting.\n\n\\begin{thm}\n\\label{ThmRDEHilbertGeneral}\nAssume one can associate to the scale $(E_n)_{n\\geq 0}$ a family of \\emph{self-adjoint} smoothing operators $\\big(J^\\epsilon\\big)_{0<\\epsilon\\leq 1}$, from $E_0$ to itself, satisfying the regularisation estimates \\eqref{EqApproximationProperties}, and let $\\bfA$ be a conservative unbounded $\\gamma$-rough driver on the scales $(E_n)_{n\\geq 0}$. Then for any $f_0\\in\\LL^2(\\RR^d)$, there exists an $\\LL^2(\\RR^d)$-valued path $f_\\bullet$, started from $f_0$, such that we have\n\\begin{equation*}\n\\delta f_{ts}(\\varphi) = f_s\\Big(A^{1,*}_{ts}\\varphi\\Big) + f_s\\Big(A^{2,*}_{ts}\\varphi\\Big) + f^\\sharp_{ts}(\\varphi)\n\\end{equation*}\nfor all $\\varphi\\in E_3$, with \n\\begin{equation*}\n\\big| f_t\\big|_0\\leq \\big| f_0\\big|_0,\n\\end{equation*}\nfor all $ t\\geq 0$, and we have, for each finite time horizon $T$,\n\\begin{equation}\n\\label{EqBoundRemainderX}\n\\Big| f^\\sharp_{ts}(\\varphi) \\Big| \\lesssim_{C_0,{\\bfA}, T,| f_0|_0}\\,\\big|\\varphi\\big|_3\\,|t-s|^{3\\gamma},\n\\end{equation}\nfor $0\\leq s\\leq t\\leq T$.\n\\end{thm}\n\n\\noindent The proof goes by approximating the unbounded rough driver $\\bfA$ by bounded rough drivers  ${\\bfA}^\\epsilon$, and by using the theory developed in Section~\\ref{SectionBoundedDrivers} to solve the equation\n\\begin{equation}\n\\label{EqApproximatedRDE}\n\\delta f^\\epsilon_{ts}(\\varphi) = f^\\epsilon_s\\Big(A^{\\epsilon,1,*}_{ts}\\varphi\\Big) + f^\\epsilon_s\\Big(\\big(A^{\\epsilon,2}_{ts}\\big)^*\\varphi\\Big) + f^{\\epsilon,\\sharp}_{ts}(\\varphi).\n\\end{equation}\nThe a priori bound \\eqref{EqFundamentalAPrioriEstimate} is used to pass to the limit as  $\\epsilon$ goes to $0$.\n\n\\medskip\n\n\\begin{Dem}\nMore concretly, let \n$$\nA^{\\epsilon,1}_{ts} := J^\\epsilon A^1_{ts}J^\\epsilon,\\quad\\quad A^{\\epsilon,2}_{ts} := J^\\epsilon A^{2,a}_{ts}J^\\epsilon - \\frac1 2 A^{\\epsilon,1}_{ts} A^{\\epsilon,1,*}_{ts},\n$$\nwith $J^\\epsilon$ understood as a continuous operator from $E_0$ to $E_{k_0+2}$, for $k_0$ big enough, and $A^{2,a}_{ts} = \\frac{1}{2}\\big(A^2_{ts}-A^{2,*}_{ts}\\big)$ stands for the antisymmetric part of $A^{2}_{ts}$. Both $A^{\\epsilon,1}$ and $A^{\\epsilon,2}$ are bounded linear operators from $E_0$ to itself, and by construction ${\\bfA}^\\epsilon = \\big(A^{\\epsilon,1},A^{\\epsilon,2}\\big)$ is a \\emph{bounded} conservative $\\gamma$-rough driver on the Banach algebra $\\textrm{L}(E_0)$. So one can denote by $M^\\epsilon_\\bullet$ the solution path in $\\textrm{L}(E_0)$ of the rough differential equation\n$$\n\\delta M^\\epsilon_{ts} = \\Big(A^{\\epsilon,1}_{ts} + A^{\\epsilon,2}_{ts}\\Big)M^\\epsilon_s + M^{\\epsilon,\\sharp}_{ts},\n$$\nprovided by the theory developed in Section~\\ref{SectionBoundedDrivers}. Given $f_0\\in E_0$, the path $f^\\epsilon_\\bullet = M^\\epsilon_\\bullet f_0$ solves the rough differential equation \\eqref{EqApproximatedRDE}. Whereas Theorem~\\ref{ThmIntegrationLinearBounded} provides an exponential control of the growth of the $E_0$-norm of $f^\\epsilon_t$, with an exponent $\\lambda^\\epsilon$ that may go to $\\infty$ as $\\epsilon$ goes to $0$, the conservative character of ${\\bfA}^\\epsilon$ ensures the uniform bound\n\\begin{equation*}\n\\big| f^\\epsilon_t\\big|_0 = \\big| f_0\\big|_0.\n\\end{equation*}\nIndeed, denoting by $(\\cdot,\\cdot)$ the scalar product on $E_0$, and setting $z^\\epsilon_t = \\big|f^\\epsilon_t\\big|^2_0$, one has\n\\begin{equation*}\n\\begin{split}\n\\delta z^\\epsilon_{ts} &= 2\\big(\\delta f^\\epsilon_{ts},f_s\\big) + \\big(\\delta f^\\epsilon_{ts}, \\delta f^\\epsilon_{ts}\\big) \\\\\n                                    &= 2\\Big(\\big(A^{\\epsilon,1}_{ts}+A^{\\epsilon,2}_{ts}\\big)f_s , f_s\\Big) + \\big(f^{\\epsilon,\\sharp}_{ts},f_s\\big) \\\\\n                                    &\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+ \\Big(\\big(A^{\\epsilon,1}_{ts}+A^{\\epsilon,2}_{ts}\\big)f_s , \\big(A^{\\epsilon,1}_{ts}+A^{\\epsilon,2}_{ts}\\big)f_s\\Big) + O\\big(|t-s|^{3\\gamma}\\big) \\\\\n                                    &= 2\\Big(A^{\\epsilon,2}_{ts}f_s , f_s\\Big) + \\Big(A^{\\epsilon,1}_{ts}f_s , A^{\\epsilon,1}_{ts}f_s\\Big) + O\\big(|t-s|^{3\\gamma}\\big)\n\\end{split}\n\\end{equation*} \nby the conservative character of $A^{\\epsilon,1}_{ts}$, which finally gives\n$$\n\\delta z^\\epsilon_{ts} = O\\big(|t-s|^{3\\gamma}\\big),\n$$\nsince the symmetric part of $A^{\\epsilon,2}_{ts}$ is $-\\frac{1}{2}A^{\\epsilon,1,*}_{ts}A^{\\epsilon,1}_{ts}$. The above equality shows that $z^\\epsilon$ is constant.\n\n\\smallskip\n\nRemark at that point that the constant \n\\begin{equation*}\nC^\\epsilon_0 := \\big\\|A^{\\epsilon,1}\\big\\|_{\\gamma\\,;\\,(-0,-1)}  + \\big\\|A^{\\epsilon,2}\\big\\|_{2\\gamma\\,;\\,(-0,-2)} +  \\big\\|A^{\\epsilon,1}\\big\\|_{\\gamma\\,;\\,(-2,-3)}  + \\big\\|A^{\\epsilon,2}\\big\\|_{2\\gamma\\,;\\,(-1,-3)}  < \\infty\n\\end{equation*}\nis not only finite but also uniformly bounded above, independently of $\\epsilon>0$. So we have, by Theorem~\\ref{ThmRegularityGain}  an $\\epsilon$-uniform upper bound on $\\big\\|f^{\\epsilon,\\sharp}\\big\\|_{3\\gamma\\,;\\,-3}$, of the form\n\\begin{equation}\n\\label{EqBondfSharpEpsilon}\n\\big\\|f^{\\epsilon,\\sharp}\\big\\|_{3\\gamma\\,;\\,-3} \\lesssim_{\\gamma,T,| f_0|_0}\\,1. \n\\end{equation}\nThese bounds ensure in particular that for each $\\varphi\\in E_3$, the functions $f^\\epsilon_\\bullet(\\varphi)$ form a bounded family of $\\gamma$-H\\\"older real-valued paths, so it has a subsequence converging uniformly to some $\\gamma$-H\\\"older real-valued function. Moreover, by weak-$\\star$ compactness, the uniform bound \\eqref{EqBondfSharpEpsilon} implies the existence of a sequence $(\\epsilon_n)_{n\\geq 0}$ converging to $0$, such that the sequence $f^{\\epsilon_n}$ converges weakly-$\\star$ in $\\LL^\\infty\\big([0,T],E_0\\big)$ to some limit $f\\in\\LL^\\infty\\big([0,T],E_0\\big)$. In particular, for each $\\varphi\\in E_3$, the sequence $f^{\\epsilon_n}(\\varphi)$ converges weakly-$\\star$ in $\\LL^\\infty\\big([0,T],\\RR\\big)$ to $f(\\varphi)$. As it has a uniformly converging subsequence, this shows the $\\gamma$-H\\\"older character of each function $f(\\varphi)$. \n\nAssuming $\\varphi\\in E_3$, it follows that one can pass to the limit in equation \\eqref{EqApproximatedRDE} in the three terms involving $f^\\epsilon_s$. The limit $f^\\sharp_{ts}(\\varphi)$ is defined as a consequence, and the bound \\eqref{EqBoundRemainderX} follows as a direct consequence of \\eqref{EqBondfSharpEpsilon}.\n\n\\smallskip \n\nIt is elementary to extend the above solution defined on $[0,T]$ to a globally defined solution satisfying the statement of the theorem.\n\\end{Dem}\n\n\\bigskip\n\nRather than working with a general scale of spaces satisfying some ad hoc conditions, we shall set \n$$\nE_n = W^{n,2}(\\RR^d),\n$$\nfor the remainder of this section on linear rough differential equation on Hilbert spaces. So we shall essentially be working from now on with rough drivers given by (at most) first order rough (pseudo-)differential operators. \n\n\n\\bigskip\n\n\n\\subsection{Tensorization}\n\\label{SubsectionTensorization}\n\nIn order to study the problem of uniqueness and further properties of solutions to general linear rough equations associated to unbounded rough drivers, we develop in this section a tensorization argument which can be seen as a rough version of the (differential) second quantisation functor in Hilbert spaces \\cite{ReedSimon}, or the variables doubling method commonly used in the theory of transport equations and conservation laws after the pioneering work of Kruzkhov \\cite{Kruzkhov}. As far as applications are concerned, we shall not restrict much our range in assuming that the rough drivers we are working with enjoy the following property. Given a bounded function $\\phi\\in W^{n_0,\\infty}$, denote by $M_\\phi$ the multiplication operator by $\\phi$; it is a bounded operator from $E_0 = \\LL^2(\\RR^d)$ to itself.\n\n\\begin{defn}\n\\label{DefnSymmetricDriver}\nAn unbounded rough driver $\\bfA$ is said to be \\textbf{\\emph{symmetric}} if the symmetric operators \n\\begin{equation*}\n\\begin{split}\n&{\\bf (i)}\\quad B^1_{{\\bfA},ts}(\\phi) = A^{1,*}_{ts}M_\\phi  + M_\\phi A^1_{ts} , \\\\\n&{\\bf (ii)}\\quad B^2_{{\\bfA},ts}(\\phi) = M_\\phi A^{2,*}_{ts}  +  A^2_{ts}M_\\phi + A_{ts}^1 M_\\phi A^{1,*}_{ts},\n\\end{split}\n\\end{equation*}\ndefine quadratic forms \n$$\ng\\mapsto \\Big(g,B^i_{{\\bfA},ts}(\\phi)g\\Big)\n$$\nthat are continuous on $E_0$, for all $0\\leq s\\leq t<T$, and for any test function $\\phi\\in W^{3,\\infty}$.\n\\end{defn}\n\nFor rough drivers ${\\bfA} = \\big(A^1,A^2\\big)$ of the form \n$$\nA^1_{ts} = X^i_{ts} V_i, \\quad\\quad A^2_{ts} = {\\bbX}^{jk}_{ts} V_jV_k,\n$$\nfor some vector fields $V_i$ on $\\RR^d$, and some weak geometric rough path $\\bfX$, the operator $B^1(\\phi)$ is the multiplication operator by $A^{1,*}_{ts}\\phi$, and the second quadratic form is of the form\n$$\n\\Big(g,B^2_{{\\bfA},ts}(\\phi)g\\Big) = \\Big(g^2 , h^\\phi_{ts}\\Big),\n$$\nfor some explicit function $h^\\phi_{ts}$ that turns $\\bfA$ into a symmetric unbounded rough driver if the vector fields $V_i$ are $\\mcC^2_b$ and $\\phi\\in W^{2,\\infty}$. \n\n\\bigskip \n \nLet now $\\bfA$ be a symmetric unbounded rough driver in the scale $(E_n)_{n\\geq 0}$, and let $f_\\bullet$ be a solution to the linear rough differential equation \n$$ \ndf_s = {\\bfA}(ds)f_s.\n$$\nConsider $f^{\\otimes 2} = f \\otimes f$, defined on $\\RR^d\\times\\RR^d$ by\n$\nf^{\\otimes 2}(x,y) = f(x)f(y)\n$; \nit satisfies the equation\n$$ \n\\delta f^{\\otimes 2} = \\Big( A^1  \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes A^1 \\Big) f^{\\otimes 2} + \\Big(A^2 \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes A^2 + \\tau (A^1 \\otimes A^1)\\Big) f^{\\otimes 2} + f^{\\otimes 2, \\sharp},\n$$\nwhere $\\mathbbm{I}$ stands for the identity map and\n$$\n\\tau (A^1 \\otimes A^1)_{t s} = A^1_{t s} \\otimes A^1_{t s}.\n$$\nSetting \n\\begin{equation*}\n\\begin{split}\n&\\Gamma_{\\bfA}^1 := A^1  \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes A^1, \\\\\n&\\Gamma_{\\bfA}^2 := A^2 \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes A^2 + \\tau (A^1 \\otimes A^1),\n\\end{split}\n\\end{equation*}\nand \n$$\n\\Gamma_{\\bfA} := \\big(\\Gamma_{\\bfA}^1, \\Gamma_{\\bfA}^2\\big), \n$$ \nwe have then \n$$\n\\delta \\Gamma_{\\bfA}^2 = \\big(A^1 A^1\\big) \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes \\big(A^1 A^1\\big) + A^1 \\otimes A^1 + \\sigma (A^1 \\otimes A^1),\n$$ \nwhere $\\sigma (v \\otimes w) = w \\otimes v$ is the exchange of the two factors in the tensor so that \n$$\n\\sigma (A^1 \\otimes A^1)_{t u s} = A^1_{us} \\otimes A^1_{tu} + A^1_{tu} \\otimes A^1_{us}.\n$$ \nIf follows that $\\Gamma_{\\bfA}$ satisfies Chen's relations \\eqref{EqChenRelations}. Endowing $E_n^{\\otimes 2}$ with its natural Hilbert space structure, we turn $\\Gamma_{\\bfA}$ into a $\\gamma$-H\\\"older unbounded rough driver in the scale $\\big(E_n^{\\otimes 2}\\big)_{n \\geq 0}$, for which $f^{\\otimes 2}$ happens to be a solution, in this scale of spaces, of the linear rough equation\n$$\ndf^{\\otimes 2}_s = \\Gamma_{\\bfA}(ds)f^{\\otimes 2}_s. \n$$\n\n\\smallskip \n\nThe goal of the doubling variable method is to use the dynamics of $f^{\\otimes 2}$ to gain some information of its behaviour near the diagonal of $\\RR^d \\times \\RR^d$ in order to control the trace of $f^{\\otimes 2}$ on the diagonal. \n\nThinking to this goal let us define a scale of spaces $\\mathcal{E}^\\nabla_n$ of test functions on  $\\RR^d \\times \\RR^d$, by requiring that they have finite $| \\cdot |^\\nabla_n$-norm. In order to introduce this norm let us define\n$$\n| \\Phi |^\\nabla_n :=\\sup_{0\\le k+l\\le n}  \\int_{\\RR^d} \\sup_{z\\in\\RR^d} \\big|\\partial_z^k \\partial_w^l\\Phi(z+w, z-w)\\big|\\, dw.\n$$ \nMoreover we introduce a blow-up transformation $T_\\varepsilon$ on test functions as follows:\n$$\nT_\\varepsilon \\Phi(x,y) =\\varepsilon^{-d} \\Phi\\big(x_+ + \\frac{x_-}{\\varepsilon}, x_+ - \\frac{x_-}{\\varepsilon}\\big) \n$$\nwhere $x_\\pm = \\frac{x \\pm y}{2}$ are coordinates parallel and transverse to the diagonal of $\\RR^d \\times \\RR^d$. Note that $T_\\varepsilon$ is a bounded operator in $\\mathcal{E}^\\nabla_0$ uniformly in $\\varepsilon$. The adjoint of this transformation for the $\\LL^2$ scalar product reads\n$$\nT^*_\\varepsilon F(x,y) = F\\big(x_+ + \\varepsilon x_- , x_+ -\\varepsilon x_-\\big).\n$$\nWe let  \n$$\n\\Gamma_{\\bfA,\\varepsilon} := T_\\varepsilon^{-1}\\Gamma_{\\bfA}T_\\varepsilon.\n$$ \nRecall the definition of the operators $ B^1_{{\\bfA},ts}(\\phi)$ and $ B^2_{{\\bfA},ts}(\\phi)$ used in the definition of a symmetric unbounded rough driver. We use below brackets $\\langle\\cdot,\\cdot\\rangle$ to denote the $\\LL^2$ scalar product. Note that $f_s$ is an element of $E_{-0} = E_0 = \\LL^2(\\RR^d)$, so $f_s^2$ is in $\\LL^1(\\RR^d)$. \n \n\\begin{defn}[Renomalizable drivers]\nThe unbounded $\\gamma$-rough driver $\\bfA$ is \\textbf{\\emph{renormalizable}} if $\\{\\Gamma_{\\bfA,\\varepsilon}\\}_{\\varepsilon>0}$  is a bounded family of unbounded $\\gamma$-rough drivers in the scale of spaces $\\big(\\mathcal{E}^\\nabla_n\\big)_{n\\geq 0}$.\n\\end{defn} \n\nThe following lemma gives flesh to the expresion 'Renormalizable driver', such as defined here. It needs to be understood in the light of di Perna-Lions' work on transport equation \\cite{diPernaLions} where a notion of renormalizable solution was first introduced. Definition~\\ref{DefnrenormalisedSolutions} will make that parallel clear in our study of the $\\LL^\\infty$ theory for the rough transport equation. Under some additional assumption stated in Definition~\\ref{DefnClosedDriver}, the present notion of renormalizable unbounded rough driver will provide in Theorem~\\ref{ThmRDEHilbertGeneralClosed} a general uniqueness result.  \n\n\\begin{lem}[Renormalisation]\n\\label{Lemmarenormalisation}\nLet $\\bfA$ be a symmetric rough driver and $f_\\bullet$ be a solution of the equation \n$$\ndf_s = {\\bfA}(ds)f_s,\n$$ \nin the initial scale of spaces $(E_n)_{n\\geq 0}$. If $\\bfA$ is renormalizable \nthen the $\\LL^1(\\RR^d)$-valued path $f^2_\\bullet$ satisfies, for all $\\phi \\in W^{3,\\infty}$, the equation\n\\begin{equation} \n\\label{EqPrimitiveGronwall}\n\\delta f^2(\\phi)_{ts} =  \\langle f_s , \\big(B^1_{{\\bfA},ts}(\\phi) + B^2_{{\\bfA},ts}(\\phi)\\big) f_s\\rangle  + O (| \\phi |_{W^{3,\\infty}} | t - s |^{3 \\gamma}) .\n\\end{equation}\n \\end{lem}\n\n\\smallskip\n \n\\noindent By polarisation the product $fg$ satisfies an equation analogous to equation \\eqref{EqPrimitiveGronwall} if both $f$ and $g$ are solutions of the equation $df_s = {\\bfA}(ds)f_s$, in the scale $(E_n)_{n\\geq 0}$.\n\n\\bigskip\n\n\\begin{Dem}\nNote that $f_\\bullet^{\\otimes 2}$ satisfies the equation\n\\begin{equation}\n\\label{EqFOtimes2}\n\\delta f^{\\otimes 2}_\\varepsilon (\\Phi)_{ts} = f^{\\otimes 2}_{\\varepsilon,s}((\\Gamma_{\\bfA,\\varepsilon}^{1,*})_{ts}\\Phi) +f^{\\otimes 2}_{\\varepsilon,s} ( (\\Gamma_{\\bfA,\\varepsilon}^{2,*})_{ts}\\Phi) +  f^{\\otimes 2,\\sharp}( T_\\varepsilon\\Phi)_{ts}\n\\end{equation}\nfor all smooth functions $\\Phi$ where $f^{\\otimes 2}_\\varepsilon = T^*_\\varepsilon f^{\\otimes 2}$.\nNote that if we show that $f^{\\otimes 2}_\\varepsilon$ is uniformly bounded in $\\mathcal{E}^\\nabla_{-0}$ then from the hypothesis that  $\\{\\Gamma_{\\bfA,\\varepsilon}\\}_{\\varepsilon>0}$ is a bounded family of unbounded $\\gamma$-rough drivers in the scale of spaces $\\big(\\mathcal{E}^\\nabla_n\\big)_{n\\geq 0}$ we also have\n $\\big\\|f^{\\otimes 2,\\sharp}(T_\\varepsilon \\Phi)\\big\\|_{3\\gamma}\\lesssim \\big| \\Phi\\big|^\\nabla_3$. As a consequence the $3\\gamma$-H\\\"older norm of the remainders $f^{\\otimes 2,\\,\\sharp}\\big(T_\\varepsilon \\Phi\\big)$ are bounded uniformly in  $\\varepsilon$ for fixed $\\Phi$. \n\nEquation \\eqref{EqPrimitiveGronwall} will come from taking in equation \\eqref{EqFOtimes2} some functions $\\Phi$ of the form\n$$\n\\Phi(x,y) = \\psi(x-y)\\,\\phi\\left(\\frac{x+y}{2}\\right),\n$$\nand by letting $\\varepsilon$ tend to $0$, after checking that some $\\varepsilon$-uniform estimates hold for the different terms in \\eqref{EqFOtimes2}.\n\n\\smallskip\n Cauchy--Schwartz inequality provides the  bound\n\\begin{equation*}\n\\begin{split}\n\\Big|f^{\\otimes 2}_s \\big(T_\\varepsilon \\Phi\\big)\\Big| &= \\left|\\int_{\\RR^d \\times \\RR^d} f_s(x_+ + \\varepsilon x_-) f_s(x_+ - \\varepsilon x_-) \\Phi(x_+ +  x_-,x_+ -  x_-)\\,dx_+ dx_- \\right|  \\\\\n&\\leq \\max_\\pm \\int_{\\RR^d \\times \\RR^d} |f_s(x_+ \\pm \\varepsilon x_-)|^2  |\\Phi(x_+ +  x_-,x_+ -  x_-)|\\,dx_+ dx_- \\\\\n&\\leq  \\max_\\pm   \\int_{\\RR^d \\times \\RR^d} |f_s(x_+ \\pm \\varepsilon x_-)|^2  \\sup_z |\\Phi(z +  x_-,z -  x_-)|\\,dx_+ dx_- \\\\\n&\\leq |f|_{\\LL^2}^2 \\int_{\\RR^d } \\sup_{z}| \\Phi(z +  w,z -  w)|\\,dw  \\\\\n&\\leq  |\\Phi|_{\\mathcal{E}^\\nabla_0} \\big|f_s\\big|_{\\LL^2}^2. \n\\end{split}\n\\end{equation*}\nwhich shows that $f^{\\otimes 2}_\\varepsilon$ is uniformly bounded in $\\mathcal{E}^\\nabla_{-0}$.\nNow, given a positive constant $\\delta$, the fact that for any smooth function $g$ which is $\\delta$-close in $\\LL^2(\\RR^d)$ of $f_s$, we have\n$$ \n\\Big|f^{\\otimes 2}_s \\big(T_\\varepsilon \\Phi\\big)- g^{\\otimes 2} \\big(T_\\varepsilon \\Phi\\big) \\Big| \\lesssim 2\\delta |f|_{\\LL^2} + \\delta^2,\n$$ \nuniformly in $\\varepsilon$, and\n$$\n\\lim_{\\varepsilon\\to 0} g^{\\otimes 2} \\big(T_\\varepsilon \\Phi) = \\int_{\\RR^d }\\big|g(x)\\big|^2 \\phi (x)\\,dx,\n$$\nshows that \n$$\nf^{\\otimes 2}_s \\big(T_\\varepsilon \\Phi\\big) \\underset{\\varepsilon\\to 0}{\\longrightarrow} f_s^2(\\phi). \n$$\nWe also have the convergence \n$$\ng^{\\otimes 2} \\left((\\Gamma_{\\bfA}^1)^*_{ts} T_\\varepsilon \\Phi\\right) = \\big( A^1_{ts} g \\otimes g + g \\otimes A^1_{ts} g, T_\\varepsilon \\Phi \\big) \\underset{\\varepsilon\\to 0}{\\longrightarrow} 2 \\int_{\\RR^d } g(x) \\big(A^1_{ts}g\\big)(x) \\phi (x)\\, dx,\n$$\nwhich we can rewrite as\n$$\ng^{\\otimes 2} \\left((\\Gamma_{\\bfA}^1)^*_{ts} T_\\varepsilon \\Phi\\right) \\underset{\\varepsilon\\to 0}{\\longrightarrow} 2\\Big( g , A^{1,*}_{ts} M_\\phi g \\Big) = \\Big(g,B^1_{{\\bfA},ts}(\\phi)g\\Big).\n$$\nUsing in addition the boundedness on $\\LL^2(\\RR^d)$ of the quadratic form associated to $B^1_{ts}(\\phi)$, one can then send $g$ to $f_s$, in $\\LL^2(\\RR^d)$, in the above convergence result and conclude that\n$$\nf^{\\otimes 2}_s \\big((\\Gamma_{\\bfA}^1)_{ts}^* T_\\varepsilon \\Phi\\big) \\to \\big( f_s , B^1_{{\\bfA},ts}(\\phi) f_s\\big). \n$$\nSimilarly, the boundedness in $\\mathcal{E}^\\nabla_0$ of the family $\\big((\\Gamma_A^2)^*_{ts}  T_\\varepsilon \\Phi\\big)_{0<\\varepsilon\\leq 1}$, together with the boundedness on $\\LL^2(\\RR^d)$ of the quadratic form associated with $B^2_{{\\bfA},ts}(\\phi)$, show that \n$$\nf^{\\otimes 2}_s \\big((\\Gamma_{\\bfA}^2)_{ts}^*\\Phi^\\epsilon\\big) \\to \\big( f_s , B^2_{{\\bfA},ts}(\\phi) f_s\\big);\n$$\nequation \\eqref{EqPrimitiveGronwall} follows, as we have the $\\varepsilon$-uniform bound $\\Big|f^{\\otimes 2,\\,\\sharp}_{ts}(T_\\varepsilon \\Phi)\\Big| \\leq  \\big|\\Phi\\big|_{3}^\\nabla \\leq \\big|\\phi\\big|_{W^{3,\\infty}}|t-s|^{3\\gamma}$.  \n\\end{Dem} \n\n\\bigskip\n\nThis result is sufficient to prove that rough linear equations driven by conservative drivers are unique if the driver is symmetric. \n\n\\begin{cor}\n\\label{CorClosedSymmCons}\nLet $\\bfA$ be a renormalizable symmetric conservative unbounded $\\gamma$-rough driver in the scale of spaces $(E_n)_{n\\geq 0}$. Then the rough linear equation\n$$\ndf_s = {\\bfA}(ds)f_s\n$$\nhas a unique solution in $\\LL^2(\\RR^d)$, started from any initial condition $f_0\\in E_0$; it satisfies $\\big|f_t\\big|_0 = \\big|f_0\\big|_0$, for all times $t$.\n\\end{cor}\n\n\\smallskip\n\n\\begin{Dem}\nIt suffices to notice that since the driver $\\bfA$ is conservative, we have $B^1_{{\\bfA},ts}({\\bf 1}) = B^2_{{\\bfA},ts}({\\bf 1}) = 0$, so it follows from equation \\eqref{EqPrimitiveGronwall} that any solution path $f_\\bullet$ has constant $\\LL^2$-norm, which proves the uniqueness claim. Existence was proved in Theorem~\\ref{ThmRDEHilbertGeneral}.\n\\end{Dem}\n\n\n\\bigskip\n\n\n\\subsection{A priori bounds for closed symmetric drivers}\n\\label{SymmetricDrivers}\n\nOne cannot use directly the renormalisation lemma to get some closed equation for $f^2$ when $\\bfA$ is non-conservative. We need for that purpose to assume that the symmetric unbounded rough driver $\\bfA$ enjoys the following property. Given that $\\bfA$ is symmetric, recall the definition of its associated family of symmetric operators $B^1_{{\\bfA},ts}(\\phi)$ and $B^2_{{\\bfA},ts}(\\phi)$ on $E_0$, indexed by $(s,t)$ and $\\phi\\in W^{3,\\infty}$, given in Definition~\\ref{DefnSymmetricDriver}.\n\n\\begin{defn}\n\\label{DefnClosedDriver}\nA \\emph{\\textbf{symmetric unbounded rough driver}} $\\bfA$ in the scale of spaces $\\big(W^{n,2}\\big)_{n\\geq 0}$, is said to be \\emph{\\textbf{closed}} if there exists some unbounded rough driver ${\\bf B}=\\big(B^1,B^2\\big)$ in the scale of spaces $\\big(W^{n,\\infty}\\big)_{n\\geq 0}$, such that we have  \n$$\n\\Big(g,B^1_{{\\bfA},ts}(\\phi)g\\Big) = \\Big(g^2,(B^1_{ts})^*\\phi\\Big), \\quad \\textrm{  and  } \\quad \\Big(g,B^2_{{\\bfA},ts}(\\phi)g\\Big) = \\Big(g^2,(B^2_{ts})^*\\phi\\Big),\n$$\nfor all $g\\in E_0$ and $\\phi\\in W^{3,\\infty}$. \n\\end{defn}\n\n\\smallskip\n\nAs an example, it is elementary to check that the unbounded rough driver ${\\bfA} = \\big(XV,\\bbX VV\\big)$ used in the rough transport equation\n\\begin{equation*}\n\\delta f_{ts} = X\\,Vf_s +  \\bbX\\,V V f_s + f^\\sharp_{ts}\n\\end{equation*}\nwith some $\\gamma$-H\\\"older weak geometric rough path ${\\bfX} = (X,\\bbX)$, is closed and symmetric if the vector fields $V = \\big(V_1,\\dots,V_\\ell\\big)$ are $C^3_b$, in which case $\\bfB = \\bfA$. Another interesting class of examples of closed symmetric unbounded rough driver in the scale of spaces $(E_n)_{n\\geq 0}$, is provided by the lift to rough drivers of $\\mcC^3_b$-semimartingale velocity fields, as given in the theory of stochastic flows. This kind of stochastic velocity fields appear in the study of Navier--Stokes equation. See the work \\cite{BailleulRiedel} for a thorough study of stochastic flows from this point of view. \n\n\\medskip\n\nBuilding on this notion of closed driver, the following statement provides amongst other things an a priori estimate on solutions of rough linear equations that plays in this setting the role played in the classical setting by a priori estimates obtained by any kind of Gronwall--type argument. The crucial point here is that no such Gr\\\"onwall machinery was available so far in a rough path--like setting; despite its elementary nature, this result may well be one of our main contributions. \n\n\\smallskip\n\n\\begin{thm}\n\\label{ThmRDEHilbertGeneralClosed}\nLet $\\bfA$ be a renormalizable  \\emph{closed symmetric} unbounded $\\gamma$-rough driver on the scales $\\big(W^{n,2}(\\RR^d)\\big)_{n\\in\\mathbb{N}}  \\newcommand{\\ZZ}{\\mathbb{Z}}  \\newcommand{\\QQ}{\\mathbb{Q}}  \\newcommand{\\HH}{\\mathbb{H}}$. Let $\\bfB$ be its associated driver, and assume that the inequality\n\\begin{equation}\n\\label{BoundPhi}\n\\Big|\\big(B^1_{t0}\\big)^*{\\bf 1} \\Big| \\vee \\Big|\\big(B^2_{t0}\\big)^*{\\bf 1} \\Big| \\leq c_t\n\\end{equation}\nholds for all times $t$, for some time-dependent mositive constant $c_t$ such that $e^{-\\lambda t}c_t$ tends to $0$ as $t$ goes to infinity, for any positive parameter $\\lambda$, so\n$$\n\\llparenthesis\\,c_\\bullet\\,\\rrparenthesis < \\infty.\n$$\n Then, given any $f_0\\in\\LL^2(\\RR^d)$, there is at most one $\\LL^2(\\RR^d)$-valued solution path $f_\\bullet$ to the equation\n\\begin{equation}\n\\label{EqLinearRDEHilbert}\n\\delta f_{ts}(\\varphi) = f_s\\Big(A^{1,*}_{ts}\\varphi\\Big) + f_s\\Big(A^{2,*}_{ts}\\varphi\\Big) + f^\\sharp_{ts}(\\varphi),\n\\end{equation}\nand we have, for each finite time horizon $T$, \n\\begin{equation}\n\\label{EqBoundRemainder}\n\\Big| f^\\sharp_{ts}(\\varphi) \\Big| \\lesssim_{{\\bfB},T, | f_0|_0}\\,\\big|\\varphi\\big|_3\\,|t-s|^{3\\gamma},\n\\end{equation}\nfor all $\\varphi\\in W^{3,2}$, and all $0\\leq s\\leq t\\leq T$. It satisfies the upper bound\n\\begin{equation}\n\\label{EqAPrioriBoundNorm}\n\\big| f_t\\big|_0\\lesssim_{{\\bfB}, t} \\big| f_0\\big|_0\n\\end{equation}\n\\end{thm}\n\n\\smallskip\n \n\\begin{Dem}\n\\noindent Let $f_\\bullet$ be a solution to the rough linear equation \\eqref{EqLinearRDEHilbert} in the scale of spaces $\\big(W^{n,2}(\\RR^d)\\big)_{n\\in\\mathbb{N}}  \\newcommand{\\ZZ}{\\mathbb{Z}}  \\newcommand{\\QQ}{\\mathbb{Q}}  \\newcommand{\\HH}{\\mathbb{H}}$. Since $\\bfA$ is closed, the $\\LL^1(\\RR^d)$-valued path $f^2_\\bullet$ happens to be a solution to the rough linear equation\n\\begin{equation*} \n\\delta f^2(\\phi)_{ts} = f^2_s\\big((B^1_{ts})^*\\phi\\big) + f^2_s\\big((B^2_{ts})^*\\phi\\big) + (f^2)^\\sharp_{ts}(\\phi),\n\\end{equation*}\nin the scale of spaces $\\big(W^{n,\\infty}(\\RR^d)\\big)_{n\\in\\mathbb{N}}  \\newcommand{\\ZZ}{\\mathbb{Z}}  \\newcommand{\\QQ}{\\mathbb{Q}}  \\newcommand{\\HH}{\\mathbb{H}}$. Denote by $C_0^{\\bf B}$ the finite constant associated to the unbounded rough driver $\\bfB$, as defined by equation \\eqref{EqDefnC0}, with $\\bfB$ in the role of $\\bfA$. It follows from the general a priori estimates on solutions of rough linear equations proved in Theorem~\\ref{ThmGeneralRegularityGain}, and the fact that $f^2$ is in $\\LL^1(\\RR^d)$, that \n\\begin{equation}\n\\label{EqBoundf2}\n\\begin{split}\n\\llparenthesis\\, (f^2)^\\sharp \\,\\rrparenthesis_{3\\gamma\\,;\\,(W^{3,\\infty})^*} &\\lesssim_{\\gamma,\\lambda}C_0^{\\bf B}\\,\\llparenthesis\\, f^2 \\,\\rrparenthesis_{(\\LL^\\infty)^*} \n\\lesssim _{\\gamma,\\lambda} C_0^{\\bf B}\\,\\llparenthesis\\, f^2_\\bullet \\,\\rrparenthesis_{\\LL^1}.\n\\end{split}\n\\end{equation}\nBut since we have the identity \n$$\nf^2_t({\\bf 1}) = f_0^2\\Big({\\bf 1} + (B^1_{ts})^*{\\bf 1} + (B^2_{ts})^*{\\bf 1}\\Big) + (f^2)^\\sharp_{t0}({\\bf  1})\n$$\nand the bound \\eqref{BoundPhi}, we also have the estimate\n\\begin{equation*}\n\\begin{split}\n\\llparenthesis\\, f^2_\\bullet \\,\\rrparenthesis_{\\LL^1} = \\llparenthesis\\, f^2_\\bullet({\\bf 1}) \\,\\rrparenthesis &\\lesssim_{\\llparenthesis\\,c_\\bullet\\,\\rrparenthesis} \\,\\big| f_0 \\big|_{\\LL^2} + \\llparenthesis\\, (f^2)^\\sharp_{\\bullet 0} \\,\\rrparenthesis_{(W^{3,\\infty})^*}  \\\\\n&\\lesssim_{\\llparenthesis\\,c_\\bullet\\,\\rrparenthesis} \\, \\Big( \\big| f_0 \\big|_{\\LL^2} + \\lambda^{3\\gamma}\\,\\llparenthesis\\, (f^2)^\\sharp \\,\\rrparenthesis_{3\\gamma\\,;\\,(W^{3,\\infty})^*} \\Big).\n\\end{split}\n\\end{equation*}\n(Note that $(f^2)^\\sharp$, in the right hand side of the above inequality, is seen as a 2-index function.) Together with the bound \\eqref{EqBoundf2}, this gives the upper bound\n$$\n\\llparenthesis\\, f^2_\\bullet \\,\\rrparenthesis_{\\LL^1} \\lesssim_{\\llparenthesis\\,c_\\bullet\\,\\rrparenthesis,\\,C_0^{\\bf B}} \\,\\big| f_0 \\big|_{\\LL^2}\n$$\nfor $\\lambda$ small enough, which implies uniqueness.\n\\end{Dem}\n\n\n\\medskip\n\n\n\\subsection{Rough transport equation}\n\\label{SubsectionL2Transport}\n\nBuilding on Theorem~\\ref{ThmRDEHilbertGeneralClosed}, one can give a complete $\\LL^2$-theory of rough transport equations \n\\begin{equation*}\n\\delta f_{ts} = X\\,Vf_s +  \\bbX\\,V V f_s + f^\\sharp_{ts}\n\\end{equation*} \ndriven by non-divergence-free vector fields $V_i$ of class $W^{3,\\infty}$.\n\n\\medskip\n\n\\begin{lem}\n\\label{LemmaGammaA}\nLet $\\bfX$ be a geometric $\\gamma$-H\\\"older rough path on $\\RR^\\ell$, and $V_1,\\dots,V_\\ell$ be $W^{3,\\infty}$ vector fields on $\\RR^d$. Then the operator $\\Gamma_{\\bfA}$ associated with ${\\bfA} = \\big(XV, \\bbX VV\\big)$ is renormalizable in the scale of spaces $\\big(\\mathcal{E}^\\nabla_n\\big)_{n\\geq 0}$.\n\\end{lem}\n\n\\smallskip \n\n\\begin{Dem}\nFor a  geometric rough path $\\bfX = (X,\\bbX)$, the operator $\\Gamma_{\\bfA}$ takes the form $\\Gamma_{\\bfA} = \\big(X  \\Gamma_{V}^{1}, \\bbX \\Gamma_{V}^{2})$, with\n\\begin{equation*}\n\\begin{split}\n&\\Gamma_{V}^{1} := V  \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes V, \\\\\n&\\Gamma_{V}^{2} := V V \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes V V + 2 (V \\otimes V) = \\Gamma_{V}^{1} \\Gamma_{V}^{1}.\n\\end{split}\n\\end{equation*}\nSo  it is enough to show that the adjoints of these operators satisfy, uniformly in $\\varepsilon$, the inequalities\n\\begin{equation}\n\\label{eq:est-gamma-v}\n|\\Gamma_{V,\\varepsilon}^{1,\\ast} \\Phi|^\\nabla_{n} \\lesssim |V|_{W^{n+1,\\infty}} |\\Phi|^\\nabla_{n+1},\n\\qquad\n|\\Gamma_{V,\\varepsilon}^{2,\\ast} \\Phi|^\\nabla_{m} \\lesssim |V|_{W^{m+2,\\infty}}^2 |\\Phi|^\\nabla_{m+2}\n\\end{equation}\nfor $n=0,2,\\,m=0,1$, for smooth test functions $\\Phi$ where $\\Gamma_{V,\\varepsilon}^{j,\\ast} = T_\\varepsilon^{-1} \\Gamma_{V}^{j,\\ast}T_\\varepsilon $ for $j=1,2$.\n\n\\smallskip\n\nWrite $V = v_k \\partial_k$ where $(v_k)_{k=1,\\dots,d}$ are the coefficients of the vector fields in the canonical basis $(\\partial_k)_{k=1,\\dots,d}$ of derivations; with these notations, we have\n$$\nV^\\ast = - v_k \\partial_k - d,\n$$ \nwhere $d := \\textrm{div} v$, is the divergence of the vector field $V$, and\n$$\n\\Gamma_{V}^{1,\\ast} = v^+_k \\partial^+_k + v^-_k \\partial^-_k + d^+ + d^-,\n$$  \nwhere, for a real-valued function $h$ on $\\RR^d$, we denote by\n$$\nh^\\pm(x,y) := h(x) \\pm h(y)\n$$ \nits symmetric and antisymmetric lift to $\\RR^d \\times \\RR^d$ and\n$$\nh^\\pm_\\varepsilon(x,y) := h(x_+ +\\epsilon x_-) \\pm h(x_+ -\\epsilon x_-)\n$$ \ntheirs blowup according to the transformation $h^\\pm_\\varepsilon = T_\\varepsilon^{-1} h^\\pm T_\\varepsilon$.\nNote that\n$$\n\\partial^+_k T_\\varepsilon = T_\\varepsilon \\partial^+_k\n\\qquad\n\\partial^-_k T_\\varepsilon = \\varepsilon^{-1} T_\\varepsilon \\partial^-_k\n$$\nso that\n$$\n\\Gamma_{V,\\varepsilon}^{1,\\ast} = T_\\varepsilon^{-1} \\Gamma_{V}^{1,\\ast}T_\\varepsilon = v^+_{k,\\varepsilon} \\partial^+_k +\\frac{ v^-_{k,\\varepsilon}}{\\varepsilon} \\partial^-_k + d^+_\\varepsilon + d^-_\\varepsilon,\n$$  \n\n\nThe first estimate in \\eqref{eq:est-gamma-v} follows from the inequalities\n$$\n\\big| a^{+}_\\varepsilon \\nabla^{+} \\Phi\\big|^\\nabla_{n}+\\big| \\varepsilon^{-1} a^{-}_\\varepsilon \\nabla^{-} \\Phi\\big|^\\nabla_{n} \\lesssim ( |a|_{W^{n,\\infty}} + |\\nabla a|_{W^{n,\\infty}}) |\\Phi|^\\nabla_{n+1}.\n$$ \nThe second inequality in \\eqref{eq:est-gamma-v} is obtained by noting that we have \n\\begin{equation*}\n\\begin{split}\n|\\Gamma_{V}^{2,\\ast} \\Phi|^\\nabla_{m} =  |\\Gamma_{V}^{1,\\ast}\\Gamma_{V}^{1,\\ast} \\Phi|^\\nabla_{m} &\\lesssim |V|_{W^{m+1,\\infty}} |\\Gamma_{V}^{1,\\ast} \\Phi|^\\nabla_{m+1} \\\\\n&\\lesssim |V|_{W^{m+1,\\infty}} |V|_{W^{m+2,\\infty}} | \\Phi|^\\nabla_{m+2}.\n\\end{split}\n\\end{equation*}\n\n\\end{Dem}\n\n\\smallskip\n\n\\begin{thm}\n\\label{ThmTransportL2}\nLet $\\bfX$ be a geometric $\\gamma$-H\\\"older rough path on $\\RR^\\ell$, and $V_1,\\dots,V_\\ell$ be $W^{3,\\infty}$ vector fields on $\\RR^d$. Then the rough transport equation \n$$\n\\delta f_s = \\big(X_{ts}\\,V + \\bbX_{ts}\\,VV\\big)f_s + f^\\sharp_{ts}\n$$ \nis well-posed.\n\\end{thm}\n\n\\smallskip\n\n\\begin{Dem}\nNotice first that the regularity assumption on the $V_i$ puts us in a position to use the a priori bounds for symmetric closed drivers stated in Theorem~\\ref{ThmRDEHilbertGeneral}, with $\\bfA$ in the role of $\\bfB$. So uniqueness is a direct consequence of the a priori bound \\eqref{EqAPrioriBoundNorm} in Theorem~\\ref{ThmRDEHilbertGeneralClosed}.\n\n\\smallskip\n\nLet now $f_0\\in E_0$ be given. We prove the existence of a solution path to rough transport equation started from $f_0$, by a classical approximation-compactness argument, relying in a crucial way on the a priori bound \\eqref{EqAPrioriBoundNorm} on the $\\LL^2$-norm of the solution to the approximate problem, and on the uniform estimate \\eqref{EqBoundRemainderX} for the remainder. \n\n\\smallskip\n\nFix a finite time horizon $T$.\nGiven that $\\bfX$ is geometric, let $\\big({\\bfX}^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ be a family of rough path lifts of smooth paths which converge to $\\bfX$ is a rough paths sense over the time interval $[0,T]$. Let also $\\big(V^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ be a family of smooth vector fields that converge to $V$ in $W^{3,\\infty}$, and let $\\big({\\bf A}^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ be their associated rough driver, defined by formula \\eqref{EqRoughDriverRoughPath} with ${\\bfX}^\\epsilon$ and $V^\\epsilon$ in place of $\\bfX$ and $V$ respectively. One can choose $\\big({\\bfX}^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ in such a way that the constant $C_0^\\epsilon$ associated with ${\\bf A}^\\epsilon$ by formula \\eqref{EqDefnC0} satisfies the inequality $C_0^\\epsilon\\lesssim C_0$, independently of $0<\\epsilon\\leq 1$. Given the smooth character of the vector fields $V^\\epsilon$, one can solve uniquely the transport equation\n\\begin{equation*}\n\\delta f^\\epsilon_{ts}(\\varphi) = f^\\epsilon_s\\big(V^{\\epsilon,\\ast} \\varphi\\big)\\,X^\\epsilon_{ts} + f^\\epsilon_s\\big(V^{\\epsilon,\\ast}V^{\\epsilon,\\ast}\\varphi\\big)\\,\\bbX^\\epsilon_{ts} + f^{\\epsilon, \\sharp}_{ts}(\\varphi), \\quad\\quad \\textrm{for } \\varphi\\in E_2,\n\\end{equation*}\nby the method of characteristics, as the above equation is actually equivalent to the ordinary differential equation\n$$ \n\\frac{df^\\epsilon_t}{dt} = f^\\epsilon_t\\big(V^{\\epsilon,\\ast}\\varphi\\big)\\,{X^\\epsilon_t}'.\n$$\nThe solutions of this problem satisfy the uniform estimates\n\\begin{equation*}\n\\big| f^\\epsilon_t \\big|_0 \\lesssim_{C_0,T}\\,\\big| f_0 \\big|_0,\n\\end{equation*}\nfor all $0\\leq t\\leq T$, as a consequence of \\eqref{EqAPrioriBoundNorm}, and we also have the uniform bound\n\\begin{equation*}\n\\sup_{0<\\epsilon\\leq 1}\\;\\big\\| f^{\\epsilon, \\sharp} \\big\\|_{3 \\gamma\\,;\\, - 3} \\lesssim_{C_0, \\gamma, T,|f_0|_0} 1,\n\\end{equation*}\nby \\eqref{EqBoundRemainderX}. These two a priori estimates are all we need to finish the proof of the theorem following word by word the end of the proof of Theorem~\\ref{ThmRDEHilbertGeneralClosed}. \n\\end{Dem}\n\n\\medskip\n\nIt is perfectly possible to extend the present theory to deal with rough linear equations with a {\\bf drift}\n$$\ndf_s = Wf_s ds + {\\bf A}(ds)f_s,\n$$\nwhere $W\\in\\textrm{L}\\big(E_{-0},E_{-2}\\big)$, such as the Laplacian operator in the $W^{n,2}(\\RR^d)$ scale of spaces. We refrain from giving the details here as this is not our main point and this does not require the introduction of new tools or ideas. This provides an alternative road to some of the results of \\cite{DiehlFrizStannat} in a slightly different setting.\n\n\n\\bigskip\n \n\n\\section{The $\\LL^\\infty$ theory of rough transport equations}\n\\label{SectionLInftyTheory}\n\nWe develop in this section an $\\LL^\\infty$ theory of the rough transport equation \n\\begin{equation}\n\\label{EqTransportEq}\n\\delta f_{ts} = X\\,Vf_s +  \\bbX\\,V V f_s + f^\\sharp_{ts}\n\\end{equation}  \nand prove its well-posed character under the assumption that the vector fields be $\\mcC^3$, for some positive constant $\\nu$. We show for that purpose that all solutions are renormalised solutions, in the sense of Di~Perna--Lions, which classically leads to uniqueness and stability results in that setting. \n\n\\medskip\n\n\\subsection{A priori estimates and existence result}\n\\label{SubsectionAPrioriEstimatesLInfty}\n\nFor developping that $\\LL^\\infty$ theory, we shall be working in the scale of Sobolev spaces \n$$\nE_n = W^{n,1}(\\RR^d),\\quad\\textrm{for } n\\geq 0,\n$$\nwith norm denoted by $|\\cdot|_n$, and in which one has regularising operators $\\big(J^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ for which estimates \\eqref{EqApproximationProperties} hold. Our minimal regularity assumptions on the vector fields will be the existence of a positive constant $C_1$ such that the inequalities\n\\begin{equation}\n\\label{EqContinuityConditionsVStar}\n\\big|V_i^*\\varphi\\big|_0 \\leq C_1 |\\varphi|_1, \\quad\\quad \\big|V_i^*V_j^*\\varphi\\big|_0 \\leq C_1 |\\varphi|_2\n\\end{equation}\nhold for all $1\\leq i,j\\leq\\ell$. These conditions hold for instance if the vector fields $V_i$ and $(V_iV_j)$ are all $\\mcC^1_b$; we write here $(V_iV_j)$ for $\\big(DV_j\\big)(V_i)$. One proves the following existence result by proceeding exactly as in the proof of Theorem~\\ref{ThmTransportL2}, using the a priori $\\LL^\\infty$-estimate\n$$\n\\big| f_t^\\epsilon \\big|_{\\LL^\\infty} = \\big|f_0^\\epsilon\\big|_{\\LL^\\infty},\n$$\nfor the regularised equation, and using Theorem~\\ref{ThmRegularityGain} to get an $\\epsilon$-uniform control on $\\big|f^{\\epsilon,\\sharp}\\big|_{3\\gamma\\,;\\,-3}$, in terms of $\\bfX$ and $\\big|f_0^\\epsilon\\big|_{\\LL^\\infty}$ only. It holds in particular if $V$ is $\\mcC^2_b$.\n\n\\smallskip \n \n\\begin{thm}[Existence for rough transport equations -- $\\LL^\\infty$ setting]\n\\label{ThmExistenceTransportEquationLInfty}\nUnder the continuity assumptions \\eqref{EqContinuityConditionsVStar} on the vector fields $V_i$, for any $f_0\\in\\LL^\\infty(\\RR^d)$, there exists an $\\LL^\\infty(\\RR^d)$-valued path $(f_t)_{t\\geq 0}$, started from $f_0$, satisfying the equation\n$$\n\\delta f_{ts}(\\varphi) = f_s\\big(V^*\\varphi\\big)\\,X_{ts} + f_s\\big(V^*V^*\\varphi\\big)\\,\\bbX_{ts} + f^\\sharp_{ts}(\\varphi)\n$$\nfor all $\\varphi\\in E_3$, and the bound\n$$ \n\\sup_{t\\geq 0} \\big| f_t\\big|_{\\LL^\\infty(\\RR^d)}\\leq \\big| f_0\\big|_{\\LL^\\infty(\\RR^d)},\n$$ \nwith a remainder $f^\\sharp(\\varphi)$ controlled by \n\\begin{equation}\n\\label{EqBoundRemainder-bis}\n\\big|f^\\sharp_{ts}(\\varphi) \\big| \\lesssim_{C_1,{\\bfX}, T, |f_0|_{\\LL^\\infty}} \\, |\\varphi|_3\\,|t-s|^{3\\gamma},\n\\end{equation}\nfor $0\\leq s\\leq t\\leq T$. \n\\end{thm} \n \n\n\\bigskip\n\n\n\\subsection{Renormalised solutions, uniqueness and stability}\n\\label{SubsectionrenormalisedSolutions}\n\nTo proceed one step further, we show that a mild strengthening of the regularity conditions imposed on the vector fields $V_i$ suffices to guarantee that all bounded solutions to the transport equation \\eqref{EqTransportEq} are actually renormalised solution, in the sense of the following definition. \n\n\\begin{defn}\n\\label{DefnrenormalisedSolutions}\nA solution $f_\\bullet$ to the transport equation \\eqref{EqTransportEq} in the scales $(E_n)_{n\\geq 0}$ is said to be a \\emph{\\bf renormalised solution} if for any function $H : \\RR\\rightarrow\\RR$, of class $\\mcC^3_b$, the path $h_\\bullet = H\\circ f_\\bullet$ is also a solution to equation \\eqref{EqTransportEq} in the same scale $(E_n)_{n\\geq 0}$.\n\\end{defn}\n\nAs expected, this property will lead below to uniqueness and stability results.\n\n\\begin{thm}\n\\label{ThmRenomalizedSolutions}\nAssume the vector fields $V_i$ are $\\mcC^3_b$. Then every solution to the transport equation \\eqref{EqTransportEq}, bounded in $ \\LL^\\infty(\\RR^d)$, is a renormalised solution.\n\\end{thm}\n \n\\medskip\n \n\\begin{Dem}  \nThe renormalisation Lemma~\\ref{Lemmarenormalisation} can be stated in the $\\LL^\\infty$ setting by chosing a slightly different scale $(\\mcF^\\nabla_n)_n$ of spaces of test functions,  with norms modelled on $\\LL^1$ \n\\begin{equation}\n\\label{EqNormNablaLInfty} \n\\big| \\varphi \\big|^\\nabla_n := \\sup_{0 \\leqslant k + \\ell \\leqslant n} \\int \\int \\Big|\\big(\\nabla^+\\big)^k \\big(\\nabla^-\\big)^\\ell \\varphi (x, y) \\Big|\\,dx dy  \n\\end{equation}\nrather than on an $\\LL^\\infty$ space used in Section~\\ref{SubsectionTensorization}. Identity \\eqref{EqPrimitiveGronwall} holds in that case with for functions $\\phi\\in W^{3,1}(\\RR^d)$, with an $O(\\cdot)$ term involving the $W^{3,1}$-norm of $\\phi$ rather than its $W^{3,\\infty}$-norm, as the proof of Lemma~\\ref{Lemmarenormalisation} works verbatim, provided we can prove that $\\Gamma_{\\bfA}$ is an unbounded rough driver in the scale of spaces $\\big(\\mcF_n^\\nabla\\big)_{n\\geq 0}$ associated with the norm \\eqref{EqNormNablaLInfty}.  (Note that we have in that case $\\big| f_s^{\\otimes 2}(T_\\varepsilon \\Phi)\\big| \\leq |\\Phi|_0^\\nabla \\big| f_s\\big|_{\\LL^\\infty}^2 = |\\Phi|_0^\\nabla \\big| f_s\\big|_{-0}^2$.)\n\n\\medskip\n\nThe proof that $\\Gamma_{\\bfA}$ is a unbounded rough driver renormalizable in the scale $(\\mcF^\\nabla_n)_n$ follows the same pattern as the proof given in Section~\\ref{SubsectionL2Transport}. We invite the reader to complete the details.\n\n\\smallskip\n\nSo it follows from the renormalisation lemma that if $f, g$ are two solutions the above argument also goes through and shows that $fg$ is also a solution, so any power $f^n$ of $f$ is also a solution, with a size of the remainder that depends only on the $\\LL^{\\infty}$ norm of $f^n$. By linearity the result can be extended to any polynomial of $f$, and by density to any continuous function $H (f)$, with $H$ defined on the interval $\\big[- \\| f \\|_{\\infty}, \\| f \\|_{\\infty}\\big]$. \n\\end{Dem}\n \n\\medskip\n\nWe can actually improve slightly this condition and require only a weak integrability for the third derivative of $V$; it provides a significant strengthening of the previous statement when the vector fields $V_i$ are divergence-free, giving some analogue of the traditional di~Perna--Lions conditions in the classical setting. Note that we do not have uniqueness of solutions for the associated rough differential equation under the conditions below.\n\n\\begin{thm}\n\\label{ThmRenomalizedSolutionsWeak}\nAssume that $V \\in \\mcC^2_b$, $\\nabla^3 V \\in L^1$ and $\\mathrm{div} V \\in \\mcC^2_b$. Then every solution to the transport equation \\eqref{EqTransportEq}, bounded in $ \\LL^\\infty(\\RR^d)$, is a renormalised solution.\n\\end{thm}\n \n\\medskip\n \n\\begin{Dem}  \nIn the proof of the renormalisation Lemma~\\ref{Lemmarenormalisation} we can use directly the general a priori estimate stated in Theorem~\\ref{ThmGeneralRegularityGain} applied to $\\Gamma_{\\bfA}$ with  $F=\\widetilde \\mathcal{E}^\\nabla_3$ and $E= \\mcF^\\nabla_0$ -- note the choice of function space for $F$. Here $\\widetilde \\mathcal{E}^\\nabla_n$ are spaces of test functions,  with norms modelled on $\\LL^\\infty$ like  $\\mathcal{E}^\\nabla_n$ but with a small change given by an additional averaging over the auxiliary variable $\\tau$ and a weight:\n\\begin{equation*}\n\\big| \\varphi \\big|^\\nabla_n := \\sup_{0 \\leqslant k + \\ell \\leqslant n} \\int  \\sup_{x\\in\\RR^d} \\Big[\\int_0^1 d\\tau \\Big|\\big(\\nabla^+\\big)^k \\big(\\nabla^-\\big)^\\ell \\varphi (x-\\tau w, x+(1-\\tau)w) \\Big|\\Big]\\, (1+|w|) dw \n\\end{equation*}\nthe reason of which will be clear below.\nIn this case we can  show that\n$$\nN_1(\\Gamma_{\\bfA,\\varepsilon}) \\lesssim (1+ |V|_{\\mcC^2_b})^2\n$$\nwhile\n$$\nN_2(\\Gamma_{\\bfA,\\varepsilon}) \\lesssim (1+|V|_{\\mcC^2_b}+|\\nabla^3 V|_{\\LL^1}+|\\mathrm{div} V|_{\\mcC^2_b})^3.\n$$\nIndeed apart from many contributions which can be estimated as in the $\\LL^2$ or as in the previous theorem, a difficult term come form the estimation of norms like $|\\Gamma_{V,\\varepsilon}^\\ast \\Gamma_{V,\\varepsilon}^\\ast  \\Gamma_{V,\\varepsilon}^\\ast  |_{\\textrm{L}(\\textrm{F},\\textrm{E})}$ of which the most singular contribution is given by $|\\Gamma_{V,\\varepsilon} \\Gamma_{V,\\varepsilon}  \\Gamma_{V,\\varepsilon}  |_{\\textrm{L}(\\textrm{F},\\textrm{E})}$. In this norm the contribution that requires more regularity to $V$ is due to  the first two vector fields $\\Gamma_{V,\\varepsilon}$ acting simultaneously on the third one giving terms of the form $\\varepsilon^{-1}|v_\\varepsilon^+ v_\\varepsilon^+ (\\nabla^2 v)^-_\\varepsilon \\nabla^- |_{\\textrm{L}(\\textrm{F},\\textrm{E})}$ and easier ones. Now expanding $(\\nabla^2 v_\\varepsilon)^-(x,y)= \\varepsilon \\int_0^1 d\\tau (\\nabla^3v)(x+\\varepsilon \\tau(y-x)) (y-x)$ we get\n$$\n\\varepsilon^{-1}|v^+_\\varepsilon v^+_\\varepsilon (\\nabla^2 v)^-_\\varepsilon \\nabla^- \\Psi|_E =\\varepsilon^{-1}  \\int \\int \\Big|(v^+_\\varepsilon v^+_\\varepsilon (\\nabla^2 v)^-_\\varepsilon  \\Psi) (x, y) \\Big| dx dy  $$\n$$\n \\lesssim  |V|_{\\LL^\\infty}^2 \\int_0^1 d\\tau \\int \\int |\\nabla^3 v(x+\\varepsilon \\tau w)| |\\Psi(x,x+w)| |w| dx dw \n $$\n$$\n \\lesssim  |V|_{\\LL^\\infty}^2 \\int_0^1 d\\tau \\int dx |\\nabla^3 v(x)| \\int dw  |\\Psi(x-\\tau w,x-\\tau w+w)| |w|\n $$\n$$\n \\lesssim  |V|_{\\LL^\\infty}^2 |\\nabla^3 V|_{\\LL^1} \\sup_{x\\in\\RR^d} \\int_0^1 d\\tau \\int dw  |\\Psi(x-\\tau w,x-\\tau w+w)| (1+|w|)\n $$\n$$\n \\lesssim  |V|_{\\LL^\\infty}^2 |\\nabla^3 V|_{\\LL^1}  |\\Phi|_{F}.\n $$\nGranted the bounds on $N_1(\\Gamma_{\\bfA})$ and $N_2(\\Gamma_{\\bfA})$ the proof continues as the proof of the previous theorem and gives the renormalisation result.\n\\end{Dem}\n \n\\medskip\n\nAs expected, Theorem~\\ref{ThmRenomalizedSolutions} on renormalised solutions to the transport equation \\eqref{EqTransportEq} comes with a number of important consequences, amongst which is an equivalent of the missing Gronwall lemma, as given by the a priori estimate \\eqref{EqGronwall} below.\n\n\\begin{thm}\n\\label{ThmUniqueness} \nAssume the vector fields $V_i$ are $\\mcC^3_b$. \\vspace{0.1cm}\n\\begin{enumerate}\n   \\item {\\bf Uniqueness --} Given an initial condition in $\\LL^\\infty(\\RR^d)$, there exists a unique solution to the transport equation which remains bounded in $\\LL^\\infty(\\RR^d)$.  \\vspace{0.2cm}\n   \\item {\\bf Stability --} Let the time horizon $T$ be finite. Let $\\Big(V^{(n)}_i\\Big)_{n\\geq 0}$, $i=1..\\ell$ and $\\Big(f^{(n)}_0\\Big)_{n\\geq 0}$ be a sequence of approximating sequences with $V^{(n)}_i$ converging to $V_i$ in $\\mcC^3_b$, and $f^{(n)}_0$ converging to $f_0$ in $\\LL(\\RR^d)$. Let also $\\Big({\\bfX}^{(n)}\\Big)_{n\\geq 0}$ be a sequence of weak geometric $\\gamma$-rough paths above smooth paths, that converge in a rough paths sense to $\\bfX$, over the time interval $[0,T]$. Then the solution paths $f^{(n)}_\\bullet$ to the transport equation associated with ${\\bfX}^{(n)}, V^{(n)}$ and $f^{(n)}_0$, converge weakly-$\\star$ in $\\LL^\\infty\\big([0,T],\\LL^\\infty(\\RR^d)\\big)$, and in $\\LL^1_\\textrm{\\emph{loc}}\\big([0,T],\\LL^\\infty(\\RR^d)\\big)$, to $f_\\bullet$.\n\\end{enumerate}\n\\end{thm}\n\n\\medskip\n\n\\begin{Dem}\n{\\bf Uniqueness --} We follow the same pattern of proof as that of Theorem~\\ref{ThmRDEHilbertGeneralClosed}. Let $f_\\bullet$ and $f'_\\bullet$ be two solution paths to equation \\eqref{EqTransportEq}, bounded in $\\LL^\\infty(\\RR^d)$, and started from the same initial condition. Let $H : \\RR\\rightarrow\\RR$ be a non-negative function, of class $\\mcC^3_b$, null at $0$ and positive elsewhere. Define the path \n$$\nh_\\bullet = H\\big(f_\\bullet - f'_\\bullet\\big);\n$$\nit is also a positive solution to the transport equation \\eqref{EqTransportEq} under the above regularity assumptions on the vector fields $V_i$, since all solutions are renormalised solution, by Theorem~\\ref{ThmRenomalizedSolutions}. Set $\\psi(x) = \\big(1+|x|^2\\big)^{-k_0}$, for $x\\in\\RR^d$, and some exponent $k_0>d$. That function satisfies \n\\begin{equation*}\n\\begin{split}\n&\\textrm{div}\\big(\\psi V\\big) = -V\\psi - (\\textrm{div}V)\\psi, \\\\\n&\\textrm{div}\\Big(\\textrm{div}\\big(\\psi V\\big)\\,V\\Big) = V^2\\psi + (\\textrm{div}V)V\\psi + \\big((V\\textrm{div}V)+(\\textrm{div}V)^2\\big)\\psi  \n\\end{split}\n\\end{equation*}\nwith \n\\begin{equation*}\n\\Big|V\\psi\\Big| \\,\\vee\\, \\Big|V^2\\psi + (\\textrm{div}V)V\\psi\\Big| \\lesssim \\psi\n\\end{equation*}\nas a consequence of the $\\mcC^1_b$ character of the vector fields $V_i$. \nDefine the scale of spaces \n$$\nE_n^\\psi := \\big\\{\\varphi = \\psi \\phi\\,;\\,\\phi \\in \\LL^\\infty\\big\\} \n$$\nwith norm \n$$\n|\\varphi|_{E_n^\\psi} := |\\phi|_{W^{n,\\infty}}\n$$ \nIt is not difficult to check that $(V X, VV \\mathbb{X})$ is a $\\gamma$-rough driver also in this scale of spaces. In this case however we have\n$$\n\\big|h_t(\\varphi)\\big| = \\big|h_t ( \\psi \\phi)\\big| \\leq |\\phi|_{\\LL^\\infty} \\big|h_t( \\psi)\\big| = |\\varphi|_{E^\\psi_0}\\big|z_t\\big|\n$$\nwhere \n$$\nz_t := h_t(\\psi), \n$$\nand so\n$$\n\\llparenthesis h_\\bullet\\rrparenthesis_{(E_0^\\psi)^*}\\leq \\llparenthesis z_\\bullet \\rrparenthesis\n$$\nBy the general a priori estimates we have that there exists $\\lambda$ and constants depending on $A$ such that\n$$\n\\llparenthesis h^\\sharp \\rrparenthesis_{3\\gamma\\,;\\,(E_3^\\psi)^*} \\lesssim \\llparenthesis z_\\bullet \\rrparenthesis\n$$\nBut now\n$$\nz_t = z_0 +  h_{0}(V^* \\psi)X_{0,t}+  h_{0}(V^*V^* \\psi)\\mathbb{X}_{0,t} + h^\\sharp_{0,t}(\\psi)\n$$\nso\n$$\n\\llparenthesis z_\\bullet \\rrparenthesis \\leq |z_0| \\big(1+ \\llparenthesis X_{0, \\bullet} \\rrparenthesis+ \\llparenthesis \\mathbb{X}_{0,\\bullet} \\rrparenthesis\\big) + \\llparenthesis h^\\sharp_{0,\\bullet} \\rrparenthesis_{(E_3^\\psi)^*}\n$$\nand since $  \\llparenthesis h^\\sharp_{\\bullet 0} \\rrparenthesis_{(E_3^\\psi)^*} \\leq \\lambda^{3\\gamma} \\llparenthesis h^\\sharp \\rrparenthesis_{3\\gamma\\,;\\,(E_3^\\psi)^*} \\leq \\lambda^{3\\gamma}  \\llparenthesis z_\\bullet \\rrparenthesis$, with $h^\\sharp$ considered as a 2-index map in its second occurence, we have for $\\lambda$ small enough\n\\begin{equation}\n\\label{EqGronwall}\n\\llparenthesis z_\\bullet \\rrparenthesis \\leq 2 |z_0| \\big(1+ \\llparenthesis X_{0, \\bullet} \\rrparenthesis+ \\llparenthesis \\mathbb{X}_{0, \\bullet} \\rrparenthesis\\big);\n\\end{equation}\nso $z_t = 0$, for all $t\\ge 0$, if $z_0 = 0$.\n\n\\bigskip\n  \n{\\bf Stability --} Denote by ${\\bfX}^{(n)}$ a smooth rough path converging to $\\bfX$ in the rough paths metric, and by $V^{(n)}_i$ a sequence of vector fields converging to $V_i$ in $\\mcC^3_b$. Let $f^{(n)}_0$ be a smooth function converging to $f_0$ in $(\\LL^1)^*$. One solves the transport equation associated with ${\\bfX}^{(n)}$ and $f^{(n)}_0$, using the elementary method of characteristics as the vector fields $V^{(n)}_i$ are sufficiently regular. It is elementary to use the uniform bound\n$$\n\\left|f^{(n)}_t\\right|_{\\LL^\\infty(\\RR^d)} \\leq \\left|f^{(n)}_0\\right|_{\\LL^\\infty(\\RR^d)} \\leq C<\\infty,\n$$ \nand the uniform a priori bound on $\\big|f^{(n),\\sharp}\\big|_{3\\gamma\\,;\\,-3}$ provided by Theorem~\\ref{ThmRegularityGain} and the convergence of ${\\bfX}^{(n)}$ to $\\bfX$, and $f^{(n)}_0$ to $f_0$, to get the existence of a subsequence of $f^{(n)}_\\bullet$ converging weakly-$\\star$ in $\\LL^\\infty\\big([0,T];\\LL^\\infty(\\RR^d)\\big)$ to some solution of the transport equation \\eqref{EqTransportEq}, bounded in $\\LL^\\infty(\\RR^d)$. Since this solution is unique, as proved above, the whole sequence $f^{(n)}_\\bullet$ converges weakly-$\\star$ to $f_\\bullet$ in $\\LL^\\infty\\big([0,T];\\LL^\\infty(\\RR^d)\\big)$. As the same conclusion holds for $\\Big(f^{(n)}_\\bullet\\Big)^2$ and $f^2$, by the renormalisation property, we classically get the convergence in $\\LL^1_\\textrm{loc}\\big([0,T];\\LL^\\infty(\\RR^d)\\big)$.\n\\end{Dem}\n\n\\bigskip\n\n\\begin{rem}\nIt may be tempting, in the light of the results exposed in Section~\\ref{SubsectionIntegrationConservative}, to try and develop an $\\LL^\\infty$ theory of differential equations driven by more general rough drivers ${\\bfA}_{ts}$ than those associated with the data of some vector fields $V_1,\\dots, V_\\ell$ and a weak geometric H\\\"older rough path over $\\RR^\\ell$, as in the transport equation \\eqref{EqTransportEq}. With a view towards the classical theory of stochastic flows, as developed by Le Jan-Watanabe, Kunita and many others, one may try, as a first step, to work with rough drivers whose first level are obtained as typical trajectories of semimartingale velocity fields. It is shown in \\cite{BailleulRiedel} that such random fields can be lifted into some objects very similar to rough drivers, under some mild regularity conditions on the semimartingale, and that the use of the approximate flow machinery introduced in \\cite{BailleulRMI} leads to some well--posedness result for some dual evolution equation \n$$\ndf_t(\\varphi) = f_t\\big({\\bf A}(dt)\\varphi\\big) .   \n$$ \n\\end{rem}\n\n\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOur aim in this paper is to take a fresh look at the logics of informational dependence and independence \\cite{HS89,HS96,Vaan}, and their compositional semantics due to Wilfrid Hodges \\cite{Hod97a,Hod97b}. We shall focus on Dependence Logic, introduced by the second author \\cite{Vaan}.\n\nThe main objective of Hodges' work was to provide a compositional model-theoretic semantics for the IF-logic of Hintikka and Sandu \\cite{HS89,HS96}, which matched their ``game-theoretical semantics''. This was achieved by lifting the standard Tarski semantics of first-order formulas, given  in terms of satisfaction in a structure with respect to an assignment to the free variables, to satisfaction by \\emph{sets of  assignments}.\n\nWe seek a deeper understanding of Hodges' construction:\n\\begin{itemize}\n\\item First and foremost, what is going on? Where does the Hodges construction come from? Is it canonical in any way? Why does it work? What structures are really at play here?\n\n\\item Because of the equivalence of Dependence Logic (or variants such as IF-logic) under this semantics to  (a significant fragment of) second-order logic, there is no hope for a completeness theorem. But we may get a useful completeness theorem with respect to a wider class of models. Understanding the general algebraic context for the semantics points the way to such a completeness notion.\n\n\\item We can  also look for \\emph{representation theorems}, with some infinitary ingredients.\n\\end{itemize}\nThe results of our investigation are quite surprising conceptually (at least to us). The main points can be summarized as follows.\n\\begin{itemize}\n\\item We find a general context for Hodges' construction. We shall not treat it in full generality here, as the general account is best stated in the language of categorical logic \\cite{Law69,Pitts}, and we wish to avoid undue technicalities. However, we will indicate the possibilities for a general algebraic semantics, as the basis for a useful completeness theorem.\n\\item We find that the natural propositional logic associated with the Hodges construction is the \\emph{logic of Bunched Implication} of Pym and O'Hearn \\cite{OHP99,Pym02}, which combines intuitionistic and multiplicative linear connectives. \n\\item This not only yields a more natural view of the strangely asymmetric notions of conjunction and disjunction in the Hodges semantics (one is intuitionistic, while ``disjunction'' is actually multiplicative \\emph{conjunction}!),  it also brings into prominence some connectives not previously considered in the setting of IF-logic or Dependence logic, in particular \\emph{intuitionistic implication}. This enables a novel analysis of the Dependence predicate of \\cite{Vaan}, as a Horn clause with respect to a more primitive predicate of single-valuedness. The well-known Armstrong axioms for functional dependence \\cite{Arm} then fall out as a standard axiomatization of intuitionistic (but not classical!) implication.\n\\item Intuitionistic implication also plays a natural r\\^ole in our version of a full abstraction theorem in the sense of Hodges.\n\\item The construction is shown to lift the interpretation of the standard quantifiers in a canonical way, so that \\emph{quantifiers are uniquely determined as the adjoints to substitution} \\cite{Law69}, just as in the standard Tarski semantics of first-order logic. This is also extended to characterizations of the dependence-friendly quantifiers of \\cite{Vaan} as adjoints.\n\\end{itemize}\n\n\\noindent The plan of the remainder of the paper is as follows. In the next section we provide background on branching quantifiers, IF-logic, dependence logic, and Hodges'  semantics. Then  in section~3 we show how the Hodges semantics is an instance of a general algebraic construction, in which the connectives of BI-logic arise naturally. In section~4, we show that the interpretation of the quantifiers in the Hodges construction is the canonical lift of the standard  interpretation of the quantifiers as adjoints, and hence is uniquely determined.  We also use the intuitionistic implication to show how the dependence-friendly quantifiers can be interpreted as certain adjoints. In section~5, we  show how the intuitionistic implication arises naturally in the proof of a full abstraction theorem. In section~6, we show how the dependence predicate can be analyzed in terms of a more primitive predicate of single-valuedness, using the intuitionistic implication. This turns the ``Armstrong axioms'' into standard theorems of intuitionistic implicational logic. The final section outlines some further directions.\n\n\\section{Dependence, Independence and Information Flow}\nWe begin with a standard example: the formal definition of continuity for a function $f : \\mathbb{R} \\longrightarrow \\mathbb{R}$ on the real numbers.\n\\[ \\forall x. \\, \\forall \\epsilon. \\, \\exists \\delta . \\, \\forall x'. \\, | x - x'| < \\delta \\; \\Rightarrow \\; | f(x) - f(x') | < \\epsilon \\, . \\]\nThis definition is often explained in current calculus courses in terms of an ``epsilon-delta game''.\\endnote{See e.g. online resources such as\\\\ \\texttt{http:\/\/library.wolfram.com\/infocenter\/MathSource\/4734\/}.} The Adversary proposes a number, $\\epsilon$, as a measure of how close we must stay to the value of $f(x)$; we must then respond with a number, $\\delta$, such that, whenever the input is within the interval $(x - \\delta, x+\\delta)$, the output does indeed pass the $\\epsilon$-test of closeness to $f(x)$. Clearly, the choice of $\\delta$ will depend on that of $\\epsilon$; the nesting of the quantifiers expresses this dependency.\n\nThis is the definition of \\emph{global continuity} of $f$, expressed in terms of \\emph{local continuity} at every point $x$. This means that the choice of $\\delta$ will depend, not only on $\\epsilon$, but on $x$ also. Now consider the definition of \\emph{uniform continuity}:\n\\[ \\forall \\epsilon. \\, \\exists \\delta . \\, \\forall x. \\, \\forall x'. \\, | x - x'| < \\delta \\; \\Rightarrow \\; | f(x) - f(x') | < \\epsilon \\, . \\]\nHere $\\delta$ still depends on $\\epsilon$, \\emph{but must be chosen independently of $x$}. This variation in dependency is tracked syntactically by the different order of the quantifiers.  Indeed,  it seems that it was only after the distinction between pointwise and uniform notions of continuity, and, especially, convergence, had been clarified in 19th-century analysis, that the ground was prepared for the introduction of predicate calculus.\n\nMore generally, dependence or independence of bounds on various parameters is an important issue in many results on estimates in number theory and analysis.\nHodges quotes a nice example from one of Lang's books \\cite{Lan64} in \\cite{Hod97a}.\n\nIntuitively, there is an evident relation between these notions and that of \\emph{information flow}. Dependence indicates a form of  information flow; independence is  the \\emph{absence} of information flow.\n\n\\subsection{Beyond first-order logic}\nIt turns out that mere rearrangement of the order of quantifiers in first-order formulas is not sufficient to capture the full range of possibilities for informational dependence and independence. This was first realized almost 50~years ago, with Henkin's introduction of  \\emph{branching quantifiers} \\cite{Hen61}. The simplest case is the eponymous \\emph{Henkin quantifier}:\n\\[ \\left( \\begin{array}{cc} \\forall x & \\exists y \\\\\n\\forall u & \\exists v\n\\end{array}   \\right) A(x, y, u, v) .\n\\]\nThe intention is that $y$ must be chosen depending on $x$, but \\emph{independently} of the choice of $u$; while $v$ must be chosen depending on $u$, but \\emph{independently} of the choice of $x$.\nThe meaning of this formula can be explicated by introducing \\emph{Skolem functions} $f$ and $g$: an equivalent formula will be\n\\[ \\exists f. \\, \\exists g. \\, \\forall x. \\, \\forall u. \\, A(x, f(x), u, g(u)) . \\]\nHere the constraints on dependencies are tracked by the dependence of the Skolem functions on certain variables, but not on others.\nNote that the Skolemized sentence is \\emph{second-order}; in fact, it belongs to the $\\Sigma_{1}^{1}$ fragment of second-order logic.\\endnote{This can be described as the fragment comprising formulas $\\exists f_{1} \\ldots \\exists f_{n}. \\, \\phi$,  where the $f_{i}$ are function variables, and $\\phi$ is a first-order formula over a signature extended by these function variables.} This second-order rendition of the meaning of the Henkin quantifier cannot be avoided, in the sense that  the Henkin quantifier strictly increases the expressive power of first-order logic, and in fact the extension of first-order logic with the Henkin quantifier is equivalent in expressive power to  the $\\Sigma_{1}^{1}$ fragment \\cite{Hen61}.\n\n\\paragraph{Examples}\n\\begin{enumerate}\n\\item Consider\n\\[ \\left( \\begin{array}{cc} \\forall x & \\exists y \\\\\n\\forall u & \\exists v\n\\end{array} \\right)  (A(x) \\rightarrow B(y)) \\; \\wedge \\; (B(u) \\rightarrow A(v)) \\; \\wedge \\; [(x = v) \\; \\leftrightarrow \\;  (y = u)].\n\\]\nThis expresses that $A$ and $B$ are equinumerous sets.\n\\item Now consider\n\\[   \\begin{array}{ll}\n\\exists v. \\, \\left( \\begin{array}{cc} \\forall x_{1} & \\exists y_{1} \\\\\n\\forall x_{2} & \\exists y_{2}\n\\end{array} \\right) (A(x_{1}) \\rightarrow A(y_{1}))   & \\wedge \\; [(x_{2} = y_{1}) \\; \\rightarrow \\;  (y_{2} = x_{1})] \\\\    & \\wedge \\; A(v)  \\; \\wedge \\;  (A(x_{1}) \\rightarrow (y_{1} \\neq v)) \\, .\n\\end{array}\n\\]\nThis expresses that $A$ is an infinite set.\n\\end{enumerate}\nThese examples show that the Henkin quantifier \\emph{is not expressible in first-order logic}.\n\n\\subsection{Further developments}\nThe next major development was the introduction of IF-logic (``inde\\-pendence-friendly logic'') by Jaakko Hintikka and Gabriel Sandu \\cite{HS89}.\nThe intention of IF-logic is to highlight informational dependence and independence. It provides a linear syntax for expressing branching quantification (and more), e.g.  the Henkin quantifier can be written in linear notation as:\n\\[ \\forall x. \\, \\exists y. \\, \\forall u. \\, (\\exists v\/x). \\, A(x, y, u, v) \\]\nThe ``slashed quantifier''  $(\\exists v\/x)$  has the intended reading ``there exists a $v$ \\emph{not depending on $x$}''. Note the strange syntactic form of this quantifier, with its ``outward-reaching'' scope for $x$.\n\n\\paragraph{Dependence Logic}\nA simplified approach was introduced by the second author, and developed extensively in the recent monograph \\cite{Vaan}.\nThe main novelty in the formulation of the logic is to use an atomic \\emph{dependence predicate\\endnote{In \\cite{Vaan}\nthe notation $=\\!\\!(x_{1}, \\ldots , x_{n}, x)$ for $D(x_{1}, \\ldots , x_{n}, x)$ is used.}} $D(x_{1}, \\ldots , x_{n}, x)$ which holds if \\emph{$x$ depends on $x_{1}, \\ldots , x_{n}$, and  only on these variables}.\nWe can then define ``dependence-friendly quantifiers'' as standard quantifiers guarded with the dependence predicate:\n\\[ (\\exists x \\setminus x_{1}, \\ldots , x_{n}). \\, \\phi \\;\\; \\equiv \\;\\; \\exists x. (D(x_{1}, \\ldots , x_{n}, x) \\; \\wedge \\; \\phi) \\, . \\]\nThis yields essentially the same expressive power as IF-logic.\n\n\\subsection{Compositionality: Hodges' Semantics}\n\nBut, what does it all mean?\nHintikka claimed that \\emph{a compositional semantics for IF logic could not be given} \\cite{Hin98}.\nInstead he gave a ``Game-Theoretical Semantics'', essentially reduction to Skolem form as above.\n\n\nWilfrid Hodges showed that it could \\cite{Hod97a,Hod97b}.\\endnote{Hintikka has apparently not conceded the point \\cite{Hin02}, although there is no argument as to the mathematical content of Hodges' results. As far as we are concerned, Hodges' semantics meets all the criteria for a compositional semantics, and is moreover fully abstract. Our concern here is to understand it better, as an interesting construction in its own right.}\n\nBefore giving Hodges' construction, it will be useful firstly to recall  Tarski's solution to the problem of how to define the truth of a sentence in a first-order structure $\\mathcal{M} = (A, \\ldots)$ with underlying set $A$.\\endnote{The classic reference is \\cite{Tar36}, but in fact the modern model-theoretic definition first appeared in \\cite{TV56}, as pointed out in Wilfrid Hodges' article on ``Tarski's Truth Definitions'' in the Stanford Encyclopedia of Philosophy, available online at \\texttt{http:\/\/plato.stanford.edu\/entries\/tarski-truth\/}, which gives an informative overview.} In order to do this, he had to deal with the more general case of open formulas. The idea was to define\n\\[ \\mathcal{M}, s \\models_{X} \\phi \\]\nwhere $X$ is a finite set of variables including those occurring free  in $\\phi$, and $s$ is an assignment of elements of $A$ to $X$.\\endnote{Explictly, an assignment is simply a function $s : X \\rightarrow A$. We write $A^{X}$ for the set of all such assignments. Older tradition was to define satisfaction relative to assignments to \\emph{all} variables, which were typically arrayed in infinite sequences. More recently, it has been understood, under the influence of categorical logic,  that to reveal the salient structure one should give the definition relative to a finite environment that  grows as quantifiers are stripped off in the recursive definition.}\nTypical clauses include:\n\\[ \\begin{array}{lcl}\n\\mathcal{M}, s \\models_{X} \\phi \\wedge \\psi & \\equiv & \\mathcal{M}, s \\models_{X} \\phi \\;\\; \\mbox{and} \\;\\; \\mathcal{M}, s \\models_{X} \\psi \\\\\n\\mathcal{M}, s \\models_{X} \\neg \\phi & \\equiv & \\mathcal{M}, s \\not\\models_{X} \\phi \\\\\n\\mathcal{M}, s \\models_{X} \\forall v. \\, \\phi & \\equiv & \\forall a \\in A. \\; \\mathcal{M}, s[v \\mapsto a] \\models_{X \\cup \\{v\\}} \\phi \\\\\n\\mathcal{M}, s \\models_{X} \\exists v. \\, \\phi & \\equiv & \\exists a \\in A. \\; \\mathcal{M}, s[v \\mapsto a] \\models_{X \\cup \\{v\\}} \\phi \\\\\n\\end{array}\n\\]\nHere $s[v\\mapsto a]$ is the assignment defined on $X \\cup \\{v\\}$ as follows: $s[v \\mapsto a](v) = a$, and $s[v \\mapsto a](w) = s(w)$ for $w \\neq v$.\n\nThe is the very prototype of a compositional semantic definition.\nVia Dana Scott, this idea led to the  use of \\emph{environments} in denotational semantics \\cite{Sco69}. Environments are nowadays ubiquitous in all forms of semantics in computer science \\cite{Win93,Mit96}.\n\n\\paragraph{Teams}\nHodges' key idea was to see that one must lift the semantics of formulas from single assignments to \\emph{sets of assignments}. Notions of dependence of one variable on others are only meaningful among a set of assignments. Hodges called these sets ``trumps''; we follow \\cite{Vaan} in calling them \\emph{teams}.\n\n\nWe consider the semantics of Dependence logic \\cite{Vaan}. Formulas are built up from standard atomic formulas and their negations and  the dependence predicates, by conjunction, disjunction, and universal and existential quantification. We shall distinguish between the usual atomic formulas (including equality statements) over the first-order signature we are working with, and the dependence formulas. In the case of the standard atomic formulas, we shall also allow their negations, and as usual refer to positive and negated atomic formulas collectively as \\emph{literals}. We shall \\emph{not} allow negations of dependence formulas; we will see later how to access negative information about dependence, using the new connectives we will introduce in the next section.\n\nThe set of all individual variables is denoted $\\mathcal{V}$.\nA \\emph{team on $X \\subseteq \\mathcal{V}$} is a set of Tarski assignments on $X$. We define the following operations on teams:\n\n\\begin{itemize}\n\\item If $T$ is a team on $X$ and $v \\in \\mathcal{V}$,  then $T[v \\mapsto A]$ is the team on $X \\cup \\{v\\}$ defined by:\n\n\\vspace{-.1in}\n\\[ T[v \\mapsto A] = \\{ t[v \\mapsto a] \\mid t \\in T \\; \\wedge \\; a \\in A \\} . \\]\n\n\n\\item If $T$ is a team on $X$, $v \\in \\mathcal{V}$, and $f : T \\longrightarrow A$, then $T[v \\mapsto f]$ is the team on $X \\cup \\{v\\}$ defined by:\n\\[ T[v \\mapsto f] = \\{ t[v \\mapsto f(t)] \\mid t \\in T \\} . \\]\n\\end{itemize}\n\n\n\n\\paragraph{The Satisfaction Relation}\n\nWe define a satisfaction relation\n\\[ \\mathcal{M}, T \\models_{X} \\phi \\]\nwhere the free variables of $\\phi$ are contained in $X$, and $T$ is a team on $X$. (In practice, we elide $\\mathcal{M}$).\n\n\nFirstly, for literals $L$  we have:\n\\[ T \\models_{X} L \\;\\; \\equiv \\;\\; \\forall t \\in T. \\, t \\models_{X} L  \\]\nwhere $t \\models_{X} L$ is the standard Tarskian definition of satisfaction of an atomic formula or its negation  in a structure with respect to an assignment.\n\n\n\n\\paragraph{Connectives and Quantifiers}\nThe  clauses for connectives and quantifiers are as follows:\n\n\\[ \\begin{array}{lcl}\nT \\models_{X} \\phi \\wedge \\psi & \\equiv & T \\models_{X} \\phi \\;\\; \\mbox{and} \\;\\; T \\models_{X} \\psi \\\\\nT \\models_{X} \\phi \\vee \\psi & \\equiv & \\exists \\, U, V. \\; ([U \\models_{X} \\phi \\;\\; \\mbox{and} \\;\\; V \\models_{X} \\psi] \\;\\; \\wedge \\;\\; [T = U \\cup V]) \\\\\nT \\models_{X} \\forall v. \\, \\phi & \\equiv & T[v \\mapsto A] \\models_{X \\cup \\{v\\}} \\phi \\\\\nT \\models_{X} \\exists v. \\, \\phi & \\equiv & \\exists f : T \\longrightarrow A. \\; T[v \\mapsto f] \\models_{X \\cup \\{v\\}} \\phi .\n\\end{array}\n\\]\n\n\\paragraph{Semantics of the dependence predicate}\nGiven a set of variables $X$ and $W \\subseteq X$, we define the following notions:\n\n\\begin{itemize}\n\\item An equivalence relation on  assignments on $X$:\n\\[ s \\simeq_{W} t \\;\\; \\equiv \\;\\;  \\forall w \\in W. \\, s(w) = t(w) . \\]\n\n\\item A function $f : A^{X} \\longrightarrow A$ \\emph{depends only on $W$}, written $f : A^{X} \\longrightarrow_{W} A$, if for some $g : A^{W} \\longrightarrow A$, $f = g \\circ p_{XW}$, where $p_{XW} : A^{X} \\longrightarrow A^{W}$ is the evident projection. Note that if such a $g$ exists, it is unique.\n\\end{itemize}\n\n\n\\noindent Now we can define:\n\\[ T \\models_{X} D(W, v) \\;\\; \\equiv \\;\\; \\forall s, t \\in T. \\, s \\simeq_{W} t \\; \\Rightarrow \\; s(v) = t(v) \\]\nNote that this expresses \\emph{functional dependence}, exactly as in database theory \\cite{Arm}.\n\n\n\\noindent An equivalent definition can be given in terms of the dependency condition on functions:\n\\[ T \\models_{X} D(W, v) \\;\\; \\equiv \\;\\; \\exists f : T \\longrightarrow_{W} A. \\, \\forall t \\in T. \\, t(v) = f(t) .  \\]\n\n\n\\noindent Strictly speaking, this is the ``positive part'' of the definition as given in \\cite{Vaan} following Hodges. There is also a negative part, which defines satisfaction for $\\phi$ as for the positive definition, but with respect to the De Morgan dual $\\phi^{d}$ of $\\phi$:\n\\[ (\\phi \\vee \\psi)^{d} = \\phi^{d} \\wedge \\psi^{d}, \\quad (\\exists v. \\, \\phi)^{d} = \\forall v. \\, \\phi^{d}, \\quad \\mbox{etc.} \\]\nThis allows for a ``game-theoretic negation'', which formally ``interchanges the r\\^oles of the players''. It is simpler, and from our perspective loses nothing, to treat this negation as a \\emph{defined operation}, and work exclusively  with formulas in negation normal form as above.\n\n\n\\noindent The theory of dependence logic: metalogical properties, connections with second-order logic, complexity and definability issues,  \\textit{et cetera}, is extensively developed in \\cite{Vaan}. However, as explained in the Introduction, many basic questions remain.\nWe shall now show how the Hodges semantics can be seen in a new light, as arising from a general construction.\n\n\\section{The Hodges construction revisited}\nAn important clue to the general nature of the construction is contained in the observation by Hodges \\cite{Hod97a} (and then in \\cite{Vaan}) that the sets of teams denoted by formulas of  IF-logic or Dependence logic are  \\emph{downwards closed}: that is, if $T \\models  \\phi $ and $S \\subseteq T$, then $S \\models \\phi$.\nThis is immediately suggestive  of well-known constructions on ordered structures.\n\n\\subsection{A general construction}\nWe recall a couple of definitions. A \\emph{commutative ordered monoid} is a structure $(M, {+}, 0, {\\leqslant})$, where $(M, {\\leqslant})$ is a partially ordered set, and $(M, {+}, 0)$ is a commutative monoid (a set with an associative and commutative operation $+$ with unit $0$), such that $+$ is monotone:\n\\[ x \\leqslant x' \\; \\wedge \\; y \\leqslant y' \\;\\; \\Rightarrow \\;\\; x+y \\leqslant x' + y' \\, . \\]\nThe primary example we have in mind is $\\pow{A^{X}}$, the set of all teams on a set of variables $X$, which we think of as the commutative ordered monoid $(\\pow{A^{X}}, {\\cup}, \\varnothing, {\\subseteq})$.\n\nA \\emph{commutative quantale} is a commutative ordered monoid where the partial order is a complete lattice, and $+$ distributes over all suprema: $m + \\bigvee_{i \\in I} m_{i} = \\bigvee_{i \\in I} (m + m_{i})$.\n\nLet $(M, {+}, 0, {\\leqslant})$ be a commutative ordered monoid. Then $\\mathcal{L}(M)$, the set of lower (or downwards-closed) sets of $M$, ordered by inclusion, is the \\emph{free commutative quantale} generated by $M$ \\cite{MS01}.\\endnote{More precisely, it is the left adjoint to the evident forgetful functor.}\n\nA downwards closed subset of a partially ordered set $P$ is a set $S$ such that:\n\\[ x \\leqslant y \\in S \\; \\Rightarrow \\; x \\in S \\, . \\]\nThus this notion generalizes the downwards closure condition on sets of teams.\n \n\\noindent The following notation will be useful.\nGiven $X \\subseteq P$, where $P$ is a partially ordered set, we define\n\\[ {\\downarrow}(X) = \\{ x \\in P \\mid \\exists y \\in X. \\, x \\leqslant y \\} \\, , \\]\nthe \\emph{downwards closure} of $X$. A set $S$ is downwards closed if and only if $S = {\\downarrow}(S)$.\n\nAs a commutative quantale, $\\mathcal{L}(M)$ is a model of intuitionistic linear logic (phase semantics \\cite{Yet,Ros90,Gir87}).\\endnote{It is also an instance of Urquhart's semilattice semantics for relevance logic \\cite{Urq72}. Mitchell and Simmons observe in \\cite{MS01} that in the case (such as ours) where the monoid is a boolean algebra, $\\mathcal{L}(M)$ is actually a model of \\emph{classical linear logic}. This does not seem apposite to our purposes here.}\nIn particular, we have\n\\[ \\begin{array}{rcl}\nA \\otimes B & = &  {\\downarrow}\\{ m + n \\mid m \\in A \\; \\wedge \\; n \\in B \\} \\\\\nA \\multimap B & = & \\{ m \\mid \\forall n. \\, n \\in A \\Rightarrow m + n \\in B \\}\n\\end{array} \\]\nWe note that when the definition of $\\otimes$, the multiplicative \\emph{conjunction}, is specialized to our concrete setting, it yields the definition of \\emph{disjunction} in the Hodges semantics!\n\nThe multiplicative implication $\\multimap$ has not been considered previously in the setting of IF-logic and Dependence logic. However, it is perfectly well defined, and is in fact uniquely specified as the adjoint of the linear conjunction:\n\\[ A \\otimes B \\leqslant C \\;\\; \\Longleftrightarrow \\;\\; A \\leqslant B \\multimap C \\, . \\]\nNote that linear implication automatically preserves downwards closure.\n\n\\subsection{What is the propositional logic of dependence?}\nIn fact, $\\mathcal{L}(M)$ carries a great deal of structure. Not\n\n only is it a commutative quantale (and hence carries an interpretation of linear logic), but it is also a \\emph{complete Heyting algebra}, and hence carries an interpretation of intuitionistic logic.\n\nWe have the clauses\n\\[ \\begin{array}{lcl}\nm \\models A \\wedge B & \\equiv & m \\models A \\; \\mbox{and} \\; m \\models B \\\\\nm \\models A \\vee B & \\equiv & m \\models A \\; \\mbox{or} \\; m \\models B \\\\\nm \\models A \\rightarrow B & \\equiv & \\forall n \\leqslant m. \\, \\mbox{if} \\; n \\models A \\; \\mbox{then} \\; n \\models B\n\\end{array}\n\\]\nThe situation where we have both intuitionistic logic and multiplicative linear logic coexisting is the setting for \\emph{BI logic}, the ``logic of Bunched Implications''  of David Pym and Peter O'Hearn \\cite{OHP99,Pym02}, which forms the basis for \\emph{Separation logic} (Reynolds and O'Hearn) \\cite{Rey02}, an increasingly influential logic for verification.\nThe construction $\\mathcal{L}(M)$ is exactly the way a ``forcing semantics'' for BI-logic is converted into an algebraic semantics as a ``BI-algebra'', \\textit{i.e.}~ a structure which is both a commutative quantale  \\emph{and} a complete Heyting algebra \\cite{POHY04}. $\\mathcal{L}(M)$ is in fact the free construction of a complete BI-algebra over an ordered commutative monoid.\n\nThis provides one reason for proposing BI-logic as the right answer to the question posed at the beginning of this subsection. The compelling further evidence for this claim will come from the natural\nr\\^ole played by the novel connectives we are introducing into the logic of dependence. This r\\^ole will become apparent in the subsequent developments in this paper.\n\n\\subsection{\\textrm{BID}-logic and its team semantics}\nWe shall spell out the extended logical language we are led to consider, and its concrete team semantics, extending the Hodges-style semantics already given in section~2.\n\nWe call the extended language \\textrm{BID}, for want of a better name. Formulas are built from atomic formulas and their negations, and  dependence formulas, by the standard first-order quantifiers, and the following propositional connectives: the intuitionistic (or ``additive'') connectives $\\wedge$, $\\vee$, $\\rightarrow$, and the multiplicative connectives $\\otimes$ and $\\multimap$.\n\n\\paragraph{Team Semantics for BI Logic}\nThe team semantics for \\textrm{BID}-logic is as follows:\n\\[ \\begin{array}{lcl}\nT \\models A \\wedge B & \\equiv & T \\models A \\; \\mbox{and} \\; T \\models B \\\\\nT \\models A \\vee B & \\equiv & T \\models A \\; \\mbox{or} \\; T \\models B \\\\\nT \\models A \\rightarrow B & \\equiv & \\forall U \\subseteq T. \\, \\mbox{if} \\; U \\models A \\; \\mbox{then} \\; U \\models B \\\\\nT \\models A \\otimes B & \\equiv & \\exists U, V. \\, T = U \\cup V \\; \\wedge \\;  U \\models A \\; \\wedge \\; V \\models B  \\\\\nT \\models A \\multimap B & \\equiv & \\forall U. \\, [U \\models A \\Rightarrow T \\cup U \\models  B ]\n\\end{array}\n\\]\nThe clauses for atomic formulas and their negations and for the dependence formulas and quantifiers are as given in section~2.\n\nAs already noted, the semantics of $\\wedge$ and $\\otimes$ coincide with those given for conjunction and disjunction in section~2. The connectives $\\vee$ and $\\rightarrow$, intuitionistic or additive disjunction and implication,  and the multiplicative implication $\\multimap$, are new as compared to IF-logic or Dependence logic.\n\n\\subsection{The semantics of sentences}\nIt is worth spelling out the semantics of sentences explicitly. By definition, sentences have no free variables, and there is only one assignment on the empty set of variables, which we can think of as the empty tuple $\\langle \\rangle$. In the Tarski semantics, there are only two possibilities for the set of satisfying assignments of a sentence, $\\varnothing$ and $\\{ \\langle \\rangle \\}$, which we can identify with \\emph{false} and \\emph{true} respectively. When we pass to the team semantics for \\textrm{BID}-logic, there are three possibilities for down-closed set of teams to be assigned to sentences: $\\varnothing$, $\\{ \\varnothing \\}$, or $\\{ \\varnothing, \\{ \\langle \\rangle \\} \\}$. Thus the semantics of sentences is \\emph{trivalent} in general.\n\nIn his papers, Hodges works only with non-empty teams, and has bivalent semantics for sentences. However, there is no real conflict between his semantics and ours. Let $\\mbox{\\textrm{BID}}^{-}$ be \\textrm{BID}-logic without the linear implication. Note that $\\mbox{\\textrm{BID}}^{-}$ properly contains Dependence logic, which is expressively equivalent to IF-logic \\cite{Vaan}.\n\n\\begin{proposition}\nEvery formula in $\\mbox{\\textrm{BID}}^{-}$-logic is satisfied by the empty team; hence in particular every sentence of $\\mbox{\\textrm{BID}}^{-}$-logic has either $\\{ \\varnothing \\}$ or $\\{ \\varnothing, \\{ \\langle \\rangle \\} \\}$ as its set of satisfying teams, and the semantics of sentences in $\\mbox{\\textrm{BID}}^{-}$-logic is bivalent.\n\\end{proposition}\n\\begin{proof}\nA straightforward induction on formulas of $\\mbox{\\textrm{BID}}^{-}$-logic.\n\\end{proof}\n\n\\noindent On the other hand, linear implication clearly violates this property. Note that the empty team satisfies $A \\multimap B$ if and only if  every team satisfying $A$ also satisfies $B$.\nWe obtain as an immediate corollary:\n\n\\begin{proposition}\nLinear implication is not definable in $\\mbox{\\textrm{BID}}^{-}$-logic, and \\textit{a fortiori}~is not definable in Dependence logic or IF-logic.\n\\end{proposition}\n\n\\subsection{The general Hodges construction}\n\nWe shall briefly sketch, for the reader conversant with categorical logic, the general form of the construction.\n\nThe standard Tarski semantics of first-order logic is a special case of Lawvere's notion of hyperdoctrine \\cite{Law69}. We refer to \\cite{Pitts} for a lucid expository account. Construing $\\mathcal{L}$ as a functor in the appropriate fashion, we can give a general form of the Hodges construction as  a functor from classical hyperdoctrines to BI-hyperdoctrines \\cite{BBTS05}.\nGiven a classical hyperdoctrine $\\mathbf{P} : \\op{\\CC} \\longrightarrow \\mathbf{Pos}$, we define a BI-hyperdoctrine  $\\mathcal{H}(\\mathbf{P})$  on the same base category by composition with the functor $\\mathcal{L}$:\n\\[ \\mathcal{H}(\\mathbf{P}) = \\mathcal{L} \\circ \\mathbf{P} : \\op{\\CC} \\longrightarrow \\mathbf{Pos} \\, . \\]\nNote that $\\mathbf{Pos}$ is an order-enriched category, and $\\mathcal{L}$ is an order-enriched functor, so it preserves adjoints, and hence in particular preserves the interpretations of the quantifiers.\nThis observation is spelled out in more detail in Proposition~\\ref{qadjprop}.\n\nThis exactly generalizes the concrete Hodges construction, which is obtained by applying $\\mathcal{H}$ to the standard Tarski hyperdoctrine.\n\nA full account will be given elsewhere.\n\n\\section{Quantifiers are adjoints in the Hodges construction}\nWe recall the team semantics for the quantifiers.\n\\[ \\begin{array}{lcl}\nT \\models_{X} \\forall v. \\, \\phi & \\equiv & T[v \\mapsto A] \\models_{X \\cup \\{v\\}} \\phi \\\\\nT \\models_{X} \\exists v. \\, \\phi & \\equiv & \\exists f : T \\longrightarrow A. \\; T[v \\mapsto f] \\models_{X \\cup \\{v\\}} \\phi .\n\\end{array}\n\\]\nWe may wonder what underlying principles dictate these definitions.\n\nTo answer this question, we firstly recall the fundamental insight due to Lawvere \\cite{Law69} that \\emph{quantifiers are adjoints to substitution}.\\endnote{See \\cite{DP02} for an introduction to adjunctions on posets.}\n\n\\subsection{Quantifiers as adjoints}\nConsider a function $f : X \\rightarrow Y$. This induces a function\n\\[ f^{-1} : \\pow{Y} \\longrightarrow \\pow{X} :: T \\mapsto \\{ x \\in X \\mid f(x) \\in T \\} . \\]\nThis function $f^{-1}$ has both a left adjoint $\\exists (f) : \\pow{X} \\longrightarrow \\pow{Y}$, and a right adjoint $\\forall (f) : \\pow{X} \\longrightarrow \\pow{Y}$. These adjoints are uniquely specified by the following conditions. For all $S \\subseteq X$, $T \\subseteq Y$:\n\\[ \\exists(f)(S) \\subseteq T \\;\\; \\Longleftrightarrow \\;\\; S \\subseteq f^{-1}(T), \\;\\; \\quad  f^{-1}(T) \\subseteq S \\;\\; \\Longleftrightarrow \\;\\;  T \\subseteq \\forall(f)(S) . \\]\nThe unique functions satisfying these conditions can be defined explicitly as follows:\n\\[ \\begin{array}{lcl}\n\\exists(f)(S) &:= & \\{ y \\in Y \\mid \\exists x \\in X. \\, f(x) = y \\; \\wedge \\; x \\in S \\}\\,,  \\\\\n\\forall(f)(S) &:= & \\{ y \\in Y \\mid \\forall x \\in X. \\, f(x) = y \\; \\Rightarrow \\; x \\in S \\}\\,.\n\\end{array}\n\\]\n\\noindent Given a formula $\\phi$ with free variables in $\\{ v_{1}, \\ldots , v_{n+1} \\}$, it will receive its Tarskian denotation $\\llbracket \\phi \\rrbracket$ in $\\mathcal{P}(A^{n+1})$ as the set of satisfying assignments:\n\\[ \\llbracket \\phi \\rrbracket = \\{ s \\in A^{n+1} \\mid s \\models_{X} \\phi \\} \\, . \\]\nWe have a projection function\n\\[ \\pi : A^{n+1} \\longrightarrow A^{n} \\; :: (a_{1}, \\ldots , a_{n+1}) \\mapsto (a_{1}, \\ldots , a_{n}) \\, . \\]\nNote that this projection is the Tarskian denotation of the tuple of terms $(v_{1}, \\ldots , v_{n})$.\nWe can characterize the standard quantifiers as \\emph{adjoints to this projection}:\n\\[ \\llbracket \\forall v_{n+1}. \\, \\phi \\rrbracket = \\forall(\\pi)(\\llbracket \\phi \\rrbracket), \\qquad  \\llbracket \\exists v_{n+1}. \\, \\phi \\rrbracket = \\exists(\\pi)(\\llbracket \\phi \\rrbracket) \\, . \\]\nIf we unpack the adjunction conditions for the universal quantifier, they yield the following bidirectional inference rule:\n\\[ \\infer=[\\quad X = \\{ v_{1}, \\ldots , v_{n} \\} \\, . ]{\\Gamma \\vdash_{X} \\forall v_{n+1}. \\, \\phi}{\\Gamma \\vdash_{X} \\phi} \\]\nHere the set $X$ keeps track of the free variables  in the assumptions $\\Gamma$. Note that\nthe usual ``eigenvariable condition'' is automatically taken care of in this way.\n\nSince adjoints are uniquely determined, this characterization completely captures the meaning of the quantifiers.\n\n\\subsection{Quantifiers in the Hodges semantics}\n\nWe shall now verify that the definitions of the  quantifiers in the Hodges semantics \\emph{are exactly the images under $\\mathcal{L}$ of their standard interpretations in the Tarski semantics}, and hence in particular that they are adjoints to substitution. Thus these definitions are \\emph{forced}.\n\n\nIt will be convenient to work with the semantic view of quantifiers, as operators on subsets. Consider formulas with free variables in $\\{ v_{1}, \\ldots , v_{n+1} \\}$. The Tarski semantics over a structure $\\mathcal{M} = (A, \\ldots)$ assigns such formulas values in $\\pow{A^{n+1}}$. We can regard the quantifiers $\\exists v_{n+1}$, $\\forall v_{n+1}$ as functions\n\\[ \\exists(\\pi), \\forall(\\pi) : \\pow{A^{n+1}} \\longrightarrow \\pow{A^{n}} \\]\n\\[ \\begin{array}{lcl}\n\\exists(\\pi)(S) & = & \\{ s \\in A^{n} \\mid \\exists a \\in A. \\, s[v_{n+1} \\mapsto a] \\in S \\} \\\\\n\\forall(\\pi)(S) & = & \\{ s \\in A^{n} \\mid \\forall a \\in A. \\, s[v_{n+1} \\mapsto a] \\in S \\}\n\\end{array}\n\\]\nFor any $m$, we define $\\mathcal{H}(A^{m}) = \\mathcal{L}(\\pow{A^{m}})$. Thus $\\mathcal{H}(A^{m})$ is the set of downwards closed sets of teams on the variables $\\{ v_{1}, \\ldots , v_{m} \\} $. This provides the corresponding ``space'' of  semantic values for formulas in the Hodges semantics.\nThe interpretation of quantifiers in that semantics is given by the following set operators:\n\\[ \\exists_{H}, \\forall_{H} : \\mathcal{H}(A^{n+1}) \\longrightarrow \\mathcal{H}(A^{n}) \\]\n\\[ \\begin{array}{lcl}\n\\exists_{H}(\\mathsf{U}) & = & \\{ T \\in \\pow{A^{n}} \\mid \\exists f : T \\rightarrow A. \\, T[v_{n+1}  \\mapsto f] \\in \\mathsf{U} \\} \\\\\n\\forall_{H}(\\mathsf{U}) & = & \\{ T \\in \\pow{A^{n}} \\mid T[v_{n+1}  \\mapsto A] \\in \\mathsf{U} \\}\n\\end{array}\n\\]\n\\noindent We extend the definition of $\\mathcal{L}$ to act on functions\\endnote{More precisely, homomorphisms of the appropriate kind. The reader familiar with category theory will see that we are really specifying the functorial action of $\\mathcal{L}$ in a particular case.} $h : \\pow{Y} \\longrightarrow \\pow{X}$:\n\\[ \\mathcal{L}(h) : \\mathcal{H}(Y) \\longrightarrow \\mathcal{H}(X) :: \\mathsf{U} \\mapsto  {\\downarrow} \\{ h(T) \\mid T \\in \\mathsf{U} \\} \\, . \\]\nIn the case that $h = f^{-1}$, where $f : X \\longrightarrow Y$, we write $\\mathcal{L}(h) = \\mathcal{H}(f)$.\n\n\\begin{proposition}\n\\label{lqprop}\nThe Hodges quantifiers are the image under $\\mathcal{L}$ of the Tarski quantifiers:\n\\[ \\exists_{H} = \\mathcal{L}(\\exists(\\pi)), \\qquad  \\forall_{H} = \\mathcal{L}(\\forall(\\pi)) \\, . \\]\n\\end{proposition}\n\\begin{proof}\nFirstly, we show that $ \\mathcal{L}(\\exists(\\pi))(\\mathsf{U}) \\subseteq \\exists_{H}(\\mathsf{U})$ for all $\\mathsf{U} \\in \\mathcal{H}(A^{n+1})$.\nSuppose that $T \\in \\mathsf{U}$. Let $T'  = \\exists(\\pi)(T)$. This means that\n\\[ \\forall t \\in T'. \\, \\exists a \\in A. \\, t[v_{n+1} \\mapsto a] \\in T \\, . \\]\nUsing the axiom of choice, there exists a function $f : T' \\longrightarrow A$ such that\n\\[ T'[v_{n+1} \\mapsto f] \\subseteq T \\in \\mathsf{U} \\, . \\]\nSince $\\mathsf{U}$ is downwards closed, this implies that $T' \\in \\exists_{H}(\\mathsf{U})$, as required.\n\n\\noindent The converse follows immediately from the fact that \n\\[ \\exists(\\pi)(T[v_{n+1} \\mapsto f]) = T \\, . \\]\nNext we show that $ \\mathcal{L}(\\forall(\\pi))(\\mathsf{U}) \\subseteq \\forall_{H}(\\mathsf{U})$. Since \n\\[ (\\forall(\\pi)(T))[v_{n+1} \\mapsto A] \\subseteq T \\, , \\]\nif $T \\in \\mathsf{U}$, then $\\forall(\\pi)(T) \\in \\forall_{H}(\\mathsf{U})$ by downwards closure. The converse follows similarly from $T \\subseteq \\forall(\\pi)(T[v_{n+1} \\mapsto A])$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{qadjprop}\nThe Hodges quantifiers are adjoints to substitution:\n\\begin{enumerate}\n\\item $\\exists_{H}$ is left adjoint to $\\mathcal{H}(\\pi)$:\n\\[ \\exists_{H}(\\mathsf{U}) \\subseteq \\mathsf{V} \\;\\; \\Longleftrightarrow \\;\\; \\mathsf{U} \\subseteq \\mathcal{H}(\\pi)(\\mathsf{V}) \\, . \\]\n\\item $\\forall_{H}$ is right adjoint to $\\mathcal{H}(\\pi)$:\n\\[ \\mathcal{H}(\\pi)(\\mathsf{V}) \\subseteq \\mathsf{U} \\;\\; \\Longleftrightarrow \\;\\; \\mathsf{V} \\subseteq \\forall_{H}(\\mathsf{U}) \\, .\\]\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nIt is straightforward to verify the adjunction conditions directly. We give a more conceptual argument.\nThere is a natural pointwise ordering on monotone functions between partially ordered sets, $h, k : P \\longrightarrow Q$:\n\\[ h \\leqslant k \\;\\; \\equiv \\;\\; \\forall x \\in P. \\, h(x) \\leqslant k(x) \\, . \\]\n$\\mathcal{L}$ is an \\emph{order-enriched functor} with respect to this ordering. Functoriality means that\n\\[ \\mathcal{L}(h \\circ g) = \\mathcal{L}(h) \\circ \\mathcal{L}(g), \\qquad \\mathcal{L}(\\id{M}) = \\id{\\mathcal{L}(M)} \\, , \\]\nwhile order-enrichment means that\n\\[ h \\leqslant k \\;\\; \\Rightarrow \\;\\; \\mathcal{L}(h) \\leqslant \\mathcal{L}(k) \\, . \\]\nThese properties imply that $\\mathcal{L}$ automatically preserves adjointness. That is, if we are given monotone maps\n\\[ f : P \\longrightarrow Q, \\qquad g : Q \\longrightarrow P \\]\nsuch that $\\id{P} \\leqslant g \\circ f$ and $f \\circ g \\leqslant \\id{Q}$, \\textit{i.e.}~ so that $f$ is left adjoint to $g$, then\n\\[ \\id{\\mathcal{L}(P)} = \\mathcal{L}(\\id{P}) \\leqslant \\mathcal{L}(g \\circ f) = \\mathcal{L}(g) \\circ \\mathcal{L}(f) \\, , \\]\nand similarly $\\mathcal{L}(f) \\circ \\mathcal{L}(g) \\leqslant \\id{\\mathcal{L}(Q)}$, so $\\mathcal{L}(f)$ is left adjoint to $\\mathcal{L}(g)$ (and of course $\\mathcal{L}(g)$ is right adjoint to $\\mathcal{L}(f)$). Combining this with Proposition~\\ref{lqprop} yields the required result.\n\\end{proof}\n\n\\subsection{The dependence-friendly quantifiers}\nWe shall also give characterizations of the dependence-guarded quantifiers as certain adjoints: this will be our first use of the intuitionistic implication.\n\nWe recall the definition of the dependence-friendly existential quantifier:\n\\[ (\\exists x \\setminus x_{1}, \\ldots , x_{n}). \\, \\phi \\;\\; \\equiv \\;\\; \\exists x. (D(x_{1}, \\ldots , x_{n}, x) \\; \\wedge \\; \\phi) \\, . \\]\nThere has not been a comparably natural notion of dependence-friendly universal quantification. According to our analysis, this is because the appropriate connective needed to express the right notion, namely intuitionistic implication, has not been available. Using it, we can define such a quantifier:\n\\[ (\\forall x \\setminus x_{1}, \\ldots , x_{n}). \\, \\phi \\;\\; \\equiv \\;\\; \\forall x. (D(x_{1}, \\ldots , x_{n}, x) \\; \\rightarrow \\; \\phi) \\, . \\]\nAs evidence for the naturalness of these quantifiers, we shall express them both as adjoints.\n\nFirstly, we recall that intuitionistic conjunction and implication are related by another fundamental adjointness \\cite{Law69}:\n\\begin{equation}\n\\label{impadjeq}\n\\mathsf{U} \\cap \\mathsf{V} \\subseteq \\mathsf{W} \\;\\; \\Longleftrightarrow \\;\\; \\mathsf{U} \\subseteq \\mathsf{V} \\rightarrow \\mathsf{W} \\, .\n\\end{equation}\nThis can be expressed as a bidirectional inference rule:\n\\[ \\infer={\\phi \\vdash \\psi \\rightarrow \\theta}{\\phi \\; \\wedge \\; \\psi \\vdash \\theta} \\, . \\]\nNext, we extend our semantic notation to the dependence-friendly quantifiers. Given $W \\subseteq \\{ v_{1}, \\ldots , v_{n} \\}$, we define $D_{W} \\in \\mathcal{H}(A^{n+1})$:\n\\[ D_{W} = \\{ T \\mid \\forall s, t \\in T. \\, s \\simeq_{W} t \\; \\Rightarrow \\; s(v_{n+1}) = t(v_{n+1}) \\} \\, . \\]\nNow we can define the semantic operators corresponding to the dependence-friendly quantifiers:\n\\[ \\exists_{W}, \\forall_{W} : \\mathcal{H}(A^{n+1}) \\longrightarrow \\mathcal{H}(A^{n}) \\]\n\\[ \\begin{array}{lcl}\n\\exists_{W}(\\mathsf{U}) & = & \\exists_{H}(D_{W} \\cap \\mathsf{U}) \\\\\n\\forall_{W}(\\mathsf{U}) & = & \\forall_{H}(D_{W} \\rightarrow \\mathsf{U})\n\\end{array}\n\\]\n\\begin{proposition}\nThe dependence-friendly existential $\\exists_{W}$ is left adjoint to the following operation:\n\\[ \\mathsf{V}  \\; \\mapsto \\; (D_{W} \\rightarrow \\mathcal{H}(\\pi)(\\mathsf{V})) \\, . \\]\nThe dependence-friendly universal $\\forall_{W}$ is right adjoint to the following operation:\n\\[  \\mathsf{V} \\; \\mapsto \\; (D_{W} \\cap \\mathcal{H}(\\pi)(\\mathsf{V})) \\, . \\]\n\\end{proposition}\n\\begin{proof}\nA direct verification is straightforward, but it suffices to observe that adjoints compose, and then to use Proposition~\\ref{qadjprop} and the adjointness~(\\ref{impadjeq}).\n\\end{proof}\n\nOf course, the analysis we have given in this sub-section  applies to any guarded quantifiers; the dependence predicates play no special r\\^ole here. The point is to show how the intuitionistic connectives round out the logic in a natural fashion. We shall apply them to a finer analysis of dependence itself in section~\\ref{depsec}.\n\n\\section{Full Abstraction}\nWe shall now prove a full abstraction result in the sense of Hodges \\cite{Hod97a}.\\endnote{As Hodges notes, he himself takes the term, and the concept, from Computer Science \\cite{Mil77,Plo77}.}\nThe point of this is to show that, even if we take sentences and their truth-values as primary,\nthe information contained in the semantics of formulas in general  is not redundant, since whenever two formulas receive different denotations, they make different contributions overall to the truth-values assigned to sentences.\n\nThe fact that such a result holds for $\\mbox{\\textrm{BID}}^{-}$-logic is notable, in that the logic is highly non-classical, while the semantics of sentences is bivalent. For \\textrm{BID}-logic, the set of possible truth values for open formulas is huge even in finite models \\cite{CH01}, while the semantics of sentences is trivalent.\n\nWhile our argument follows that of Hodges \\cite{Hod97a}, we find a  natural r\\^ole for the intuitionistic implication, and can give a very simple proof, while Hodges' argument goes through the correspondence with the game-theoretical semantics.\n\nTo formalize  full abstraction, we introduce the notion of a \\emph{sentential context} with respect to a set of variables $X$. This is a formula with an occurrence of a ``hole'' $[\\cdot]$ such that inserting a formula with free variables in $X$ into the hole yields a sentence.\nNow consider two formulas $\\phi$ and $\\psi$ of \\textrm{BID}-logic, with free variables in $X$. We say that the formulas are \\emph{semantically equivalent} if they have the same denotations, \\textit{i.e.}~ the same sets of satisfying teams, in all interpretations with respect to all structures. We say that $\\phi$ and $\\psi$ are \\emph{observationally equivalent} if for all sentential contexts $C[\\cdot]$ for $X$, $C[\\phi]$ and $C[\\psi]$ are assigned  the same truth values in all interpretations. The fact that semantic equivalence implies observational equivalence follows immediately from the compositional form of the semantics. The converse is \\emph{full abstraction}.\\endnote{While this notion is perfectly consistent with usage in Computer Science, one very important tensioning ingredient in the programming language context is missing, namely correspondence with an independently defined operational semantics \\cite{Mil77,Plo77}.}\n\n\\begin{proposition}\nThe team semantics is fully abstract for any sublanguage of \\textrm{BID}-logic containing universal quantification and intuitionistic implication.\n\\end{proposition}\n\\begin{proof}\nSuppose that $\\llbracket\\phi \\rrbracket \\setminus \\llbracket \\psi \\rrbracket$ in some interpretation contains a team $T$. Extend the language with a  relation symbol $R$, and the interpretation by assigning ${\\downarrow}(T)$ to $R$. Then use the context\n\\[ C[\\cdot] \\; \\equiv \\; \\forall v_{1}, \\ldots , \\forall v_{n}. \\, (R(v_{1}, \\ldots , v_{n}) \\rightarrow [\\cdot]) \\]\nwhere the free variables in $\\phi$ and $\\psi$ are contained in $\\{ v_{1}, \\ldots , v_{n} \\}$.\nThen $C[\\phi]$ is true (satisfied by the empty tuple), since for every team $T'$ satisfying $R(v_{1}, \\ldots , v_{n})$, $T' \\subseteq T$, and hence by assumption and downwards closure, $T'$ satisfies $\\phi$. This means that all teams  over  $\\{ v_{1}, \\ldots , v_{n} \\}$ satisfy the implication $R(v_{1}, \\ldots , v_{n}) \\rightarrow \\phi$, and hence $\\langle \\rangle $ satisfies $C[\\phi]$. On the other hand, $C[\\psi]$ is not satisfied by the empty tuple, since $T$ satisfies $R(v_{1}, \\ldots , v_{n})$, while $T$ does not satisfy $\\psi$ by assumption.\n\\end{proof}\nNote that the use of the intuitionistic implication in relativizing to those teams satisfying the precondition $R(v_{1}, \\ldots , v_{n})$ is exactly what is needed.\n\n\\section{Analyzing Dependence}\n\\label{depsec}\n\nWe now turn to the dependence predicate itself. Since it encapsulates the ``jump'' from  first-order to second-order semantics, we cannot be too hopeful about taming it axiomatically\\endnote{See \\cite{MR1867954} for details on this.}. But it turns out that we can give a finer analysis in \\textrm{BID}-logic.\n\nConsider the following ``trivial'' case of dependence:\n\\[ C(v) \\; \\equiv \\; D(\\varnothing, v) \\, . \\]\nThis expresses that $v$ depends on nothing at all, and hence has a fixed value --- functional dependency for the constant function. Semantically, this is the following simple special case of the semantics of dependence:\n\\[ T \\models_{X} C(v) \\; \\; \\equiv \\;\\; \\forall t_{1}, t_{2} \\in T. \\, t_{1}(v) = t_{2}(v) \\, . \\]\nUsing the intuitionistic implication, we can \\emph{define} the general dependence predicate from this special case:\n\\begin{equation}\n\\label{depeq}\nD(W, v) := \\left(\\bigwedge_{w \\in W} C(w) \\right) \\; \\rightarrow \\; C(v) \n\\end{equation}\n\n\\begin{proposition}\n\\label{depcprop}\nThe definition of $D$ from $C$ is semantically equivalent to the definition given previously:\n\\[ T \\models_{X} D(W, v) \\;\\; \\equiv \\;\\; \\forall s, t \\in T. \\, s \\simeq_{W} t \\; \\Rightarrow \\; s(v) = t(v) .) \\]\n\\end{proposition}\n\\begin{proof}\nThis is just an exercise in unwinding the definitions. Note that the intuitionistic implication lets us range over all subsets of the team which are in a single equivalence class under $\\simeq_{W}$, and require that $v$ is constant on those subsets.\n\\end{proof}\n\n\\subsection{Armstrong Axioms}\nThe current stock of plausible axioms for the dependence predicates is limited to the\n\\emph{Armstrong axioms} from database theory \\cite{Arm}. These are a standard complete set of axioms for functional   dependence. They can be given as follows.\n\n\\begin{center}\n\\begin{tabular}{ll}\n(1)\n & Always $D(x,x)$. \\\\\n(2) & If $D(x,y,z)$, then $D(y,x,z)$. \\\\\n(3) & If $D(x,x,y)$, then $D(x,y)$. \\\\\n(4) & If $D(x,z)$, then $D(x,y,z)$. \\\\\n(5) & If $D(x,y)$ and $D(y,z)$, then $D(x,z)$. \\\\\n\\end{tabular}\n\\end{center}\n\n\\noindent However, in the light of our analysis, the Armstrong axioms simply fall out as standard properties of  implication and conjunction.\\endnote{Formal connections between the Armstrong axioms and  propositional logic  were made by Fagin \\cite{Fag}. He only considered Horn clauses, so the distinction between intuitionistic and classical logic was not apparent. Nevertheless, the passage to two-element subsets in the ``Semantic proof of the Equivalence Theorem'' in \\cite{Fag} implicitly involves similar reasoning to Proposition~\\ref{depcprop}.}\nIf we  set $p = C(x)$, $q = C(y)$, $r = C(z)$, and use (\\ref{depeq}) to translate the Armstrong axioms into purely implicational form, we see that they  correspond to the following:\n\n\\begin{center}\n\\begin{tabular}{ll}\n(1)\n & $p \\rightarrow p$. \\\\\n(2) & $(p \\rightarrow q \\rightarrow r) \\rightarrow (q \\rightarrow p \\rightarrow r)$. \\\\\n(3) & $(p \\rightarrow p \\rightarrow q) \\rightarrow (p \\rightarrow q)$. \\\\\n(4) & $(p \\rightarrow r) \\rightarrow (p \\rightarrow q \\rightarrow r)$. \\\\\n(5) & $(p \\rightarrow q) \\rightarrow (q \\rightarrow r) \\rightarrow (p \\rightarrow r)$. \\\\\n\\end{tabular}\n\\end{center}\nThese are the well-known axioms $\\textbf{I, C, W, K, B}$ respectively\\endnote{Axiom (4) as given generalizes the standard \\textbf{K} axiom  $p \\rightarrow q \\rightarrow p$, which is obviously derivable from (1) and (4) by substitution and Modus Ponens.}  \\cite{CF58}  --- which form a complete axiomatization of \\emph{intuitionistic} (but not classical!) implication.\\endnote{Under the (Curry part of the) Curry-Howard correspondence, they correspond to a well-known functionally complete set of \\emph{combinators} \\cite{CF58}.}\nA standard example of a classically valid implicational formula which is \\emph{not} derivable from these axioms is \\emph{Peirce's law}: $((p \\rightarrow q) \\rightarrow p) \\rightarrow p$.\n\n\nThus we have reduced the understanding of the dependence predicate to understanding of the, \\textit{prima facie}~simpler, constancy predicate $C$.\n\n\\section{Further Directions}\nIn this final section, we shall sketch a number of further directions.\nDetailed accounts are under development, and  will appear elsewhere.\n\n\\subsection{Completeness}\nPredicate BI-logic is a well developed formalism, with a proof theory which is sound and complete relative to an algebraic semantics \\cite{Pym02}. Since \\textrm{BID}-logic is a special case, we have a sound ambient inference system. Of course this is not complete for the intended semantics for \\textrm{BID}-logic  --- and cannot be.\nWe may hope to obtain completeness for some smaller class of models, possibly on the lines of the Henkin completeness theorem for higher-order logic \\cite{Hen50}.\n\n\\subsection{Diagrams}\nNow fix a particular interpretation in a structure $\\mathcal{M}$  with universe $A$. Consider the following construction. We  introduce constants for each $a \\in A$, the usual first order diagram (all true atomic sentences), and the following  infinitary axiom:\n\\[ \\forall v. \\, \\bigotimes_{a \\in A} (v=a) \\, . \\]\nWe can \\emph{define} the predicate $C$ (and hence dependence $D$) by the following infinitary formula:\n\\[ C(v) := \\bigvee_{a \\in A} (v=a) \\, . \\]\nNote how the two different connectives (one additive, the other multiplicative) feature naturally.\n\nThis gives a logical (albeit infinitary) characterization of dependence.\n\n\\subsection{Representation}\nWe can also consider representation theory for the structures $\\mathcal{H}(X) = \\mathcal{L}(\\mathcal{P}(X))$. We seek lattice-theoretic properties of these structures which suffice to characterize them.\n\nFirstly, we note that the down-closures of single teams are exactly the \\emph{complete join-primes} of the lattice:\n\\[ a \\leqslant \\bigvee_{i} b_{i} \\;\\; \\Rightarrow \\;\\; \\exists i. \\, a \\leqslant b_{i} . \\]\nMoreover, these join-primes order generate, \\textit{i.e.}~ every element is the join of the join-primes below it.\nAll of this structure is in terms of the intuitionistic disjunction.\n\n\nNext, we note that the join-primes are closed under $\\otimes$, which is  moreover idempotent on the join-primes, endowing them with the structure of a semilattice.\nThis is very different to the semilattice structure given by intuitionistic disjunction: e.g.\n\\[ {\\downarrow}(T_{1}) \\vee {\\downarrow}(T_{2}) = {\\downarrow}(\\{T_{1}, T_{2}\\}) \\neq  {\\downarrow}(T_{1} \\cup T_{2}) = {\\downarrow}(T_{1}) \\otimes {\\downarrow}(T_{2}) \\, . \\]\nThe \\emph{double singletons} are exactly the complete atoms in this semilattice, which is complete atomic in the usual sense.\n\n\nSyntactically, assuming names for elements, we can describe these atomic join-primes in the lattice of propositions over variables $v_{1}, \\ldots , v_{n}$ as\n\\[  (v_{1} = a_{1}) \\; \\wedge \\; \\cdots \\; \\wedge \\;  (v_{n} = a_{n}). \\]\nThese are of course the tuples. (Downclosures of) arbitrary teams are then described by expressions\n$\\bigotimes_i A_i$,  where $A_i$ ranges over such atoms. Arbitrary elements are joins (intuitionistic disjunctions) of such elements. So there is a normal form for general elements:\n\\[ \\bigvee_{i} \\bigotimes_{ij} A_{ij} \\, . \\]\nMoreover, from the lattice-theoretic properties it is easily shown that the ordering between such normal forms agrees with the set inclusion ordering.\n\n\n\n\\subsection{Expressiveness}\n\\newcommand{\\rel}[1]{\\mbox{rel}(#1)}\nOne of the defining characteristics of Dependence Logic as well as IF-logic is that they can be expressed in Existential Second Order Logic, $\\Sigma^1_1$, and conversely, every $\\Sigma^1_1$ definable property of structures can be expressed with a sentence of Dependence Logic. Both are true even on finite structures. To see what this connection with $\\Sigma^1_1$ means let us adopt the notation that if $T$ is a team on a set $X$ of variables, then $\\rel{T}$ is the corresponding relation.\nHodges~\\cite{Hod97b} associates with every formula $\\phi$ of IF-logic (equivalently, of Dependence Logic) with free variables in the set $X=\\{x_1,...,x_n\\}$ an Existential Second Order sentence $\\tau_\\phi(R)$, with $R$ an $n$-ary predicate symbol, such that in any model $\\mathcal{M}$ and for any team $T$ on $X$ the following holds: \\begin{equation}\\label{sol}\\mathcal{M},T\\models_X\\phi\\iff (\\mathcal{M},\\rel{T})\\models\\tau_\\phi(R).\\end{equation} Conversely, if $\\Phi$ is any Existential Second Order sentence, then there is a sentence $\\phi$ of Dependence Logic such that the following holds for all models $\\mathcal{M}$: $$\\mathcal{M}\\models\\Phi\\iff\\mathcal{M},\\{\\langle\\rangle\\}\\models\\phi.$$\nVirtually all model theoretic properties of Dependence Logic follow from this relationship with $\\Sigma^1_1$, for example, the Compactness Theorem, the downward and upward L\\\"owenheim-Skolem Theorems, the Interpolation Theorem, and the fact that every sentence $\\phi$ in Dependence Logic for which there exists a ``negation\" $\\psi$ such that for all $\\mathcal{M}$ $$\\mathcal{M}\\models\\phi\\iff\\mathcal{M}\\not\\models\\psi,$$ is actually first order definable\\endnote{See e.g. \\cite{Vaan} for details.}. Also the interesting fact that the class of properties of finite structures expressible in Dependence Logic is exactly NP follows from this. Because of these connections it is quite interesting to ask whether the extensions $\\mbox{\\textrm{BID}}^{-}$ and $\\textrm{BID}$ can likewise be embedded in $\\Sigma^1_1$, the existential fragment of Second Order Logic.\n\nNow the question arises which semantics one should use. To be able to compare results with Dependence Logic and IF-logic, we use the full semantics familiar from \\cite{Hod97a} and \\cite{Vaan}.\n\n\\begin{proposition}\nThere is no translation of  any extension of  Dependence Logic containing either intuitionistic implication or linear implication into existential second order $\\Sigma^1_1$. The same is true on finite models, assuming NP$\\ne$co-NP.\n\\end{proposition}\n\n\\begin{proof} Let $\\phi(x_1)$ be a formula of Dependence Logic in the empty vocabulary such that for any team $T$: $\\mathcal{M},T\\models\\phi(x_1)$ if and only if $A$ is infinite\\endnote{The free variable $x_1$ plays no role in this.}. Let $\\bot$ denote a sentence in the empty vocabulary, only satisfied by the empty team, e.g. $\\forall x.x=x\\wedge\\neg x=x$.\nSuppose there were an Existential Second Order sentence $\\tau(R)$ such that\na model $\\mathcal{M}$ and a team $T$ on $\\{x_1\\}$ satisfy $\\phi(x_1)\\to\\bot$ if and only if $(\\mathcal{M},\\rel{T})$ satisfies $\\tau(R)$. If $\\mathcal{M}$ is any finite model and $T=\\{s\\}$, where $s(x_1)\\in M$, then  $\\mathcal{M},T\\models\\phi(x_1)\\to\\bot$, whence $(\\mathcal{M},\\{s(x_1)\\})\\models\\tau(R)\\wedge\\exists x_1R(x_1)$. By the Compactness Theorem of Existential Second Order Logic, $\\tau(R)\\wedge\\exists x_1R(x_1)$ has an infinite model $(\\mathcal{M}',\\rel{T'})$. Thus $\\mathcal{M}'$ and the team $T'$ satisfy $\\phi(x_1)\\to\\bot$. Moreover, $T'\\ne\\varnothing$. By the definition of the semantics of $\\to$, since $T'$ satisfies $\\phi(x_1)$ in $\\mathcal{M}'$, $T'$ must satisfy $\\bot$, a contradiction.\n\nLet us then consider finite models. It is easy to write down   a formula $\\phi(x_1)$ of Dependence Logic in the vocabulary of graphs such that for any team $T\\ne\\varnothing$: $\\mathcal{M},T\\models\\phi(x_1)$ if and only if $\\mathcal{M}$ is 3-colorable. Let $\\bot$ be as above. If $\\mathcal{M}$ is any graph that is not 3-colorable and $T=\\{s\\}$, where $s(x_1)\\in M$, then  $\\mathcal{M},T\\models\\phi(x_1)\\to\\bot$. On the other hand, suppose $\\mathcal{M}$ is 3-colorable, but $\\mathcal{M}$ and some team $\\{s\\}$ satisfy $\\phi(x_1)\\to\\bot$.  By the definition of the semantics of $\\to$, since $\\{s\\}$ satisfies $\\phi(x_1)$ in $\\mathcal{M}$, $\\{s\\}$ must satisfy $\\bot$, a contradiction. Thus a graph $\\mathcal{M}$ and a team $\\{s\\}$ satisfy $\\phi(x_1)\\to\\bot$ if and only if $\\mathcal{M}$ is not 3-colorable. Suppose now there were an Existential Second Order sentence $\\tau(R)$ such that\na graph $\\mathcal{M}$ and a team $T$ satisfy $\\phi(x_1)\\to\\bot$ if and only if $(\\mathcal{M},\\rel{T})$ satisfies $\\tau(R)$. Then we could check if a graph $\\mathcal{M}$ is not 3-colorable by checking if  $\\tau(R)$ is satisfied by $\\mathcal{M}$ and and a team $\\{s\\}$, where $s$ can be any assignment. The latter is NP, so we get NP=co-NP.\n\nThe same argument can be used to show that $\\multimap$ leads outside of $\\Sigma^1_1$: Suppose $\\phi(x_1)$ is as above and there is an Existential Second Order sentence $\\tau(R)$ such that\na model $\\mathcal{M}$ and a team $T$ satisfy $\\bot \\, \\wedge \\, (\\phi(x_1)\\multimap\\bot)$ if and only if $(\\mathcal{M},\\rel{T})$ satisfies $\\tau(R)$. If $\\mathcal{M}$ is any finite model and $T=\\varnothing$, then  $\\mathcal{M},T\\models\\bot \\, \\wedge \\, (\\phi\\multimap\\bot)$, whence $(\\mathcal{M},\\varnothing)\\models\\tau(R)$. By the Compactness Theorem of Existential Second Order Logic, $\\tau(R)$ has an infinite model $(\\mathcal{M}',\\rel{T'})$. Thus $\\mathcal{M}'$ and the team $T'$ satisfy $\\bot \\, \\wedge \\, (\\phi\\multimap\\bot)$. In particular, $T'=\\varnothing$ and $\\varnothing$ satisfies $\\phi\\multimap\\bot$. Since in this model any $\\{s\\}$ satisfies $\\phi$, by the definition of the semantics of $\\multimap$, $\\{s\\}$ satisfies $\\bot$, a contradiction.\n\\end{proof}\n\n\n\\noindent The proof actually shows that $\\textrm{BID}$ fails to satisfy the Compactness Theorem. A similar argument shows that $\\textrm{BID}$ fails to satisfy the Downward L\\\"owenheim Skolem Theorem. \n\n\\begin{proposition}\nThere is a translation of $\\textrm{BID}$ into Full Second Order Logic.\n\\end{proposition}\n\n\\begin{proof} We follow \\cite{Hod97b} (see also \\cite{Vaan}) and present only the additions needed over and above Dependence Logic and IF-logic:\n\\begin{eqnarray*}\n\\tau_{\\phi\\multimap\\psi}(R) &=&\\forall S(\\tau_\\phi(S)\\to\\forall U(\\forall \\vec{x}(U(\\vec{x})\\leftrightarrow(S(\\vec{x})\\vee\nR(\\vec{x})))\\to\\tau_\\psi(U)))\\\\\n\\tau_{\\phi\\to\\psi}(R) &=&\\forall S(\\forall \\vec{x}(S(\\vec{x})\\to\nR(\\vec{x}))\\to(\\tau_\\phi(S)\\to\\tau_\\psi(S))) \\, .\n\\end{eqnarray*}\\end{proof}\n\n\\noindent In conclusion, we may say that $\\mbox{\\textrm{BID}}^{-}$ and $\\textrm{BID}$ seem to have a more robust and uniform algebraic structure than Dependence Logic and IF-logic. We anticipate that this is reflected also in an effective proof theory, still to be developed. On the other hand the price of this seems to be that ``nice\" model theoretic properties are lost, at least in the full semantics. Perhaps there are some underlying, hitherto unidentified, reasons why logics developed for dependence cannot simultaneously have a ``nice\" model theory and effective proof theory. After all, we know from Lindstr\\\"om's Theorem (\\cite{MR0244013}) that there are intrinsic obstacles to having model-theoretically defined extensions of first order logic with both nice proof theory and nice model theory. However, we have a trivalent logic, unlike the setting considered by Lindstr\\\"om. So it is too early to say whether there are general reasons why $\\textrm{BID}$ does not satisfy Compactness and other model theoretic properties familiar from Dependence Logic, or whether we  have just not found the right concepts yet.\n\n\\theendnotes\n\n\\bibliographystyle{klunamed}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMicrostructure of materials are meticulously engineered to cater the ever progressing and more demanding needs. \nProperties, once thought to be irreconcilable, are increasingly combined through appropriate combination of phases in the microstructures~\\cite{jacques2001developments,huang2017multiphase,kumar2020carbon}.\nEven though microstructures characterised by more than one chemically-distinct phases have been around for a considerable period of time, the distribution of the constituent phases engenders a unique category of multiphase materials. \nFor instance, in polycrystalline pearlitic steel, the combination of ferrite and cementite, in a specific phase-fraction, is observed in all grains, despite being separated by the interfaces (grain boundaries)~\\cite{zhang2011microstructure,hwang2019microstructure}.\nSuch grains can be treated as chemically homogeneous, despite the presence of two distinct phases. \nHowever, on the other hand, there are specialised steels, wherein the grains are not chemically homogeneous but are exclusively associated with one of the constituent phases~\\cite{liljas2008development}. \nIn other words, instead of martensite and ferrite co-existing in all grains of the polycrystalline microstructure, individual grains assume a phase, thereby establishing a chemical inhomogeneity in the system~\\cite{armas2002mechanical}.\nPolycrystalline materials, characterised by these chemically-distinct grains, are qualified contextually as multiphase, and depending on number of constituent phases, these materials are referred to duplex, triplex and such. \n\nBesides steel, multiphase microstructures are established in wide-range of materials to achieve desired combination of properties~\\cite{filip2003effect,gollapudi2011microstructure}.  \nThe extensive applicability of two- and three-phase titanium alloys is primarily due to properties which is a direct consequence of their multiphase microstructure. \nThe prevalence of multiphase polycrystalline arrangement in high-entropy alloys is a principal reason for their characteristic behaviour~\\cite{tang2015tensile,liu2019fatigue}.  \nIn ceramics, mechanical properties including fracture toughness are noticeably enhanced by two-phase microstructure.\nDuplex system of alumina and silicon carbide is a prime example of such behaviour-enhanced ceramics~\\cite{jang1995effect,lutz1991k}. \nMoreover, it has been reported that the introduction of an additional phase, which institutes a triplex microstructure, exacts a more preferred response from ceramics to an imposed mechanical conditions~\\cite{neuman2017high,feng2020effect}. \nSimilar favourable effects of multiphase microstructure are observed in composite materials as well~\\cite{do2008microstructure}. \n\nIn addition to the characteristic feature of the phases associated with the multiphase materials, the emerging properties are also considerably influenced by the volume fraction of the constituent phases, $i.e$ phase-fraction, and average grain size. \nConsequently, to achieve desired behaviour, multiphase materials with varying phase-fraction ranging from minimal minor-phase (close to 1\\%) to equi-fraction (50-50 in duplex), and appropriate grain sizes, are fabricated~\\cite{sternitzke1997structural,fan1997computer}. \nWhile material-specific processing techniques are employed for introducing the necessary phase-fraction, the required grain sizes are obtained through a rather well-known microstructural transformation called grain growth~\\cite{yu2021high}. \nThe mechanism of grain growth in multiphase systems is significantly different single-phase polycrystalline microstructure.\nIn polycrystalline materials with chemically-homogeneous grains, the local diffusion of atoms across the grain boundaries, which consequently leads to its migration, ultimately governs the grain growth. \nIn other words, in single-phase systems, the grain growth is primarily dictated by the movement of the interface (or) grain boundaries. \nHowever, in multiphase microstructures, the association of grains to a specific constituent phase adds an additional facet to the conventional grain growth. \nBesides reducing the number of grains, grain growth in multiphase systems conserves the characteristic volume-fraction of the phases. \nTherefore, the curvature-driven transformation, which minimises the overall grain-boundary energy, in multiphase materials is, in principle, a combination of coarsening and grain growth~\\cite{cahn1991stability}. \nTypifying features of both coarsening and grain growth, which respectively are preservation of volume fraction and growth of larger grains at the expense of smaller ones, are simultaneously observed in multiphase materials. \nFor this reason, the energy-minimising curvature-driven transformation in multiphase systems is at times referred to as concurrent grain growth and coarsening. \nCorrespondingly, as opposed to interface migration, the long-range diffusion of atoms dictates grain growth in multiphase polycrystalline systems~\\cite{fan1997diffusion}. \nRegardless of the mechanism, owing to the influence of the grain size on the properties rendered by the materials, grain growth in multiphase systems have been extensively analysed~\\cite{guo2009microstructural,liu2015synergetic}. \nMoreover, it is conceivable, and indeed reported that, a change in the average grain-size of a multiphase material, while employed in an application, results in undesired behaviour. \nIn other words, besides the direct effect of grain-size on the properties of multiphase materials, its life in an application is dictated by the rate at which the grains grow.  \nSolid-oxide fuels with triplex microstructure are prime examples of this relation between the growth kinetics and life of a material~\\cite{lei2017phase}. \nConsidering this influence of grain size, investigations have been geared towards understanding the kinetics of grain growth in multiphase polycrystalline system.  \n\nExperimental techniques generally pose definite practical difficulties when adopted for comprehensive analyses of grain growth.\nTherefore, theoretical approaches have long since been adopted to complement, and extend, the existing understanding~\\cite{saito1992monte,kawasaki1989vertex,anderson1984computer}.\nThese theoretical studies, particularly one involves multiphase-field models, have been offering critical insights on mechanism and kinetics of grain growth in multiphase system, which are consistent with the experimental observation~\\cite{krill2002computer,perumal2017phase,mckenna2014grain}. \nExisting investigations, and resulting understanding, of grain-growth kinetics in multiphase polycrystalline systems can broadly be categorised into two. \nOne relates material-specific parameters, including diffusivities and grain-boundary energy anisotropies, to the rate of grain growth~\\cite{ravash2014three,ravash2017three2}, while the other focuses on the effect of microstructural features like phase-fractions~\\cite{fan1997computer,yadav2016effect}.\nWhile the material-specific studies are, in principle, inherently  bound by the choice parameter(s), the analyses involving phase-fraction offer more generalised insights that are relevant to a wide-range of multiphase polycrystalline systems. \n\nIn-keeping with the change in grain-growth mechanism, the investigations focusing on microstructural features unravel that the introduction of second phases significantly reduces the kinetics of overall evolution. \nMoreover, it is realised that the expression capturing the temporal change in the average radius of the polycrystalline system reflects coarsening kinetics, $\\bar{R}^n\\propto t$ with $n=3$, irrespective of the volume-fraction of the second phase~\\cite{fan1997diffusion,ohnuma1999computer}. \nEven though the major and minor phase(s) - categorised based on volume-fraction- , and the entire polycrystalline system as a whole, evolve at a rate that complies with the coarsening power-law, the kinetic-coefficient ($k$) varies with the  phases, particularly their volumes~\\cite{fan1997computer}.\nFor instances, in a duplex microstructure characterised by unequal volume-fraction of constituent phases, the grains of the major phase, with higher phase-fraction, grow at the noticeably faster rate when compared to the minor phase. \nThis disparity in the kinetics of the grain growth between the phases is indicated by the difference in the kinetic-coefficient ($k$).\nThe relatively sluggish growth of the minor phase(s) is attributed to the increased distance between the respective chemically-similar grains, and complex diffusion-pathways that facilitate the growth, when compared to the proximity of the major phases. \nFurthermore,  attempts have been quantify the effect of volume-fraction on the kinetic-coefficients of major- and minor-phases by comparing the evolution of the microstructures with varying phase-fractions~\\cite{yadav2016effect}. \nDespite these apparent advancements, there continues to exist certain aspects to the grain-growth kinetics that prevent a comprehensive understanding of the evolution of multiphase polycrystalline systems. \n\nIn addition to the characteristic nature, and volume-fraction, of the phases, the properties of multiphase systems depend on the \\textit{overall grain-size} of the polycrystalline microstructure.\nHowever, studies aimed at explicating the grain-growth rate in multiphase materials primarily report, and discuss, on the difference in the kinetics exhibited by the major- and minor-phases, while largely overlooking the evolution of the entire microstructure~\\cite{fan1997topological,jang1995effect}. \nMoreover, the collective influence of the growth rate of major- and minor-phases on the temporal change in overall average size of the grains is yet to be convincingly understood. \nBesides, in three (or more) phase systems, the interdependency between the grain-growth rate of different phases have not been thoughtfully considered so far. \nAmongst others, in the present work, attempts are made to address the aforementioned questions on the grain-growth kinetics of the overall multiphase microstructure through appropriate statistical analysis. \n\n\n\\section{Devising \\lq multidimensional\\rq \\thinspace dataset}\n\nPersuasive understanding on how the evolution of grains of an individual phase relates to the overall grain growth in a multiphase system, can hardly be gained from analysing of a few microstructures.  \nTo that end, a \\lq multidimensional\\rq \\thinspace dataset is developed by modelling grain growth exhibited by duplex and triplex microstructures in the multiphase-field framework~\\cite{perumal2018phase,amos2019understanding}. \nThermodynamical underpinnings of the adopted model is relatively well-established, and is often acclaimed to be a computationally-efficient alternate~\\cite{amos2020grand,amos2020distinguishing}. \nEven though the complete formulation of the present approach~\\cite{perumal2019concurrent,perumal2020quadrijunctions}, its ability to incorporate different modes of mass transfer, and recover sharp-interface solutions is exhaustively discussed elsewhere ~\\cite{amos2020multiphase,hoffrogge2021multiphase}, a brief outline focusing on the key aspects in rendered here. \n\n\\subsection{Multicomponent multiphase-field model}\n\nMultiphase-field approach is characterised by the introduction of scalar variable(s) which distinguishes the different phases in the system. \nConventionally, while modelling polycrystalline microstructures, these variables, called phase-field ($\\phi(\\vv{x},t)$), are employed to differentiate various grains, and their spatio-temporal evolution translates to grain growth. \nAcross the grain boundary separating two grains, phase-field assumes a value of $\\phi=0$ in one grain, while the other is realised by a constant non-zero value, which is generally $\\phi=1$. \nThe diffuse region, wherein the value of the phase-field gradually varies, $\\phi(\\vv{x},t)\\in(0,1)$, describes the grain boundary.  \nGiven that, in most cases, the grains in the polycrystalline systems are chemically homogeneous, phase-field typifying the various grains are introduced as a tuple, which is expressed as \n\\begin{align}\\label{eq:mp1a}\n\\mbox{{\\bm{$\\phi$}}}=\\left \\{\\phi_{1},\\phi_{2},\\dots,\\phi_{N}\\right \\},\n\\end{align}\nwhere $N$ denotes the total number of grains in the system. \nHowever, since the present approach models grain growth in multiphase systems comprising of $N$ phases, wherein numerous grains are associated each phase, the corresponding phase-field is extended and written as\n\\begin{align}\\label{eq:mp2}\n\\mbox{{\\bm{$\\phi$}}}=\\Big\\{ \\underbrace{ \\{ \\phi_{\\alpha}^{1},\\phi_{\\alpha}^{2}\\dots\\phi_{\\alpha}^{q_{\\alpha}} \\}}_{\\vv{\\phi_{\\alpha}}}, \\underbrace{ \\{ \\phi_{\\beta}^{1},\\phi_{\\beta}^{2}\\dots\\phi_{\\beta}^{q_{\\beta}} \\}}_{\\vv{\\phi_{\\beta}}}\\dots \\underbrace{ \\{ \\phi_{N}^{1},\\phi_{N}^{2}\\dots\\phi_{N}^{q_{N}} \\}}_{\\vv{\\phi_{N}}} \\Big\\}.\n\\end{align}\nMoreover, in this formulation, number of grains sharing a given phase is represented by $q_i$ where $i\\in\\{\\alpha,\\beta,\\dots,N\\}$.\nBy assigning appropriate concentration, grains of a given phase are distinguished from the rest, $\\{\\mbox{{\\bm{$\\phi$}}}(\\mbox{{\\bm{$c$}}})=\\mbox{{\\bm{$\\phi$}}}_\\alpha(\\mbox{{\\bm{$c$}}}) | \\mbox{{\\bm{$c$}}}=\\vv{c}_\\alpha\\}$.  \nSimilar to phase-field, in order to encompass the $K$-different chemical components, the concentration is introduced as a tuple.\nCorrespondingly, concentration of a random grain $m$ associated with phase-$\\alpha$ reads\n\\begin{align}\\label{eq:conc1}\n\\mbox{{\\bm{$c$}}}^{\\alpha}_{m}=\\left\\{c^{\\alpha}_{m:i},c^{\\alpha}_{m:j},\\dots,c^{\\alpha}_{m:K}\\right\\}.\n\\end{align}\nThe homogeneity in the chemical composition of the grains associated with a given phase, say $\\alpha$, yields\n\\begin{align}\\label{eq:conc2}\n\\mbox{{\\bm{$c$}}}^{\\alpha}_{1}=\\dots=\\mbox{{\\bm{$c$}}}^{\\alpha}_{m}=\\dots=\\mbox{{\\bm{$c$}}}^{\\alpha}_{q_{\\alpha}}\\equiv\\mbox{{\\bm{$c$}}}^{\\alpha}=\\left\\{c^{\\alpha}_{i},c^{\\alpha}_{j},\\dots,c^{\\alpha}_{k}\\right\\}.\n\\end{align}\nTo ensure that the volume-fraction of the phases remains unaltered despite the evolution, equilibrium composition is assigned to the corresponding phases.\nIn other words, since the current approach attempts to model grain growth in multiphase system, $\\mbox{{\\bm{$c$}}}^{\\alpha}$ represents a tuple of equilibrium composition that characterises phase-$\\alpha$.\n\nFollowing the conventional framework, the overall energy-density of the multiphase polycrystalline system is formulated as the combination interface (grain boundaries) and bulk (grain) contribution~\\cite{provatas2011phase}. \nCorrespondingly, by incorporating the appropriate phase-field and concentration, the energy-density of the system comprising of $N$-phases and $K$ chemical components is written as \n\\begin{align}\\label{eq:functional1}\n\\cal{F}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}}) & =\\cal{F}_{\\text{int}}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})+\\cal{F}_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}}) \\\\ \\nonumber\n&=\\int_{V}f_{\\text{int}}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})+f_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})\\diff V,\n\\end{align}\nwhere $f_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})$ and $f_{\\text{int}}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})$ are the respective energy-contribution of grain and grain boundary with $V$ representing the volume. \nThe interface contribution, $f_{\\text{int}}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})$, akin to most multiphase-field techniques, comprises of a gradient- and potential-energy term. \nGrain-boundary energy densities, and corresponding anisotropies, are introduced to the system through the interface-energy contribution~\\cite{tschukin2017concepts}. \nWhile multi-well potentials are generally employed to penalise phase-field, and ensure its bounds, in the present work, an obstacle-type potential operating in combination with Gibbs simplex is involved~\\cite{amos2018phase}. \nFurthermore, the energy contributions from the grains, $f_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})$, which is reasonably assumed to be insignificant while modelling grain growth in single-phase system, is formulated as the interpolation of the energy contribution of the individual grains, $f(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})=\\sum_{\\alpha}^{N}\\sum_{m}^{q_{\\alpha}}f^{\\alpha}_{m}(\\mbox{{\\bm{$c$}}}^{\\alpha})h(\\phi_{\\alpha}^{m})$.\nGiven that the contribution of the individual grains is dictated by the characteristic equilibrium-composition of the associated phases, the volume of the phases during grain growth is preserved by the energy-density term, $f_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})$~\\cite{amos2018globularization,amos2020limitations}.  \n\nThe spatio-temporal evolution of phase-field, which translates to grain growth, is formulated by considering phenomenological minimisation of the overall energy-density of the system.\nCorrespondingly, the evolution of a random grain $m$, which is associated with phase-$\\alpha$, is dictated by\n\\begin{align}\\label{eq:ph_evo1}\n\\tau\\epsilon\\frac{\\partial \\phi_{\\alpha}^{m}}{\\partial t} & = -\\frac{\\partial \\cal{F}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})}{\\partial \\phi_{\\alpha}^{m}}\\\\ \\nonumber\n&=\\epsilon\\left[ \\vv{\\nabla} \\cdot \\frac{\\partial a(\\mbox{{\\bm{$\\phi$}}}, \\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})}{\\partial \\vv{\\nabla} \\phi_{\\alpha}^{m}} - \\frac{\\partial a(\\mbox{{\\bm{$\\phi$}}}, \\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})}{\\partial \\phi_{\\alpha}^{m}} \\right] - \\frac{1}{\\epsilon} \\left[ \\frac{\\partial w(\\mbox{{\\bm{$\\phi$}}})}{\\partial \\phi_{\\alpha}^{m}} \\right] - \\left[ \\frac{f^{\\alpha}_{m}(\\mbox{{\\bm{$c$}}}^{\\alpha},\\phi_{\\alpha}^{m})}{\\partial \\phi_{\\alpha}^{m}} \\right] - \\Lambda,\n\\end{align}\nwhere the Lagrange multiplier $\\Lambda$ is introduced to ensure that the summation of phase-fields at any point in the system is $1$.\nMoreover, in the above evolution equation, while $a(\\mbox{{\\bm{$\\phi$}}}, \\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})$ and $w(\\mbox{{\\bm{$\\phi$}}})$ correspond to the gradient and potential energy terms, the parameter dictating interface-width, and its stability during the migration, are denoted by $\\epsilon$ and $\\tau$, respectively.\n\nIn multiphase-field models, wherein the energy contribution of the bulk phases are described based on the dependent-concentration, the corresponding driving-force, which emerges from $\\frac{f^{\\alpha}_{m}(\\mbox{{\\bm{$c$}}}^{\\alpha},\\phi_{\\alpha}^{m})}{\\partial \\phi_{\\alpha}^{m}}$ in Eqn.~\\eqref{eq:ph_evo1}, and dictates the evolution of phase-field, can be viewed as the difference in the Legendre transform of the free-energy densities.\nThis understanding forms the basis of the grand-potential approach, and when consistently extended assumes chemical potential as the continuous and dynamic variable replacing phase-dependent concentration~\\cite{plapp2011unified}. \nGiven its computational efficiency, this approach is adopted in the present work, and the driving-force dictating phase-field evolution is formulated by treating chemical potential as the dynamic variable. \nThe temporal evolution of the chemical potential, which principally governs the bulk driving-force in phase-field evolution, is written as\n\\begin{align}\\label{chempot_ev}\n\\frac{\\partial \\mu_{i}}{\\partial t}=\\left \\{ \\vv{\\nabla}\\cdot\\left[ \\sum_{j=1}^{K-1} \\vv{M}(\\mbox{{\\bm{$\\phi$}}}) \\vv{\\nabla} \\mu_{j}\\right ] - \\sum_{\\alpha}^{N} \\sum_{m}^{q} c_{i}^{\\alpha}\\frac{\\partial \\phi_{\\alpha}^{m}}{\\partial t} \\right \\} \\left [ \\sum_{\\alpha}^{N} \\sum_{m}^{q} h(\\phi_{\\alpha}^{m}) \\frac{\\partial c_{m:i}^{\\alpha}}{\\partial \\mu_{j}} \\right ]_{ij}^{-1},\n\\end{align}  \nwhere $\\mu_{i}$ denotes the continuous chemical-potential of component-$i$. \nThe mobility of the migrating elements, in the multicomponent setup, is dictated by matrix $\\vv{M}(\\mbox{{\\bm{$\\phi$}}})$ which also facilitates the incorporation surface diffusion~\\cite{amos2020multiphase,hoffrogge2021multiphase}. \nThe phase-dependent concentration of component-$i$ in random grain $m$ belonging to phase-$\\alpha$, and the corresponding interpolation function, are represented by $c_{m:i}^{\\alpha}$ and $h(\\phi_{\\alpha}^{m})$, respectively.\nThe evolution of the different microstructures with varying phase-fractions, in this work, are modelled by solving the Eqns.~\\eqref{eq:ph_evo1} and ~\\eqref{chempot_ev}.\n\n\\subsection{Simulation setup}\n\nExisting studies unravel that, unlike Zener pinning~\\cite{fan1998numerical}, the overall trend in grain-growth kinetics exhibited by the individual phases, and entire microstructure, of the multiphase system are largely independent of the dimensionality of the simulation domain~\\cite{fan1997computer,poulsen2013three,yadav2016effect}.\nIn other words, in both two- and three-dimensional setup, similar disparity in the evolution kinetics of  major-, minor- and equifraction phases, in relation to overall growth rate, has been reported. \nTherefore, in the present work, the grain growth in various multiphase-polycrystalline systems are modelled in two-dimensional framework. \nMoreover, irrespective of the variation in phase-fraction, two-dimensional domains of similar configurations are adopted for all numerical studies.\n\nTwo-dimensional domains considered for the current investigations are uniformly discretised into $2048 \\times 2048$ cells of identical dimension, $\\Delta$x = $\\Delta$y = $5 \\times 10^{-7}$m, through the finite-difference scheme.\nPolycrystalline microstructures comprising of approximately $10000$ grains are instituted over the discretised domain through Voronoi tessellation. \nThese grains are associated to the constituent phases by assigning the characteristic chemical composition.\nSince the grains are equiaxed with almost similar size, the required phase-fraction in the microstructure is achieved by relating the appropriate number of randomly distributed grains to the corresponding phases.\nIn other words, duplex microstructure comprising of 33$\\%$ minor phase is devised, in the initial stages, by assigning the respective chemical composition to the one-third of the grains randomly. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{simplex}\n    \\end{tabular}\n    \\caption{  A simplex, analogous to ternary isotherm, depicting all-possible phase-fractions in duplex and triplex microstructures, along with single-phase systems (vertices). The section that encompasses the varied microstructures considered in this work is distinguished, and each  point referring to a specific system is highlighted and designated as $s1$, $s2$, and such. \n    \\label{fig:simplex}}\n\\end{figure}\n\nGiven that the principal focus of this work is to understand the evolution kinetics of  different phases in relation to each other, and to the overall grain-growth rate exhibited by the entire microstructure, a rather straightforward distinction is made between the phases. \nWhile a binary system with two chemical component, $\\tilde{i}$ and $\\tilde{j}$, is considered for establishing duplex microstructure, three-phase microstructure is construed in the framework of ternary system with components $\\tilde{i}$, $\\tilde{j}$ and $\\tilde{k}$.\nIn duplex system, $\\alpha$-phase is a $\\tilde{i}$-rich phase with equilibrium composition of $c^{\\alpha}_{\\tilde{i}:eq}=0.9$, and $c^{\\alpha}_{\\tilde{j}:eq}=0.9$ characterises matrix $\\gamma$-phase, wherein both concentrations are expressed in mole fraction.\nConcentration of the solvent in $\\alpha$- and $\\gamma$-phase remains unaltered in the triplex system, while the remnant content is equally partitioned between solutes, $\\{\\tilde{j},\\tilde{k}\\}$ and $\\{\\tilde{i},\\tilde{k}\\}$, respectively.\nThe $\\beta$-phase, exclusively introduced in the three-phase systems, is characterised by composition $c^{\\beta}_{\\tilde{i}:eq} = c^{\\beta}_{\\tilde{j}:eq} = 0.05$.\nSince the present investigation is primarily interested in microstructural features like phase-fractions, diffusivities of the components are assumed to be identical, and unit matrix is correspondingly incorporated in the formulation. \nMoreover, the energy-densities of the grain boundaries, irrespective of the chemical-composition of the grains they separate, is treated as isotropic and unity. \nThe length scale parameter, $\\epsilon$, is appropriately defined such that the diffuse interface is of constant thickness comprising of four cells~\\cite{mittnacht2021morphological}. \n\nThe temporal evolution of the dynamic variables, phase-field and chemical potential, that dictate the microstructural changes in the multiphase polycrystalline system, are solved over the homogeneous cells of the two-dimensional domain by forward-marching Euler's scheme. \nIn order to ensure that the computational resources are optimally used, the domain is decomposed into smaller segments, and dealt simultaneously, through Message Passing Interface (MPI). \nThe complexity of the numerical treatment is reduced by suitably non-dimensionalising the input parameters, and incorporating them as dimensionless values~\\cite{amos2020multiphase}.\n\n\\subsection{Varying phase-fractions}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{systems}\n    \\end{tabular}\n    \\caption{ Microstructure corresponding to each point in the section of the simplex that includes homogeneous systems along with duplex and triplex microstructures with varying volume-fraction of the constituent phases.  \n    \\label{fig:systems}}\n\\end{figure}\n\nIn applications, depending on the material need, multiphase systems with varying degree of phase-fractions are employed. \nThough investigating every combination of the phase-fraction would be redundant, convincing level of understanding can only be gained by considering relatively increased number of phase-fractions. \nParticularly, since the present study adopts statistical techniques to explicate the kinetic relation between the evolving phases, the accuracy of the outcome depends on the wealth of information (data) available. \nTo that end, in this work, grain growth in twenty different systems, which encompasses  one homogeneous, five duplex and fourteen triplex microstructures, is modelled, and \\lq multidimensional\\rq \\thinspace dataset is built by monitoring the temporal evolution of the grains. \n\nAs opposed to random consideration of different volume-fractions of phases, a systematic choice of various phase-fraction is made from a 2-simplex.\nThe simplex, in its entirety, along with the section focused for the current study is shown in Fig.~\\ref{fig:simplex}. \nThe points within, and on, the 2-simplex can be interpreted in a manner akin to the ternary isotherm.\nCorrespondingly, while the three vertices indicate the homogeneous microstructure of phase-$\\alpha$, -$\\beta$ and -$\\gamma$, the duplex microstructures are encapsulated by the edges joining the vertices. \nAny point within the simplex represents triplex system, with phase-fraction dictated by its position.\nAs illustrated in Fig.~\\ref{fig:simplex}, a section of the simplex emanating from the vertex characterising the homogeneous $\\gamma$-microstructure is considered for the present analyses. \nThis section of the simplex renders a wide-variety of polycrystalline systems ranging from single phase homogeneous to triplex with equifraction of constituent phases. \nMoreover, owing to the configuration of the section in Fig.~\\ref{fig:simplex}, $\\gamma$ -phase acts as the matrix for the duplex and triplex microstructures with unequal volume-fractions of phases. \nMultiphase microstructures corresponding to the different points of the simplex-section is collectively illustrated in Fig.~\\ref{fig:systems}. \nThough the volume-fraction of the minor-phases can be as low as 5$\\%$, the grains associated with these phases hardly occupy a position on the grain boundary. \nIn other words, despite the low volume-fraction and reduced size, the grains of the minor phases seldom render an influence analogous to the particles in Zener pinning. \nMoreover, the reduced size of the grains associated with minor phases, in the initial stages of the grain growth, is consistent with experimental observations~\\cite{liu2015synergetic,ritasalo2013microstructural,praveen2016exceptional} and existing theoretical studies~\\cite{fan1997computer,yadav2016effect}. \n\n\\section{Results and discussion}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.65\\textwidth]{duplex_validation}\n    \\end{tabular}\n    \\caption{  Temporal change in the average radius of the phase-$\\alpha$ and $-\\gamma$ grains, $\\bar{R}_\\alpha(t)$ and $\\bar{R}_\\gamma(t)$, and the entire microstructure,  $\\bar{R}(t)$, for homogeneous, and duplex systems characterised by $50\\%$ and $10\\%$ of phase-$\\alpha$. \n    \\label{fig:duplex_validation}}\n\\end{figure}\n\nBy monitoring the grain growth exhibited by the homogeneous and multiphase systems, a \\lq multidimensional\\rq \\thinspace dataset is devised which essentially comprises of temporal change in the average radius of the individual phase-associated grains, and the entire microstructure as a whole.\n(A dataset can truly be multidimensional only when its dependent variable is governed by more than one independent variable. \nConsidering that explicating the multivariate nature of the overall grain-growth kinetics in multiphase system is the ultimate outcome of the current study, the term \\lq multidimensional\\rq \\thinspace is written within quotes.)\nThis dataset is analysed through comfortably realised statistical techniques to unravel the effect of the individual phases on the evolution kinetics of the entire system.  \nSince the present study focuses primarily on the kinetics of the grain growth, the topological changes and the distribution of the grains are largely overlooked. \n\n\\subsection{Duplex microstructures}\n\nDuplex microstructure comprises of two distinct phases, and is characterised by grains associated with one of these constituent phases, $\\alpha$ and $\\gamma$. \nDespite the inhomogeneity in the concentration distribution, in duplex microstructures, given that the grain growth occurs in a continuum, the temporal evolution of one phase, and its corresponding kinetics, is inherently coupled with the other. \nThis is illustrated and discussed in Appendix.\nIn other words, the growth rate exhibited by the duplex microstructures with varying phase-fractions can be convincingly expressed by considering kinetics of the only one of the evolving phases. \nTherefore, the aim of the present investigation in duplex microstructure reduces to identifying which of the phases, major or minor, principally governs the kinetics of overall grain-growth. \n\n\\subsubsection{Comparison with single phase microstructure}\n\nBefore proceeding to realise the degree of influence rendered by the different phases on the overall growth-kinetics exhibited by the duplex microstructure, rather straightforward investigations are pursued to verify the outcomes of the present approach in relation to the existing reports~\\cite{fan1997computer,fan1997diffusion}. \nEven though,  the outcomes of the modelling technique in relation to the established theories and observations have already been reported elsewhere~\\cite{amos2020multiphase}, certain relevant aspects are analysed in a statistical framework here. \n\nIn Fig.~\\ref{fig:duplex_validation}, the progressive change in the average radius of homogeneous and two duplex microstructures with time are presented. \nMoreover, the temporal increase in the average radius of the phase-associated grains are monitored, and included in this illustration. \nWhile $\\bar{R}(t)$ represents the average radius of the entire polycrystalline microstructure, the corresponding parameter for the grains of  phase-$\\alpha$ and -$\\gamma$ in duplex microstructures is respectively denoted by $\\bar{R}_\\alpha(t)$ and $\\bar{R}_\\gamma(t)$.\nMoreover, the multiphase microstructures in this, and subsequent, discussions are described based on the volume fraction of the minor phase. \nFor instance, $\\alpha 10$ indicates duplex microstructure with $10\\%$ phase-$\\alpha$, while equifraction system are denoted $\\alpha 50$.\n\nIrrespective of the nature of the microstructure, homogeneous or otherwise, Fig.~\\ref{fig:duplex_validation} shows a continual increase in the average radius reflecting the grain growth exhibited by the system. \nWith the introduction of a second-phase in the microstructure, a significant decrease is observed in the rate at which the radius increases with time. \nThis noticeable change in the kinetics is predominantly due to the change in grain-growth mechanism, which is governed by the long-range diffusion of the chemical components in duplex microstructure.\nMoreover, in system with equal volume-fraction of phases, the temporal increase in average radius of the entire microstructure and individual phases are largely identical with marginal deviation. \nOn the other hand, in $\\alpha 10$, significant disparity is noticed in the rate at which the major-phase grains evolve when compared to the minor-phase. \nFig.~\\ref{fig:duplex_validation} illustrates that the increase in the average radius of the entire duplex microstructure lies in between the growth exhibited by the individual phases.\nThe difference on the growth kinetics between the phases, exclusively in the duplex microstructure characterised by unequal volume-fraction, is due to the corresponding distribution of the phases. \nOwing to its reduced volume, the grains of the minor phase in $\\alpha 10$ microstructure are considerably separated when compared to major-phase grains. \nTherefore, the diffusion path, which the chemical components need to transverse to achieve grain growth, is longer, and more convoluted.\nConsequently, the growth rate exhibited by the minor-phase grains is significantly lower the grains of phase-$\\gamma$. \nSo far, no convincing argument has been made on how the evolution kinetics of the entire duplex microstructure relates to the different growth-rates adhered-to by the major- and minor-phase grains, and to that end, this becomes the primary focus of the current analysis. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.85\\textwidth]{duplex_power}\n    \\end{tabular}\n    \\caption{  Correlation coefficient characterising the proportionality between time and the average radius of individual phase-grains and entire microstructure, raised to different exponents,  for homogeneous and duplex systems $\\alpha50$ and $\\alpha10$.  \n    \\label{fig:duplex_power}}\n\\end{figure}\n\n\\subsubsection{Grain growth kinetics}\n\nEven though the microstructures illustrated in Fig.~\\ref{fig:duplex_validation} render a progressive increase in the average radius with time, owing to the difference in the governing mechanism, the exponent of the power law capturing the growth kinetics vary depending on the nature of the system~\\cite{fan1997computer,fan1997diffusion}. \nWhile the exponent $n=2$ characterises grain growth in homogeneous system, evolution kinetics of the individual phases, and the duplex microstructure as a whole, largely follow relation $\\bar{R}^3(t) \\propto t$.\nIn order to affirm that the evolution of the entire microstructure, and its corresponding phases, adhere to the power law,  the temporally varying average radius is raised to different exponents, $n=\\{1,2,3,4\\}$, and related to time. \nThe correlation coefficient (Pearson) characterising the relation between the average radius with the various exponent and time is ascertained, and graphically represented in Fig.~\\ref{fig:duplex_power}. \nIn this illustration, $<\\bar{R}^n,t>$ denotes the correlation coefficient between the average radius raised to order n and time. \nFig.~\\ref{fig:duplex_power} shows that, in homogeneous system ( $\\alpha 0$), maximum correlation is observed when $n=2$, thereby indicating that the grain growth in this microstructure adheres to the power law, $\\bar{R}^2(t) \\propto t$.\nFurthermore, correlation coefficient relating the average radius of the individual phases, and entire duplex microstructures, with time, is higher when $n=3$, which implies that the evolution in the multiphase systems complies to the established power law~\\cite{fan1997diffusion}. \nEven though it might appear that,  in Fig.~\\ref{fig:duplex_power}, for equifraction duplex system ($\\alpha 50$) , the maximum correlation is observed in $n=4$.\nHowever, given the marginal difference when compared to $n=3$, such consideration leads to \\lq overfitting\\rq \\thinspace, thus returning to $\\bar{R}^3(t) \\propto t$ as the statistically sound relation. \n\n\\subsubsection{Ascertaining governing factor}\n\nAs shown in Fig.~\\ref{fig:duplex_validation}, the change in the average radius of the entire microstructure with time, in duplex systems with unequal volume-fraction of phases, invariably lies  between curves representing the increase in the radius of the individual phase-grains. \nMoreover, existing works unravel that, in radius versus time plot, the relative position of the overall microstructure curve varies with respect to the phase-associated grains curves, as the phase-fraction changes~\\cite{fan1997computer,yadav2016effect}.\nWhile the effect of phase-fraction on the evolution of the individual phases are convincingly elucidated, its influences on the growth rate of entire duplex microstructure is yet to be sufficiently addressed. \nIn order to identify this influence, more particularly, to answer the resulting question on evolution kinetics of which phase, major or minor, principally effects the growth rate of entire duplex microstructure, each system is statistically analysed.\nStatistical programming language R is employed for these, and all other relevant, investigations in this work. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.9\\textwidth]{rdq_duplex}\n    \\end{tabular}\n    \\caption{  Coefficient of determination, quantifying the effect of  growth kinetics of individual phase-grains on the overall grain-growth rate of the entire system, is  estimated using Eqn.~\\eqref {coeff_deter} for different duplex microstructure with varying phase-fraction.\n    \\label{fig:rdq_duplex}}\n\\end{figure}\n\nFrom the dataset comprising of  temporally varying average radius, the growth rate exhibited by the individual phase-grains ($d\\bar{R}_{\\alpha}\/dt$ or $d\\bar{R}_{\\gamma}\/dt$) and entire microstructure ($d\\bar{R}\/dt$), at every instance $(t)$, is determined for each duplex system. \nSubsequently, by treating the growth rate of the individual phase-grains, and entire microstructure, as \\textit{response} and \\textit{predictor} variable, respectively, the corresponding kinetics are related to the each other.\nFrom the emerging relation, the \\textit{coefficient of determination}, $\\chi$, for the combination of an individual phase-grains and overall microstructure is estimated.\nScatter plots that illustrates the dependency of $d\\bar{R}_{\\alpha}\/dt$ or $d\\bar{R}_{\\gamma}\/dt$ and $d\\bar{R}\/dt$ are included in \\textcolor{red}{Appendix2}.\n\nFor a given duplex system, following the conventional description, the coefficient of deterioration considering the growth kinetics of minor phase-$\\alpha$ ($d\\bar{R}_{\\alpha}\/dt$) and entire microstructure ($d\\bar{R}\/dt$) is calculated by \n\\begin{align}\\label{coeff_deter}\n \\chi_{\\alpha}=\\frac{\\chi^{\\text{SST}}_{\\alpha}-\\chi^{\\text{SSE}}_{\\alpha}}{\\chi^{\\text{SST}}_{\\alpha}},\n\\end{align}\nwhere $\\chi^{\\text{SST}}_{\\alpha}$ is estimated by treating the instantaneous growth-rate of overall microstructure as univariate parameter, and summing-up the squares of the disparity (error) between the individual values and the mean.  \nOn the other hand, $\\chi^{\\text{SSE}}_{\\alpha}$ represents the sum of the squared differences between the datapoints and regression line relating the instantaneous kinetics of the $\\alpha$-phase grains and overall duplex-microstructure. \nBased on the description of the coefficient of determination in Eqn.~\\eqref{coeff_deter}, $\\chi_{\\alpha}$ can be viewed as a parameter that quantifies the effect of $\\alpha$-grains growth-kinetics on the evolution rate of entire microstructure.  \nTherefore, in addition to $\\chi_{\\alpha}$, the corresponding parameter that realises the influence of the major-phase growth-kinetics ($d\\bar{R}_{\\gamma}\/dt$) on the overall evolution rate, $\\chi_{\\gamma}$ is appropriately determined for all the different duplex microstructures considered in this investigation. \n\nCoefficients of determination separately quantifying the role of $d\\bar{R}_{\\alpha}\/dt$ and $d\\bar{R}_{\\gamma}\/dt$ in overall growth-rate exhibited by the microstructure, $\\chi_{\\alpha}$ and $\\chi_{\\gamma}$, is calculated for different duplex systems with varying phase-fractions and plotted in Fig.~\\ref{fig:rdq_duplex}. \nThe variation observed in the coefficients of determination, across the different duplex systems, unravels that the effect of individual phase-grains on the overall growth kinetics is primarily dependent on phase-fraction of the microstructure. \nIn a duplex system characterised equal volume-fraction of phases, identical coefficients of determination implies that both $\\alpha$- and $\\gamma$-grains similarly influence the evolution kinetics of the entire microstructure. \nOn the other hand, noticeable disparity between $\\chi_{\\alpha}$ and $\\chi_{\\gamma}$ is observed in duplex microstructures with varying volume-fraction of constituent phases. \nMoreover, Fig.~\\ref{fig:rdq_duplex} shows that this inequality in the coefficients of determination becomes more pronounced with increase in the difference  between the volume-fraction of the phases in duplex microstructure.\nWhile the coefficient of determination pertaining to major phase-$\\gamma$ exhibits a relatively marginal change, and continues to remain noticeably greater, $\\chi_{\\alpha}$ progressively decrease with reduction in the volume-fraction of the corresponding minor-phase grains. \nIn other words,  Fig.~\\ref{fig:rdq_duplex} unravels that, in duplex systems with unequal volume-fraction of phases, the overall growth rate of the entire microstructure ($d\\bar{R}\/dt$) is primarily influenced by the evolution kinetics of the major-phase grains ($d\\bar{R}_{\\gamma}\/dt$). \nFurthermore, it is evident from the illustration that the dominance of the major phase in effecting the overall growth kinetics becomes more definite with increase in the corresponding volume-fraction (or decrease the amount of minor phase).\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.9\\textwidth]{dupR_sep}\n    \\end{tabular}\n    \\caption{  Progressive increase in the average radius of phase-$\\alpha$ and -$\\gamma$ grains, $\\bar{R}_{\\alpha}^3$ and $\\bar{R}_{\\gamma}^3$,  with time for various duplex systems with characteristic phase-fraction.\n    \\label{fig:dupR_sep}}\n\\end{figure}\n\n\\subsubsection{Verifying the statistical claim}\n\nIn order to substantiate the understanding rendered by the analyses based on coefficient of determination, the temporal change in the average radius of the phase-associated grains ($\\bar{R}_{\\alpha}$ or $\\bar{R}_{\\gamma}$) and entire microstructure ($\\bar{R}$) are studied in a conventional manner.\nIn  Fig.~\\ref{fig:dupR_sep}, the progressive increase in the average radius of  major- and minor-phase grains with time, in duplex systems with varying phase-fractions, are cumulatively presented. \nSince the evolution of  different duplex microstructures are considered together, for the ease of distinction, temporal change in $\\bar{R}_{\\alpha}^3$ and $\\bar{R}_{\\gamma}^3$ is adopted for this illustration.\n\nConsistent with the mechanism of evolution, it is observed that the minor phase-grains in system with the minimal volume-fraction ($\\alpha 10$) grows at a least rate.\nHowever, the growth kinetics of the these grains noticeably increase as the corresponding phase gains more volume in the microstructure. \nAccordingly, in duplex systems with unequal phase-fractions, minor-phase grains of $\\alpha 40$ microstructure exhibits highest growth-rate, followed by $\\alpha 33$ and $\\alpha 16$. \nOn the other hand, the evolution kinetics of major-phase grains are minimal in  $\\alpha 40$ system, and significantly increase in $\\alpha 33$ and $\\alpha 16$ as the volume fraction of the phase-$\\alpha$ reduces. \nMoreover, the maximum growth-rate in $\\gamma-$phase grains is observed in microstructure with minimal volume of minor phase, $\\alpha 10$.\nOwing to the influence of volume fraction, which governs the kinetics through the diffusion paths, the disparity in the temporal change in average radius of the major- and minor-phase grains becomes more evident as the inequality in phase-fraction increases. \nIn other words, as shown in Fig.~\\ref{fig:dupR_sep}, the progressive change in $\\bar{R}_{\\alpha}$ and $\\bar{R}_{\\gamma}$ with time is notably far-apart in $\\alpha 10$ system when compared to the rest. \nNevertheless, this separation gets reduced with the increase in the volume-fraction of minor phase-$\\alpha$.\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.7\\textwidth]{dupR_conso}\n    \\end{tabular}\n    \\caption{ Temporal change in the average radius of entire duplex systems, $\\bar{R}^3$, with varying  phase-fractions during grain growth.\n    \\label{fig:dupR_conso}}\n\\end{figure}\n\nWhile Fig.~\\ref{fig:dupR_sep} can be appropriately discussed to affirm the consistency of the present approach, equipped with the understanding on how evolution of individual phase-grains relate to the growth rate of entire microstructure, the general trend in the overall growth kinetics can be predicted from it. \nAnalyses based on coefficient of determination, in Fig.~\\ref{fig:rdq_duplex}, unravels that the growth-rate of duplex microstructure is predominantly influenced by evolution kinetics of major-phase grains. \nMoreover, the effect of the minor-phase grains decreases as their corresponding volume-fraction reduces.\nAdopting these insights, and given that in Fig.~\\ref{fig:dupR_sep} $\\gamma$-grains of $\\alpha 10$ exhibit maximum growth rate, it can be predicted that the overall growth kinetics of the corresponding microstructure will be noticeably greater than the other duplex systems considered in this study. \nFurthermore, it can also be stated that, since the volume fraction of minor-phase continues to be significantly lower in $\\alpha 16$ and  $\\alpha 33$, the overall growth will be dominated by the $\\gamma$-phase grains, and their kinetics will correspondingly follow the $\\alpha 10$ microstructure. \nFinally, considering that the volume of phase-$\\alpha$ is close to major-phase in $\\alpha 40$,   based on Fig.~\\ref{fig:rdq_duplex}, it can be suggested that this duplex microstructure will exhibit the least rate of evolution.\n\nIn order to verify the accuracy of the above predictions, emerging from the understanding of coefficient of determination, the overall growth-rate exhibited by different duplex microstructures is cumulatively presented in Fig.~\\ref{fig:dupR_conso}.\nIt complete adherence to the prediction, it is observed that, in duplex microstructures with unequal phase-fraction, maximum and minimum growth-rate respectively pertains to $\\alpha 10$ and $\\alpha 40$ microstructure. \nAdditionally, the kinetics of evolution exhibited by $\\alpha 16$ and  $\\alpha 33$ lie in between the maximum and minimum, with the former noticeably greater than the later. \nUltimately, Fig.~\\ref{fig:dupR_conso} affirms that, in duplex systems characterised by unequal volume-fraction of constituent phases, the overall grain-growth kinetics is primarily governed by the evolution-rate of the major-phase grains.\nThis influence of the major-phase grains gets increasingly dominant with increase in its volume-fraction. \n\n\\subsubsection{Phase fraction and growth rate}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.7\\textwidth]{duplexK}\n    \\end{tabular}\n    \\caption{  Change in the kinetics coefficients that governs grain growth with the variation in the phase-fractions of evolving duplex microstructures. \n    \\label{fig:duplexK}}\n\\end{figure}\n\nHaving realised that, for a given duplex microstructure, the overall grain-growth rate ($d\\bar{R}\/dt$) is predominantly dictated by the kinetics adhere-to by the major-phase grains ($d\\bar{R}_{\\gamma}\/dt$), attempts are made to relate the varying phase-fraction to the observed growth-rate across different systems. \nIntroducing chemical-inhomogeneity amongst the grains, in a polycrystalline system, shifts the governing mechanism, irrespective of the volume-fraction of the phases. \nKinetics rendered by this mechanism in multiphase microstructure complies with the power-law characterised by the exponent $n=3$.\nConsidering that the exponent remains unaltered reflecting the governing mechanism, despite the varying phase-fraction in multiphase systems, the disparity in the rate of grain growth can only be understood from the kinetic coefficient, $k$, that relates the average radius to time. \n\nBy studying the evolution of different duplex systems considered in this study, the corresponding kinetic coefficients are ascertained. \nKinetic coefficients associated with duplex microstructures with varying phase-fractions are illustrated in Fig.~\\ref{fig:duplexK}. \nSince the growth rate of the duplex microstructure are principally dictated by the evolution of major-phase grains, the corresponding volume-fraction is considered for this representation. \nFig.~\\ref{fig:duplexK} indicates that with increase in the volume of a major phase in duplex system, grain growth in the system occurs at an higher rate. \nIn order definitively understand the influence of the $\\gamma$-fraction on the kinetic coefficient exhibited by the corresponding duplex microstructure, non-linear regression technique is adopted and a relation expressed as \n\\begin{align}\\label{Dpf_k}\nk_{\\text{dup}} = A_{\\text{dup}} + B_{\\text{dup}}\\exp(C_{\\text{dup}}\\dot V_{\\gamma}),\n\\end{align}\nis realised, where $V_{\\gamma}$ is the volume fraction of the major phase-$\\gamma$.\nThe constants $A_{\\text{dup}}$, $B_{\\text{dup}}$ and $C_{\\text{dup}}$, for the present consideration, respectively assume the value of $537.3$, $3.8\\times 10^{-3}$, and $0.13$. \nIn the above relation, it is vital to note that $k_{\\text{dup}}$ indicates the kinetic coefficient adhered-to by the entire duplex system during grain growth, not the evolving major- or minor-phases. \n\n\\subsection{Triplex microstructures}\n\nTriplex systems are characterised by the association of  individual grains, in the polycrystalline setup, to one of the three constituent phases. \nCorresponding microstructure, in this study, comprises of phases $\\alpha$, $\\beta$ and $\\gamma$, with $\\gamma$ largely acting as the matrix or major-phase. \nIn the existing works, unlike duplex systems, very few three-phase microstructures with varying phase-fractions are analysed~\\cite{ravash2017three2}.\nThis limited consideration of triplex microstructure can largely be attributed to the computational burden associated with it. \nMoreover, conventionally, the grain-growth kinetics of the triplex microstructure are discussed by focusing on the evolution of the individual phase-grains without sufficiently relating it the overall system.\nIn the present study, on the other hand, grain growth in fourteen different three-phase microstructures, with varying phase-fractions, are examined to elucidate with statistical certainty how the evolution of the individual phase-grains effects the growth kinetics of entire triplex microstructure. \n\n\\subsubsection{Grain growth kinetics}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.75\\textwidth]{triplex_power}\n    \\end{tabular}\n    \\caption{  Correlation coefficient characterising the relation between time and average radius of individual phase-grains and entire microstructure, raised to different power ($n=1,2,3$ and $4$), for two triplex systems, $\\alpha39 \\beta22$ and $\\alpha33\\beta33$. \n    \\label{fig:triplex_power}}\n\\end{figure}\n\nDespite the difference in the number of phases, grain growth in both duplex and triplex systems are fundamentally governed by the same mechanism. \nTherefore, the grain growth in three-phase systems, which is dictated by the diffusion of chemical components, adheres to the power law with the exponent, $n=3$. \nIn order to ensure that the grain growth in triplex systems is accurately modelled by the present approach, a validation technique adopted for duplex microstructures, is extended. \nCorrespondingly, the temporally-varying average radius of the individual phase-grains, $\\bar{R}_{\\alpha}$, $\\bar{R}_{\\beta}$ and  $\\bar{R}_{\\gamma}$, and overall microstructure, $\\bar{R}$, are raised to different powers ($n=\\{1,2,3,4\\}$) and related to time, $t$.\nCorrelation coefficient characterising the different relations are ascertained for two triplex microstructures, $\\alpha33\\beta33$ and $\\alpha39\\beta22$,  and are graphically illustrated in Fig.~\\ref{fig:triplex_power}. \n\nIt is evident in Fig.~\\ref{fig:triplex_power} that correlation coefficient relating the average radius to the time is highest when $n=3$ for both individual phases, and overall microstructure.\nThe maximum correlation exhibited by the cube of the different average radii, $\\bar{R}_{\\alpha}$, $\\bar{R}_{\\beta}$,  $\\bar{R}_{\\gamma}$ and $\\bar{R}$, with time implies that the growth of the individual phase-grains, and the entire microstructure, are predominantly governed by the long-range diffusion of the chemical components. \n\n\\subsubsection{Ascertaining governing factor}\n\nConsidering that grain growth in both duplex and triplex system are predominantly dictated by the diffusion of the chemical components, phase-fraction renders identical influence on the evolution of individual phases. \nIn other words, when certain phase(s) assumes minor volume-fraction in the three-phase microstructure, owing to relative increase in the length, and complexity, of the diffusion path, the growth of the corresponding grains are stunted. \nOn the other hand, the evolution kinetics is enhanced when the volume of the phase(s) is dominant in the multiphase systems. \nApart from these generalised understanding, existing report rarely offer any further insights on the grain-growth kinetics in triplex microstructures.\nParticularly, similar to duplex system, sufficient consideration has not been rendered to relate the growth kinetics of the individual phases to the evolution of the entire triplex system. \nTo that end, in this analysis, the impact of the growth rate of  individual phase-grains on that entire three-phase microstructure is examined by ascertaining the corresponding coefficient of determination. \n\nThe instantaneous growth rate for constituent phase-grains, $d\\bar{R}_{\\alpha}\/dt$, $d\\bar{R}_{\\beta}\/dt$ and $d\\bar{R}_{\\gamma}\/dt$, along with the entire triplex microstructure, $d\\bar{R}\/dt$, is determined by monitoring temporal change in the respective parameter. \nThese instantaneous growth kinetics of the individual phase-grains are related to that of the entire microstructure, and the corresponding coefficient of determination is estimated through the Eqn.~\\eqref{coeff_deter}.\nFor each system, three distinct coefficients of determination, $\\chi_{alpha}$,  $\\chi_{beta}$ and $\\chi_{gamma}$, are estimated, reflecting the characteristic feature of the triplex microstructure. \nThese coefficients of determinations are related to the phase-fractions of the microstructure, and illustrated in  Fig.~\\ref{fig:triALL_Rsq}. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{triALL_Rsq}\n    \\end{tabular}\n    \\caption{  Change in coefficient of determination, typifying the influence of growth rate of individual phase-grains on the evolution kinetics of entire microstructure, with variation in the volume fraction of constituent phases.\n    \\label{fig:triALL_Rsq}}\n\\end{figure}\n\nIn the triplex systems considered in the present study, the volume-fraction of phase-$\\gamma$ reaches as low as 33\\% despite being the major phase. \nSuch volume-fraction of phase-$\\gamma$ is noticed in triplex microstructure characterised by equifraction of phases. \nFurthermore, in some system like $\\alpha 45 \\beta 10$, the phase-$\\alpha$ assumes a volume-fraction of $45\\%$, in spite of being one of the minor phases.\nThis, and similar, understanding of phase-fraction is vital to investigate the coefficients of determination presented in Fig.~\\ref{fig:triALL_Rsq}. \n\nFig.~\\ref{fig:triALL_Rsq} unravels that, even though the volume-fraction of the major-phase varies noticeably across different triplex systems, the corresponding coefficient of determination, $\\chi_{gamma}$, continues to remain the high. \nGiven that phase-$\\gamma$ stays as a major-phase, despite the change in phase-fractions, the high values of $\\chi_{gamma}$ can be attributed to the dominant volume of the respective grains.\nIn other words, analogous to the duplex microstructure, the evolution kinetics of major-phase grains offer relatively greater influence on the overall growth-rate exhibited by the entire triplex microstructure. \nFurthermore, Fig.~\\ref{fig:triALL_Rsq} suggests that  the coefficient of determination of the minor-phases, $\\alpha$ and $\\beta$, noticeably increases as their corresponding volume-fraction raises. \nParticularly, as the volume of phase-$\\alpha$ gets as dominant as $\\gamma$, in a triplex system, identical coefficient of determination is rendered, $\\chi_{alpha} = \\chi_{gamma}$.\nOn the other hand, when the volume of the phases are minimal, the respective coefficient of determination assumes least value. \nUltimately, it is evident from Fig.~\\ref{fig:triALL_Rsq} that the influence of the individual phase-grains on the overall growth kinetics depends largely on the corresponding volume-fraction. \nIn a triplex system with unequal volume-fraction of phases, the growth rate of the entire microstructure, $d\\bar{R}\/dt$, is predominantly governed by the evolution of the major-phase grains, $d\\bar{R}_{\\gamma}\/dt$.\nWhen volume of two phases are dominant in a three-phase system, the growth rate of both these phase-grains offer identical influence on evolution of the microstructure. \nThe contribution of a given phase-grains to the overall evolution kinetics, $d\\bar{R}\/dt$, becomes least, when its volume-fraction are minimal. \nBased on the understanding rendered by Fig.~\\ref{fig:triALL_Rsq},  as demonstrated for duplex systems (Fig.~\\ref{fig:dupR_conso}), the growth-kinetics of a triplex microstructure, in relation to others with varying phase-fractions, can be predicted from the temporal change in the average radius of the corresponding phase-associated grains. \n\n\\subsubsection{Interdependency in the evolving phases}\n\nIn duplex systems, since the grains of the polycrystalline microstructure are associated with either of the two  constituent phases, the evolution of a particular phase-grains, and its kinetics, are inherently bound to the other. \nHowever, the same interdependency cannot be expected in triplex systems, wherein the grains can be associated with one of the three possible phases. \nMoreover, in three-phase microstructures, level of influence offered by one evolving phase-grains on the rest of the phase-associated grains has not been conscientiously addressed yet.\nTherefore, by examining the temporal change in the average radius of a particular phase-grains in relation to the others, the interdependency exhibited between the phases, in triplex microstructures, during grain growth is elucidated. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{rate_corr}\n    \\end{tabular}\n    \\caption{  Correlation coefficient explicating the interdependency between the growth rate of different phase-grains, $d\\bar{R}_{\\alpha}\/dt$, $d\\bar{R}_{\\beta}\/dt$ and $d\\bar{R}_{\\gamma}\/dt$, during grain growth of four different triplex system with varying phase-fractions. \n    \\label{fig:rate_corr}}\n\\end{figure}\n\nInstead investigating all the fourteen triplex microstructure to explicate the effect of one evolving phases on the other, the systems are categorised based on phase-fraction, and one microstructure from each category is analysed. \nSince the coefficient of determination, which quantifies the effect of  an evolving-phase grains on the growth of the overall triplex microstructure, depends on volume fraction, the phase-fraction based grouping is deemed reasonable. \nApart from the equifraction system, $\\alpha 33 \\beta 33$, wherein all the constituent phases largely occupy similar volume, the remaining systems can be categorised as \\lq equi-major \\rq \\thinspace, \\lq equi-minor \\rq \\thinspace and \\lq non-equifraction \\rq \\thinspace triplex microstructures. \nWhile, in equi-major system, volume of one of the minor phase is equal to that of the major-phase  ($V_\\alpha= V_\\gamma$), the volume fraction of minor phases are identical in equi-minor microstructures ($V_\\alpha= V_\\beta$).\nMoreover, the non-equifraction system stand in direct contrast to the equifraction microstructure, and characterised by totally unequal volume-fraction of the constituent phases ($V_\\alpha\\neq V_\\beta \\neq V_\\gamma$). \n\nIn order understand the degree of interdependency between the evolving phases, in addition to equifraction microstructure, grain growth exhibited systems  $\\alpha 36 \\beta 22$, $\\alpha 45 \\beta 10$ and $\\alpha08\\beta08$ pertaining to non-equifraction, equi-major and equi-minor, respectively, are analysed.\nGiven that the primary focus of  the present investigation is \\textit{not} to quantify the effect of an evolving-phase grains on the rest, but rather to qualitatively realise the degree of interaction between two phases, during grain growth in a triplex systems, coefficient of determination is not estimated. \nHowever, alternatively, the growth rate of different phase-associated grains are estimated, $d\\bar{R}_{\\alpha}\/dt$, $d\\bar{R}_{\\beta}\/dt$ and $d\\bar{R}_{\\gamma}\/dt$, and are related to  (or plotted against) each other.\nCorrelation coefficient characterising the relation between the growth rate of two phase-grains are realised for pre-determined triplex microstructures, and graphically illustrated in Fig.~\\ref{fig:rate_corr}. \n\nBefore elucidating level of interaction between two evolving phase-grain in a given triplex microstructure, based on Fig.~\\ref{fig:rate_corr}, it is exceedingly critical to realise the variation in the range of correlation coefficient across the different systems. \nParticularly, as opposed to the maximum value, which remains constant at unity, the least value of the correlation coefficient changes with phase-fraction. \nCorrespondingly, Fig.~\\ref{fig:rate_corr} unravels that the lowest correlation-coefficient in the equifraction system is maximum ($0.84$) when compared to the rest of the triplex microstructures, and it is respectively followed by non-equifraction ($\\alpha 36 \\beta 22$) and equi-major ($\\alpha 45 \\beta 10$) microstructures, with the absolute minimal exhibited by the equi-minor system ($\\alpha08\\beta08$). \nThis significant disparity in the lowest value of correlation coefficient emphasis the importance of considering the context, $i.e$ the correlation-coefficient range, while interpreting the interaction between two-phases in a given microstructure during grain growth.\nIn other words, apparently least interdependency between the two-phase grains in a equifraction system would translate to a strong interaction in the context of equi-minor triplex microstructure. \n\nThe graphical representation of correlation coefficient, in Fig.~\\ref{fig:rate_corr}, that indicates level of influence the growth rate of one evolving phase-grain has on the other,  during grain growth in triplex microstructures, unravels few similarities and dissimilarities across the systems with varying phase-fractions. \nIt is evident in this illustration that, irrespective of the nature of the three-phase system, the correlation coefficient relating the growth rate of $\\alpha$- and $\\beta$-phase grains are minimal, within a given microstructure. \nIn other words, during grain growth in a triplex system, the evolution kinetics of one minor-phase grains imposes the least effect on growth rate of other low-volume phase-grains.\nDespite being equifraction, largely owing the manner in which the triplex microstructure is initialised, such effect is also observed in $\\alpha 33 \\beta 33$ microstructure.\nHowever, it is vital to note that the least correlation in equifraction system is tantamount to noticeable interaction in relation to other triplex microstructures. \nTherefore, in  $\\alpha 33 \\beta 33$ system, the evolution kinetics of one-phase grains is generally interlinked with grains of the other phases, but this interaction is least between phase-$\\alpha$ and -$\\beta$.\n\nWithin a triplex system, during grain growth, Fig.~\\ref{fig:rate_corr} suggests that minimal interdependency following two minor-phases is observed between the minor-phase and the matrix grains, irrespective of the phase-fractions. \nThe only exception is the equi-minor system wherein the volume-fraction of the minor-phases are identical. \nFurthermore, in all triplex systems, the growth rate of minor-phase grains with relatively greater volume-fraction when compared to the other ($V_{\\alpha} > V_{\\beta}$ ) is strongly coupled with the evolution of the major-phase grains. \nUltimately, the study of interdependency between the rate of the evolving phase-grains during grain growth in triplex systems, using correlation coefficients, unravel that a general trend is observed in three-phase microstructures irrespective of the phase-fractions. \nIf we distinguish the constituent phases as minor- , inter- and major-phase depending on their corresponding volume fraction, which in the present study is $\\alpha$,$\\beta$ and $\\gamma$ respectively, then the least interaction during grain growth is exhibited by the minor- and inter-phase grains.\nWhile the growth rate of major-phase grains are considerably interlocked with the inter grains, the effect of the minor phase on the matrix is comparatively lower.\nIn other words, in a given triplex microstructure, the level of influence offered by the evolution rate of one phase-grains on the other, during grain growth, is primarily dictated by the their corresponding volume fractions. \nWhen the volume fraction of two phases are minimal, their degree of interaction is also minimal, whereas a considerable dependency is noticed when the volume of the two phases are dominant in view of the third. \n\nThis analysis on the interdependency of the kinetics of evolving phases in triplex system, during grain growth, unravels that, for expressing the overall growth rate of a three-phase microstructure, the two ideal variables, with least \\textit{multicollinearity}, are the evolution rate of the grains of minor phases. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.9\\textwidth]{triALL_fitting}\n    \\end{tabular}\n    \\caption{  Effect of phase-fraction on the kinetic coefficient governing the rate of grain growth in triplex systems.\n    \\label{fig:triALL_fitting}}\n\\end{figure}\n\n\\subsubsection{Phase fractions and growth rate}\n\nA principal insight rendered by the present study is that, in grain growth, the evolution rate of a multiphase system is primarily governed by the volume-fraction of its constituent phases through their respective growth kinetics. \nTherefore, by relating the phase-fraction of  fourteen different triplex microstructures, considered in this work, to its corresponding growth kinetics, an attempt to extract a generalised expression. \nUnlike in duplex system, since triplex microstructures are characterised by three constituent phases,  the corresponding evolution rate is dictated by two independent variables, $i.e$ volume fractions. \nBy directly relating the volume fraction, instead of the evolution rate of individual phase-grains, to the growth kinetics exhibited by the triplex systems, the question of multicollinearity is obviated.\n\nBased on the proportionality exhibited by the phase-fraction and growth kinetics in duplex microstructure, multiple non-linear regression is employed to relate the volume-fraction of constituent phases to the corresponding grain growth-rate exhibited by the entire three-phase microstructures. \nStrictly owing to the different phase-fractions considered in the present analysis, volume fraction phase-$\\alpha$ and -$\\gamma$ are considered as the independent variables. \nExpression rendered by multiple non-linear regression that relates phase-fractions to the kinetic coefficient governing the grain-growth rate of  entire three-phase microstructure reads\n\\begin{align}\\label{Tpf_k}\n k_{\\text{tri}}=A_{\\text{tri}} + B_{\\text{tri}} \\exp(C_{\\text{tri}}^{\\alpha}V_{\\alpha}+C_{\\text{tri}}^{\\gamma}V_{\\gamma}),\n\\end{align}\nwhere $V_{\\alpha}$ and $V_{\\gamma}$ correspond to the volume fraction of phase-$\\alpha$ and -$\\gamma$.\nMoreover, $A_{\\text{tri}}$, $B_{\\text{tri}}$, $C_{\\text{tri}}^{\\alpha}$ and $C_{\\text{tri}}^{\\gamma}$ are constants whose respective values are $268.2$, $2.2 \\times 10^{-7}$, $0.19$ and $0.22$. \nIn Fig.~\\ref{fig:triALL_fitting} kinetic coefficient adhered to different triplex microstructures during grain growth is plotted against the corresponding volume fraction of phase-$\\alpha$ and -$\\gamma$.\nThe curve reflecting Eqn.~\\eqref{Tpf_k} is included in this illustration.\nNoticeably good agreement between the numerical results and the curve indicates that Eqn.~\\eqref{Tpf_k} convincingly relates the kinetic coefficients characterising the grain growth in isotropic triplex systems to the phase-fractions.  \n\n\\section{Conclusion}\n\nGrain growth in polycrystalline systems can be desirous under certain conditions, while unwelcomed in others. \nFor instance, grain growth is induced during processing technique to establish required average grain-size, while noticeable measures are generally taken to avoid it during a given application. \nThe subjective role of grain growth extends beyond homogeneous polycrystalline system to multiphase microstructures as well. \nTherefore, it becomes vital to understand the grain-growth kinetics exhibited by highly-applicable multiphase polycrystalline microstructures associated with duplex and triplex systems. \nParticularly, generalised insights that aide in comprehending the growth rate of multiphase microstructures with varying phase-fraction are exceedingly critical, as they can be adopted for wide-range of systems and application. \nTo that end, in this study, the grain-growth kinetics of duplex and triplex systems are studied by employing approachable statistical techniques. \n\nConventionally, the grain-growth kinetics associated with multiphase systems are discussed by considering the evolution rate of individual phase-grains and entire-microstructure separately. \nSuch treatments rarely offer much insights on how the growth kinetics of individual phases relate to the overall evolution rate exhibited by the entire system. \nTherefore, in the present work, statistical tools are employed to realise the effect of growth rate of a given phase-associated grains on the overall kinetics of evolving microstructure. \nCorrespondingly, it is unraveled that, during grain growth in multiphase systems, the influence of the growth kinetics of a phase-grains on the evolution of entire system depends on its volume fraction. \nIn other words, in systems with varying volume fraction of constituent phases, the evolution kinetics of the major-phase grains predominantly govern the overall growth rate exhibited the microstructure. \nEven though, when focusing on a specific system, it might appear that the temporal change in the average radius of the entire multiphase system lies close the evolution of the corresponding radius of the minor-phase grains, when viewed in relation to microstructures of varying phase-fractions, the dominance of growth kinetics of the major-phase grains gets increasingly evident. \nThis holistic understanding that encompasses multiphase systems with varying phase-fractions, though arrived from  statistical treatments, in the present work, it is vindicated through conventional representation and corresponding discussions.\nMoreover, in addition to effecting the role of individual phase-grains on the evolution kinetics of entire multiphase system, phase fraction also dictates the interdependency between growth rates of different phase-grains. \nDuring grain growth, evolution kinetics of grains of two phases are largely independent when their corresponding volume-fractions in the multiphase system are minimal.\nOn the other hand, noticeable interaction is observed in the grain-growth rate of two phases that dominant the multiphase microstructure. \n\nUnderstanding rendered by the present investigation can be exploited for various purposes, however, one critical utilisation would be to alter the grain-growth rate of a given multiphase system, with a definite phase-fraction, by appropriately, and exclusively, varying the evolution kinetics of the major-phase grains. \nTo that end, in the upcoming works, attempts will be made to substantiate the approach of modifying the grain-growth rate in multiphase systems by employing the dominant influence of the major-phase grains.\n\n\\section*{Appendices}\n\n\\subsection*{Appendix 1: Interdependency in duplex systems}\\label{sec:app1}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{app1}\n    \\end{tabular}\n    \\caption{  The average radius of phase-$\\alpha$ grains, at a given instance, is plotted against the corresponding radius of matrix-phase grains for duplex microstructure with equal volume-fraction of phases. In the subplots, for the same system, the growth rate of individual phase-grains are related, and instantaneous average radius of phase-$\\alpha$ and -$\\gamma$ grains are plotted for duplex microstructure with $10\\%$ alpha.\n    \\label{fig:app1}}\n\\end{figure}\n\nIn a polycrystalline system, irrespective of its nature, the continuum is established by the multiple grains present it. \nDuring the grain growth, despite the continual disappearance of the grains, the continuum is sustained by the growth the surviving grains. \nThis characteristic feature of the grain growth introduces interdependency between the evolving grains. \nCorrespondingly, in duplex systems, wherein the grains are associated with one of the two-constituent phases, the evolution of phase-$\\alpha$ grains are inherently linked the grains of phase-$\\gamma$. \nEven though the interaction between the phase-associated grains, during grain growth of a duplex system, can be theoretically conceived, to explicate it with a statistical certainty, the average radius of $\\alpha$-grains, at a given time $t$, is plotted against corresponding of radius of $\\gamma$ grains in Fig.~\\ref{fig:app1} for two-phase microstructure with equal volume-fraction of phases, $\\alpha 50$.\nThe trend in this illustration indicates a inherent interlocking between the evolution of the phase-$\\alpha$ and -$\\gamma$ grains during the grain growth of equifraction duplex-microstructure.\nIn addition to the $\\alpha 50$ microstructure, the average radius of constituent phase-grains, at a given instance, are ascertained for duplex system with $10\\%$ minor phase-$\\alpha$.\nSimilar to equifraction system, these instantaneous average radii of phase-$\\alpha$ and -$\\gamma$ grains are plotted against each other, and illustrated in Fig.~\\ref{fig:app1} as a subplot.\nDespite the change in the phase-fraction, in $\\alpha 10$microstructure as well, a definite interaction between the radius of major- and minor-phase grains is evident. \n\nBesides the average radius of the phase-$\\alpha$ and -$\\gamma$ at a given instance, using the same approach, the relation between the kinetics of grain growth associated with these phases can be explicated. \nCorrespondingly, the evolution kinetics of the $\\alpha$-grains are related to that of the $\\gamma$ ones, for equifraction duplex microstructure, and are included as a subplot in Fig.~\\ref{fig:app1}. \nThis illustration unravels that even though there exists a perceivable interdependency between the instantaneous kinetics of phase-$\\alpha$ and -$\\gamma$ grains, it is not as straightforward as the average radius\n \n\\subsection*{Appendix 2: Effect of individual phase-grains kinetics on growth-rate of entire system}\\label{sec:app1}\n \n \\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{app2conso}\n    \\end{tabular}\n    \\caption{  The instances grain-growth rate exhibited by an individual phase-grains are plotted with respect to that of the entire duplex microstructure with equal volume fraction of phases. In (a) the kinetics of $\\alpha$-grains are related to the grain-growth rate of the entire microstructure, while $\\gamma$-grains evolution kinetics are considered (b). The corresponding outcomes for $\\alpha 10$ microstructure with $10\\%$ of minor phase-$\\alpha$ are included as subplots. \n    \\label{fig:app2conso}}\n\\end{figure}\n\nOne of the primary aim of the present investigation is to realise the effect of individual phases on grain-growth rate of entire two-phase microstructure. \nParticularly, the role of evolution kinetics of a given phase-grains on the overall growth rate of duplex system. \nTo that end, the kinetics of evolution exhibited by phase-$\\alpha$ grains, at a given instance, is related to the overall growth rate of equifraction duplex-system, and plotted in Fig.~\\ref{fig:app2conso}a. \nThe graphical illustration relating the evolution rate of  phase-$\\alpha$ grains and duplex microstructure with $10\\%$ minor phase is included as a subplot. \n\nFig.~\\ref{fig:app2conso}a unravels that the growth rate of phase-$\\alpha$ grains imposes a definite influence on the overall kinetics exhibited by the equifraction-duplex system, $\\alpha 50$, during grain growth. \nOn the other hand, the subplot of the corresponding illustration, which pertains to two-phase microstructure with $10\\%$ of phase-$\\alpha$, indicates a relation between $d\\bar{R}_{\\alpha}\/dt$ and $d\\bar{R}\/dt$, it is not as definite as one noticed in the equifraction microstructure.  \nIt is the degree of inter-relation between the kinetics of individual phase-grains, and overall growth-rate of a given duplex microstructure, which varies with the phase-fraction, is realised by \\textit{coefficient of determination}. \nIn other words, while the coefficient of determination relating the  $d\\bar{R}_{\\alpha}\/dt$ and $d\\bar{R}\/dt$ will be higher for equifraction duplex microstructure, in $\\alpha 10$ system it will assume a relatively low value reflecting a not so definite relation between the kinetics.\n\nIn Fig.~\\ref{fig:app2conso}b the grain-growth kinetics of phase-$\\gamma$ grains and entire duplex system, with equal volume-fraction of phases, is plotted against each other. \nAs a subplot, the corresponding relation between $d\\bar{R}_{\\gamma}\/dt$ and $d\\bar{R}\/dt$ for $\\alpha 10$ duplex microstructure with $10\\%$ of phase-$\\alpha$ is illustrated. \nUnlike the influence of phase-$\\alpha$ on overall growth-kinetics in Fig.~\\ref{fig:app2conso}a, a highly definite relation is observed between the evolution rate of phase-$\\gamma$ grains and entire microstructure in both equifraction and $\\alpha 10$ duplex microstructure. \nConsequently, the corresponding values of coefficient of determination will largely be independent of the phase-fraction.\n\n \n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe quest for the QCD chiral critical end point (CEP) in the phase diagram, \ntogether with the nature of the phase transition between hadron matter and quark \ngluon plasma (QGP), are open questions that have attracted the attention of the \nphysical community for some years \\cite{Halasz:1998qr}. Remarkable theoretical \nand experimental efforts \\cite{Brambilla:2014jmp} are being made to unveil the \nrich details of the QCD phase structures \\cite{Gupta:2011wh}. Experimentally, \none of the main goals of the heavy ion collision (HIC) program is the possible \nexistence and location of the CEP on the QCD phase diagram, with great \ndevelopments over the last years \n\\cite{Abelev:2009bw,Aggarwal:2010cw,Adamczyk:2013dal,Aduszkiewicz:2015jna}. \n\nIn relativistic HIC, the measurement of fluctuations of conserved \nquantities, such as baryon, electric charge, and strangeness number, play a \nmajor role in the experimental search for the CEP. Indeed, experimental \nmeasurements of cumulants of net-proton (proxy for net-baryon), net-charge, and \nnet-kaon (proxy for net-strangeness) are expected to carry significant amounts \nof information on the medium created by the collision (for a review, see \n\\cite{Friman:2011pf,Asakawa:2015ybt,Braun-Munzinger:2015hba,Luo:2017faz}).\nFluctuations are studied by measuring event-by-event fluctuations: a given \nobservable is measured on an event-by-event basis and its fluctuations are \nstudied for the ensemble of events \\cite{Braun-Munzinger:2015hba}.\n\nParticularly relevant are the cumulants of the net-baryon number because a \nsecond-order phase transition occurs at the CEP, resulting in divergences of\ncorrelation lengths for a static system of infinite size. \nThe cumulants of the baryon number thus diverge at the CEP \n\\cite{Stephanov:1998dy,Stephanov:1999zu}.\nThe study of the kurtosis \\cite{Stephanov:2011pb} and the skewness \n\\cite{Asakawa:2009aj} for the net-baryon number fluctuation distributions is \nessential as they are related to higher-order cumulants that can be extracted \nfrom event-by-event fluctuations in HIC experiments. \nOnce they are constituted by ratios of cumulants they are independent of the \nvolume of the system.\n\nThe study of fluctuations of conserved charges (baryon number, electric charge, \nand strangeness) at finite temperature and density has been done by using the \n(2+1) flavor Nambu$-$Jona-Lasinio (NJL) model in \n\\cite{Luo:2017faz,Chen:2015dra,Fan:2016ovc,Fan:2017kym}. \nBy using the (2+1) Polyakov$-$Nambu$-$Jona-Lasinio (PNJL) model, these \nfluctuations were investigated at finite temperature in \n\\cite{Fu:2009wy,Fu:2010ay,Bhattacharyya:2010ef,Bhattacharyya:2014uxa,Shao:2017yzv} \nand at finite temperature and density in \n\\cite{Fu:2010ay,Shao:2017yzv,Liu:2017qyc}.\n\nOther models have been employed to study higher-order baryon number \nsusceptibilities at finite temperature and density like the Polyakov-loop \nextended quark-meson model \\cite{Almasi:2017bhq}, where the influence of \nrepulsive vector-interactions on this fluctuations was also analyzed, the \nhybrid quark-meson-nucleon model \\cite{Marczenko:2017huu}, or the  SU(3) flavor \nparity-doublet quark-hadron model \\cite{Mukherjee:2016nhb} where the \nhigher-order baryon number susceptibilities near the chiral and the nuclear \nliquid-gas transitions were investigated.\n\nThe eventual location of the CEP can be affected by several conditions such as \nthe presence of external magnetic fields or the strangeness and isospin content \nof the medium \n\\cite{Costa:2013zca,Costa:2015bza,Rechenberger:2016gim}. \nThe study of the CEP location has been undertaken using different versions of \nthe NJL and PNJL models. In particular, it was shown that the presence of \nrepulsive vector interactions affects strongly the position of the CEP. The \nrole played by them were analyzed in detail in \n\\cite{Fukushima:2008wg,Costa:2015bza}. \nThe calibration of these models at high densities requires the existence of \nexperimental data or neutron star observables.\nParticularly relevant is the introduction of repulsive interactions, namely the\nvector-isoscalar terms, that seems to be necessary to describe $2M_\\odot$ \nhybrid stars \\cite{Pereira:2016dfg}.\n\nThe chiral restoration of strange quarks may play an important role \ninside neutron stars. \nIn particular, if this transition occurs at densities that can be found inside \ncompact stars, pure quark matter \\cite{Pereira:2016dfg}, or, exotic quark \nphases such as the color-flavor-locked (CFL) phase could be realized in their \ninterior \\cite{Alford:2001zr}. \nBesides, a phase transition could also have an important effect on the \nmean-free path of neutrinos in a protoneutron star as discussed in \n\\cite{Gulminelli:2013qr}. The cooling of protoneutron stars during the first \nseconds is essentially driven by the neutrinos that diffuse out of the star. \nA phase transition would give rise to a opalescence like phenomena reducing a \nlot the neutrino mean-free path, and, therefore, allowing for a much larger \ninteraction of neutrinos with matter.\n\n\nIn this work, we study the phase diagram using the (2+1)-flavor \nNambu--Jona-Lasinio model coupled to the Polyakov loop, designated as PNJL \nmodel, from the point of view of the kurtosis and skewness of net-baryon number \nfluctuations. \nIt is expected that in HIC the fireball evolves along isentropes, lines with \nconstant entropy per baryon, and, therefore, we analyze how these quantities, \nas well as the velocity of sound, behave along isentropes. \nOur main objective is to identify the similarities and differences of a QCD \nphase diagram which has a CEP or not, namely when a \nsufficiently strong repulsive vector interaction is taken into account.\n\nThe model is succinctly reviewed in Sec. \\ref{sec:model}, while the results are \ndiscussed in Sec. \\ref{sec:Results}.\nFinally we draw our conclusions in Sec. \\ref{sec:conclusions}. \n\n\\section{Model and formalism}\n\\label{sec:model}\n\nThe Lagrangian density for the Polyakov extended Nambu--Jona-Lasinio (PNJL) \nmodel reads\n\\begin{eqnarray}\n{\\cal L} &=& {\\bar{q}} \\left[i\\gamma_\\mu D^{\\mu}-\n\t{\\hat m}_c \\right ] q ~+~ {\\cal L}_\\text{sym}~+~{\\cal L}_\\text{det}~\n  +~{\\cal L}_\\text{vec} \\nonumber\\\\\n&+& \\mathcal{U}\\left(\\Phi,\\bar\\Phi;T\\right),\n\t\\label{Pnjl}\n\\end{eqnarray}\nwhere the quark field is represented by $q = (u,d,s)^T$ in flavor space, and \n${\\hat m}_c= {\\rm diag}_f (m_u,m_d,m_s)$ is the corresponding (current) mass \nmatrix.\nThe ${\\cal L}_\\text{sym}$ and ${\\cal L}_\\text{det}$ denote, respectively, \nthe scalar-pseudoscalar and the 't Hooft six-fermion interactions\n\\cite{Klevansky:1992qe,Hatsuda:1994pi},\n\\begin{align}\n\t{\\cal L}_\\text{sym}&= G_s \\sum_{a=0}^8 \\left [({\\bar q} \\lambda_ a q)^2 + \n\t({\\bar q} i\\gamma_5 \\lambda_a q)^2 \\right ] \\\\\n\t{\\cal L}_\\text{det}&=-K\\left\\{{\\rm det} \\left [{\\bar q}(1+\\gamma_5)q \\right] + \n\t{\\rm det}\\left [{\\bar q}(1-\\gamma_5)q\\right] \\right \\}.\n\\end{align}\nThe vector interaction is given by \\cite{Mishustin:2000ss}\n\\begin{equation} \n{\\cal L}_\\text{vec} = - G_V \\sum_{a=0}^8  \n\\left[({\\bar q} \\gamma^\\mu \\lambda_a q)^2 + \n ({\\bar q} \\gamma^\\mu \\gamma_5 \\lambda_a q)^2 \\right]. \n\\label{p1} \n\\end{equation}\nThe effective gluon field is given by \n$A^\\mu = g_{strong} {\\cal A}^\\mu_a\\frac{\\lambda_a}{2}$, where\n${\\cal A}^\\mu_a$ represents the SU$_c(3)$ gauge field.\nThe spatial components are neglected in Polyakov gauge at finite temperature,\ni.e., $A^\\mu = \\delta^{\\mu}_{0}A^0 = - i \\delta^{\\mu}_{4}A^4$. \nThe Polyakov loop value is defined as the trace of the Polyakov line,\n$ \\Phi = \\frac 1 {N_c} {\\langle\\langle \\mathcal{P}\\exp i\\int_{0}^{\\beta}d\\tau\\,\nA_4\\left(\\vec{x},\\tau\\right)\\ \\rangle\\rangle}_\\beta$,\nwhich is the order parameter of the $\\mathbb{Z}_3$ \nsymmetric\/broken phase transition in pure gauge.\nFor the pure gauge sector we use the following effective potential  \\cite{Roessner:2006xn},\n\\begin{eqnarray}\n\t& &\\frac{\\mathcal{U}\\left(\\Phi,\\bar\\Phi;T\\right)}{T^4}\n\t= -\\frac{a\\left(T\\right)}{2}\\bar\\Phi \\Phi \\nonumber\\\\\n\t& &\n\t+\\, b(T)\\mbox{ln}\\left[1-6\\bar\\Phi \\Phi+4(\\bar\\Phi^3+ \\Phi^3)\n\t-3(\\bar\\Phi \\Phi)^2\\right],\n\t\\label{Ueff}\n\\end{eqnarray}\nwhere \n$a\\left(T\\right)=a_0+a_1\\left(\\frac{T_0}{T}\\right)+a_2\\left(\\frac{T_0}{T}\\right)^2$, \n$b(T)=b_3\\left(\\frac{T_0}{T}\\right)^3$. \nIts parametrization values are $a_0 = 3.51$, $a_1 = -2.47$, $a_2 = 15.2$, \nand $b_3 = -1.75$ \\cite{Roessner:2006xn}, while the critical temperature is set \nto $T_0=210$ MeV.\nThe divergent ultraviolet sea quark integrals are regularized by a sharp cutoff \n$\\Lambda$ in three-momentum space.\nFor the NJL model parametrization, we consider:\n$\\Lambda = 602.3$ MeV, $m_u= m_d=5.5$ MeV, $m_s=140.7$ MeV, \n$G_s \\Lambda^2= 1.835$, and $K \\Lambda^5=12.36$ \\cite{Rehberg:1995kh}.\\\\\n\nFluctuations of conserved charges, such as the baryon number, provide vital \ninformation on the effective degrees of freedom and on critical phenomena.\nThey behave characteristically in a thermal equilibrium medium.\nIf there is a CEP in the phase diagram of strongly interacting matter, these \nfluctuations are then expected to provide characteristic signatures that, \nhopefully, can be experimentally observed.\nFor a static system of infinite size, the fluctuations of baryon number diverge \nat the CEP (second-order phase transition point). \nHowever, the created medium in HIC experiments has both finite size and \nlifetime that restricts its correlation length and, instead of divergent \nfluctuations, only moderate enhancements are expected.\nFluctuations of conserved charge are characterized by their cumulants or \nsusceptibilities.\nThe present work focuses on the baryon number charge susceptibilities. \nThe n$th$-order net-baryon susceptibility is given by\n\\begin{equation}\n \\chi_B^n(T,\\mu_B)= \\frac{\\partial^n\\pc{P(T,\\mu_B)\/T^4}}{\\partial(\\mu_B\/T)^n}.\n\\end{equation}\nDifferent susceptibility ratios $\\chi_B^n(T,\\mu_B)\/\\chi_B^m(T,\\mu_B)$\nare calculated in order to eliminate the volume dependence, allowing for a \ncomparison with experimental observables. \nIn this work, we analyze the following ratios\n\\begin{equation}\n\t\\frac{\\chi_B^4(T,\\mu_B)}{\\chi_B^2(T,\\mu_B)}=\\kappa\\sigma^2,\\quad\n\t\\frac{\\chi_B^3(T,\\mu_B)}{\\chi_B^1(T,\\mu_B)}=\\frac{S_B\\sigma^3}{M},\n\\label{eq:ratios}\n\\end{equation}\nwhere $M=VT^3\\chi_B^1$ is the mean, $\\sigma^2=VT^3\\chi_B^2$ the variance, $S_B$ \nthe skewness, and $\\kappa$ is the kurtosis of the net-baryon number \ndistribution. \n\n\\section{Results}\\label{sec:Results}\n\\begin{figure*}[t!]\n\t\\centering\n\n\t\\includegraphics[width=0.8\\linewidth]{1.pdf}\n\t\\caption{The light-quark condensate \n\t\t$\\ev{u\\bar{u}}(T,\\mu_B)\/\\ev{u\\bar{u}}(0,0)$ for $G_V=0$ (left) and \n\t\t$G_V=0.72G_s$ (right). \n\t\tThe following information is displayed: the CEP (dot), the chiral \n\t\tfirst-order phase transition boundary (solid line), and both the chiral \n\t\t(dashed line) and deconfinement (dotted line) crossover boundaries.} \n\t\\label{fig:1}\n\\end{figure*}\n\nWe analyze, herein, the net-baryon susceptibilities on the $(T,\\mu_B)$ plane.\nTwo PNJL models are analyzed: (i) a model with no vector interactions $G_V=0$,\nwhich predicts a CEP; \nand (ii) a model with vector interactions $G_V=0.72G_s$, which predicts no CEP.\nWe want to discuss what distinguishes these two scenarios. \nIn the following, symmetric quark matter is considered: \n$\\mu_u=\\mu_d=\\mu_s=\\mu_q=\\mu_B\/3$, where $\\mu_{i}$ are the chemical potential of \neach quark flavor and $\\mu_B$ is the baryonic chemical potential.\n\nThe hydrodynamical expansion of a HIC fireball is expected to follow  \ntrajectories of constant entropy per baryon, $s\/\\rho_B$, known as isentropes. \nThese trajectories contain important information on the adiabatic evolution of \nthe system. \nIt is thus interesting to analyze the susceptibility ratios \n[Eqs. (\\ref{eq:ratios})] along different isentropes \\cite{Costa:2010zw}.\nIt is important to note that while the net charge and the net strangeness are \nnot constrained in the present work; in a HIC, however, the ratio of electric \ncharge over baryon number is $Q\/\\rho_B\\simeq0.4$ and no net strangeness is \nproduced, $n_s=0$.\\\\\n\nThe phase diagrams for the chiral and deconfinement transitions are presented in \nFig. \\ref{fig:1}. The (normalized) light-quark condensate value\n$\\ev{u\\bar{u}}(T,\\mu_B)\/\\ev{u\\bar{u}}(0,0)$ is shown, where  \n$\\ev{u\\bar{u}}(0,0)$ is the vacuum value (due to isospin symmetry \n$\\ev{u\\bar{u}}=\\ev{d\\bar{d}}$).\nThe $G_V=0$ model predicts a CEP at \n$(T^{\\mbox{\\footnotesize{CEP}}},\\mu_B^{\\mbox{\\footnotesize{CEP}}})=(133\\, \\text{MeV},862\\,\\text{MeV})$, \nwhile the $G_V=0.72G_s$ model has no CEP, and the (approximate) \nchiral restoration is thus attained via an analytic transition (crossover) over \nthe whole phase diagram. \nThe chiral (dashed line) and deconfinement (dotted line) crossover boundaries \nare determined by the location $(T,\\mu_B)$ of the maximum of the order \nparameter susceptibilities (the point where fluctuations are largest).\nIt is interesting that the crossover boundaries show similar behavior for both \nmodels:\nthe gap between the deconfinement and chiral crossovers reduces with increasing \n$\\mu_B$ and becomes zero for some $\\mu_B$ values, which turns out to be near the \nCEP for $G_V=0$, above which they separate and follow distinct paths.\n\nBoth boundaries, the chiral  phase transition and the deconfinement phase \ntransition boundaries, are determined from the peaks of the susceptibility. The \ncrossing of the deconfinement and  chiral phase transitions has already been \nobserved before \\cite{Costa:2011fh} and it is possible to identify the crossing \nfrom the calculation of the susceptibility peaks at fixed temperatures: before \nand after the crossing they are two distinguishable peaks. At the crossing, \nthat stretches along a finite range of temperatures,  the two peaks overlap. \nThe crossing region includes part of the chiral crossover for both models, and, \nin the case of the model with a CEP, also the CEP, and part of the first-order \nphase transition.\n\nWe thus conclude that, due to the mixing between the gluonic and quarkionic \ndegrees of freedom, the chiral phase transition has a strong influence on \ndeconfinement transition. This is reflected on the behavior of the \ndeconfinement transition at the light quark and the strange quark chiral \ntransition. At the light quark transition, the crossing temperature is not much \naffected, but the crossing chemical potential is tightly connected with the \nposition of the chiral transition and the crossing follows, as referred above,  \nthe chiral crossover or both the chiral crossover and first-order transition. \nAs a consequence, the crossing occurs at a much larger chemical potential for \nthe $G_V=0.72 G_s$ model. A similar interconnection is observed at the strange \nchiral crossover in Fig. 2 and 3 in the $G_V=0$ model, where the deconfinement \ntransition presents a kink.\n\t\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[width=0.65\\linewidth]{2.pdf}\n\t\\caption{The net-baryon number susceptibilities \n\t\t$\\chi^3_B$ (top) and $\\chi^4_B$ (bottom) for $G_V=0$ (left) and \n\t\t$G_V=0.72G_s$ (right).\n\t\tThe following information is displayed: the CEP (black dot), the first-order\n\t\tphase chiral transition boundary (black solid line), both the chiral (black \n\t\tdashed line) and deconfinement (black dotted line) crossover boundaries,\n\t\tand the $s\/\\rho_B=\\{0.5,1,5,10,14\\}$ isentropic trajectories (dark green \n\t\tdotted-dashed lines) are also shown, which appear in the counterclockwise\n\t\tdirection, respectively.} \n\t\\label{fig:2}\n\\end{figure*}\nWe show  the $\\chi_B^3$ (top) and $\\chi_B^4$ (bottom) susceptibilities in Fig \n\\ref{fig:2}. \nTo a better understanding of their dependencies in the $(T,\\mu_B)$ plane, \nthe following features are also displayed:\nthe CEP (black dot), the first-order chiral phase transition boundary (black \nsolid line), and both the chiral (black dashed line) and the deconfinement \n(black dotted line) crossover boundaries.\nFurthermore, the isentropic trajectories (dark green dashed-dotted lines), \ni.e., paths along which the entropy density per baryon, $s\/\\rho_B$, is fixed, \nare also shown for $s\/\\rho_B=\\{0.5,1,5,10,14\\}$. \nThe two last trajectories cross the crossover line above the CEP of the $G_V=0$ \nmodel.\n\n\\begin{figure*}[!htbp]\n\t\\centering\n\t\\includegraphics[width=0.65\\linewidth]{3.pdf}\n\t\\caption{The net-baryon number susceptibility ratios\n\t\t$\\chi^4_B\/\\chi^2_B$ (top) and $\\chi^3_B\/\\chi^1_B$ (bottom) \n\t\tfor $G_V=0$ (left) and $G_V=0.72G_s$ (right).\n\t\tThe following information is displayed: the CEP (black dot),\n\t\tthe first-order phase chiral transition boundary (black solid line), both \n\t\tthe chiral (black dashed line) and deconfinement (black dotted line) \n\t\tcrossover boundaries, and the $s\/\\rho_B=\\{0.5,1,5,10,14\\}$ isentropic \n\t\ttrajectories (dark green dotted-dashed lines) are also shown, which appear\n\t\tin the counterclockwise direction, respectively.} \n\t\\label{fig:3}\n\\end{figure*}\n\nThe first three values allows us to discuss the phase diagram at low $T$ and \nhigh $\\mu_B$, where the (approximate) chiral restoration of the strange quark \noccurs.\nFor the $G_V=0$ model, the susceptibilities exhibit a nonmonotonic dependence \nnear the CEP, whose behavior strongly depends on the direction on which the CEP \nis approached. \nThe susceptibilities diverge at the CEP, with the divergence being stronger\nas higher susceptibilities orders are considered.\nAn interesting result is present at low $T$ and high $\\mu_B$. \nDespite the transition for the strange quark being just a crossover,\nand, therefore, without any nonanalytic behavior, a similar CEP \nstructure is seen at $\\mu_B\\approx1500$ MeV for the susceptibilities. \nThis indicates that a slight change on the model parametrization might induce \na first-order phase transition for the strange quark, and a corresponding CEP. \nThe $\\chi_B^3$ and $\\chi_B^4$ values for the $G_V=0.72G_s$ model show precisely \nthis behavior for the light quark sector: even though there is no CEP, and the \nchiral transition occurs via a crossover over the whole phase diagram, the \nnonmonotonic behavior of the susceptibilities is still present, as discussed \nwithin the NJL model \\cite{Fan:2017kym}.\nThe study of a scenario with a hypothetical negative \n$T^{\\mbox{\\footnotesize{CEP}}}$ for the light CEP, obtained by varying the \nvalue of the anomaly-induced six-fermion term, $K$, was done in \n\\cite{Chen:2016sxn}; it was shown that the magnitude of the susceptibilities \nalso changes significantly if a hypothetical negative temperature CEP is taken \ninto account.\n\nThe ratios $\\chi_B^4\/\\chi_B^2$ and $\\chi_B^3\/\\chi_B^1$ are shown in \nFig. \\ref{fig:3}. The sudden decrease near the deconfinement pseudocritical \ntemperature (dotted black line) indicates that both quantities are valuable \nsignatures of deconfinement transition.\nAs noted in \\cite{Fu:2009wy}, the statistical confinement, provided by the \nPolyakov loop (at low temperatures, when $\\Phi,\\bar{\\Phi}\\rightarrow 0$, \ncontributions coming from one- and two-quark states are suppressed, while \nthree-quark states are not \\cite{Hansen:2006ee}), is essential to obtain a \nlow-temperature limit for the susceptibility ratios that is consistent with \nthe hadron resonance gas model.\nThe results for the $G_V=0.72G_s$ model clearly show that the nonmonotonic \nbehavior of $\\chi_B^3$ and $\\chi_B^4$, which signals the presence of a\ncritical behavior, is still present even in the absence of a CEP. The \nnonmonotonic behavior persists, with a smaller intensity, up to almost the same \ntemperature as for the $G_V=0$ model. To make this feature clear, we show the \nnegative region of $\\chi^4_B\/\\chi^2_B$ in Fig. \\ref{fig:4}. \n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{4.pdf}\n\t\\caption{The region $\\chi^4_B\/\\chi^2_B<0$ for $G_V=0$ (blue) and \n\t$G_V=0.72G_s$ (red) models.\n\t\t} \n\t\\label{fig:4}\n\\end{figure}\nDespite the strong vector interaction used, it is remarkable that, for the \n$G_V=0.72G_s$ model (red), the $\\chi^4_B\/\\chi^2_B<0$ region extends from zero \nup to temperatures similar with the ones obtained for $G_V=0$ (blue). \nThis indicates that, at higher temperatures, both models are not discernible   \nexclusively from the sign change of $\\chi^4_B\/\\chi^2_B$. \nActually, a region with $\\chi^4_B\/\\chi^2_B<0$ is still present (at lower \ntemperatures though) even when the vector interaction strength is increased up \nto $G_V\\approx1.4G_s$. \nIf instead of looking at the whole negative region of $\\chi^4_B\/\\chi^2_B<0$, \none considers the stronger fluctuation region $\\chi^4_B\/\\chi^2_B<-200$ the \nfollowing pattern is seen:\nwhile this region extends to a range of $\\Delta \\mu_B\\approx100$ MeV and \n$\\Delta T\\approx20$ MeV for $G_V=0$, we get $\\Delta \\mu_B\\approx20$ MeV and \n$\\Delta T\\approx60$ MeV for $G_V=0.72G_s$.  \nThese different ranges on $T$ and $\\mu_B$ for the two different \nscenarios could help distinguish them, taking only into account the behavior of \nthe fluctuations. It should, however, be recalled that if the CEP exists, for \nmoderate temperatures and high enough baryonic density the line of first-order \ntransition could be crossed during the evolution of the fireball, giving rise \nto effects like multifragmentation \n\\cite{Mishustin:2006ka,Braun-Munzinger:2015hba}. \nThis would be a region where our no CEP model would present fluctuations \nsimilar to the ones existing in a model with CEP, above the CEP.\n\n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width=0.80\\linewidth]{5.pdf}\n\t\\caption{Isentropic trajectories $s\/\\rho_B=\\{0.5,1,5,10,14\\}$ for \n\tthe $G_V=0$ (solid lines) and the $G_V=0.72G_s$ (dotted lines) models.} \n\t\\label{fig:5}\n\\end{figure}\n\nTo complete the discussion, in the following we analyze the  \nisentropic trajectories with a small $s\/\\rho_B$ within the two scenarios.\n\nThe comparison of the isentropic trajectories between $G_V=0$ (solid lines) \nand $G_V=0.72G_s$ (dashed lines) models is in Fig. \\ref{fig:5}.  \nThe trajectories differ for high values of $T$ and $\\mu_B$, i.e., as soon as \nthe system becomes denser enough for the vector interactions to set in.\nTwo features that distinguish the $G_V=0$ model from the $G_V=0.72G_s$\nis the behavior of the trajectories near the CEP and the existence of a \nunstable spinodal region. \nThe trajectories with low $s\/\\rho_B$ values get enclosed into the unstable \nspinodal region when crossing the first-order phase transition to the chiral \nbroken phase. \nAs the system enters into the unstable spinodal region, the rapid formation of \nfragments of high density matter that occur should enhance the baryon number \nfluctuations \\cite{Braun-Munzinger:2015hba}.\nDue to the absence of spinodal region for the $G_V=0.72G_s$ model, such effect does not occur and the susceptibilities have an analytic behavior.\n\n\nIn Fig. \\ref{fig:6}, we show the $\\chi_B^4\/\\chi_B^2$ (top) and \n$\\chi_B^3\/\\chi_B^1$ (bottom) values along the $s\/\\rho_B=14$ (left) and the \n$s\/\\rho_B=10$ (right) isentropes (these isentropic trajectories are shown in \nFig. \\ref{fig:2} and \\ref{fig:3}).\nAs the value $s\/\\rho_B$ of the isentropic trajectory decreases, we are covering \na higher $\\mu_B$ region on the phase diagram. \nAs we move from $s\/\\rho_B=14$ to $s\/\\rho_B=10$, we are then approaching a \nregion of higher baryon fluctuations that reflects the vicinity of a CEP.\nWhile the fluctuations of $\\chi_B^4\/\\chi_B^2$ and $\\chi_B^3\/\\chi_B^1$\ngrow with decreasing $s\/\\rho_B$,\nthey also become constrained to a smaller temperature region (this is clear \nthrough the shape of the blue region in Fig. \\ref{fig:4}). \nThe decreasing gap between the chiral and deconfinement transitions with \nincreasing $\\mu_B$, which vanishes at the CEP (see Figs. \\ref{fig:2} and \n\\ref{fig:3}), is also reflected in the fluctuations: for $s\/\\rho_B\\geq14$ a two \npeak structure is present on the left side of the fluctuation (for \n$s\/\\rho_B=14$, a small bump at $T\\approx 150$ MeV is barely seen).\n\n\nThis two peak structure, which reflects the deconfinement\/chiral restoration \ngap, is clearer when a vector interaction is included. \nThe fluctuations for the $G_V=0.72G_s$ model, over the same isentropic \ntrajectories, are shown in Fig. \\ref{fig:7}.\nThe fluctuations along the $s\/\\rho_B=14$ trajectory show a two peak structure \nagain on the left side of the fluctuation. \nDespite the existence of a sign change of $\\chi_B^4\/\\chi_B^2$ (top) for both \nmodels (also seen in Fig. \\ref{fig:4}), their intensity is weaker for the \n$G_V=0.72G_s$ model, allowing one to notice the effect of the deconfinement\ntransition on the fluctuations ratios.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{6.pdf}\n\t\\caption{The values of $\\chi_B^4\/\\chi_B^2$ (top)\n\tand $\\chi_B^3\/\\chi_B^1$ (bottom) as a function of temperature along the isentropes\n\t$s\/\\rho_B=14$ (left) and $s\/\\rho_B=10$ (right) for the $G_V=0$ model.} \n\t\\label{fig:6}\n\\end{figure}\n\nLet us now focus on the crossover region at low temperatures, i.e., low \n$s\/\\rho_B$ values, for the $G_V=0.72G_s$ model.\nIn Fig. \\ref{fig:8}, we display the values of $\\chi^4_B\/\\chi^2_B$ (red) and \n$\\chi^3_B\/\\chi^1_B$ (blue) along the isentropes $s\/\\rho_B=5$ (top) and \n$s\/\\rho_B=1$ (bottom). The $(T,\\mu_B)$ dependence of the isentropic \ntrajectories can be seen in Figs. \\ref{fig:2} and \\ref{fig:3}. \nThe fluctuations increase strongly as lower isentrope values are considered. \nThe large fluctuation of $\\chi^4_B\/\\chi^2_B$ for $s\/\\rho_B=1$ reflects the \ncrossing of the isentropic trajectory with the chiral crossover line at \n$T\\approx 40$ MeV.\nThe features obtained at these low $s\/\\rho_B$ values are similar with the ones \nof the model with CEP but at $s\/\\rho_B=10$ and $14$; i.e., we get similar \nfluctuation amplitudes for a much lower $T$.\n \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{7.pdf}\n\t\\caption{The values of $\\chi_B^4\/\\chi_B^2$ (top)\n\tand $\\chi_B^3\/\\chi_B^1$ (bottom) as a function of temperature along the isentropes\n\t$s\/\\rho_B=14$ (left) and $s\/\\rho_B=10$ (right) for the $G_V=0.72G_s$ model.} \n\t\\label{fig:7}\n\\end{figure}\n\n\n\\begin{figure}[!b]\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{8.pdf}\n\t\\caption{The value of $\\chi^4_B\/\\chi^2_B$ (red) and $\\chi^3_B\/\\chi^1_B$ (blue) \n\t\tas a function of temperature along the isentropic trajectories $s\/\\rho_B=5$ (left) and $s\/\\rho_B=1$ \n\t\t(right) for the $G_V=0.72G_s$ model.} \n\t\\label{fig:8}\n\\end{figure}\n\n\\begin{figure*}[!tb]\n\t\\centering\n\t\\includegraphics[width=0.65\\linewidth]{9.pdf}\n\t\\caption{Sound velocity squared $v_s^2$ as a function \n\t\tof temperature (left) and baryon chemical potential (right) along the \n\t\tisentropic trajectories $s\/\\rho_B=10$ (top) and $s\/\\rho_B=1$ (bottom) for \n\t\t$G_V=0$ (blue) and $G_V=0.72G_s$ (red) models. \n\t\tThe squares and circles indicate the location of\n\t\tthe chiral and the deconfinement pseudocritical boundaries, respectively.} \n\t\\label{fig:9}\n\\end{figure*}\n\n\\begin{table*}[t!]\n\\centering\n\\begin{tabular}{|c||c|c|c||c|c|c||c|c|c|}\n\\hline\n\\multirow{2}{*}{$s\/\\rho_B$} & $T^{\\chi}$ & $\\mu_B^{\\chi}$ & \\multirow{2}{*}{$(v_s^2)^{\\chi}$} & $T^{\\Phi}$ & $\\mu_B^{\\Phi}$ & \\multirow{2}{*}{$(v_s^2)^{\\Phi}$} & $T^{min}$ & $\\mu_B^{min}$ & \\multirow{2}{*}{$(v_s^2)_{min}$} \\\\\n & [MeV] & [MeV] & & [MeV] & [MeV] & & [MeV] & [MeV] &  \\\\\n\\hline\n\\multicolumn{10}{|c|}{$G_V=0$ } \\\\\n\\hline\n8 & 133 & 861 & 0.018 & 133 & 863 & 0.018 & 133 & 863 & 0.018 \\\\\n\\hline\n10 & 145 & 790 & 0.036 & 143 & 796 & 0.032 & 141 & 803 & 0.031 \\\\\n\\hline\n12 & 153 & 727 & 0.053 & 148 & 738 & 0.040 & 147 & 738 & 0.039 \\\\\n\\hline\n14 & 161 & 667 & 0.070 & 153 & 672 & 0.044 & 150 & 668 & 0.042 \\\\\n\\hline\n20 & 176 & 525 & 0.10 & 161 & 504 & 0.045 & 150 & 668 & 0.042 \\\\\n\\hline\n\\multicolumn{10}{|c|}{$G_V=0.72G_s$}  \\\\\n\\hline\n0.1 & 5 & 1172 & 0.0049 & $-$ & $-$ & $-$ & 6 & 1173 & 0.0016 \\\\\n\\hline\n1 & 37 & 1165 & 0.019 & $-$ & $-$ & $-$ & 43 & 1166 & 0.016 \\\\\n\\hline\n10 & 144 & 931 & 0.088 & 140 & 906 & 0.069 & 134 & 859 & 0.060 \\\\\n\\hline\n12 & 153 & 870 & 0.10 & 147 & 819 & 0.067 & 141 & 768 & 0.059 \\\\\n\\hline\n14 & 161 & 814 & 0.11 & 152 & 737 & 0.064 & 174 & 689 & 0.056 \\\\\n\\hline\n20 & 185 & 554 & 0.18 & 162 & 512 & 0.048 & 159 & 494 & 0.042 \\\\\n\\hline\\end{tabular}\n\\caption{The sound velocity squared at the chiral \n\t\t$(v_s^2)^{\\chi}=(T^{\\chi},\\mu_B^{\\chi})$ and deconfinement \n\t\t$(v_s^2)^{\\Phi}=(T^{\\Phi},\\mu_B^{\\Phi})$ pseudocritical boundaries for\n\t\tseveral isentropes $s\/\\rho_B$. The minimum of $v_s^2$ [$(v_s^2)_{min}$] and\n\t\tits location $(T^{min},\\mu_B^{min})$ is also presented. }\n\\label{Table1}\n\\end{table*}\n\nFinally, we have determined the square of the sound velocity, \n$v_s^2=dP\/d{\\cal E}|_{s\/\\rho_B=\\mbox{const.}}$, along two isentropic trajectories. \nThe sound velocity plays a central role in the hydrodynamical evolution of \nmatter created in HIC being very different in the different stages of the \nexpansion. It affects, among others, the momentum distribution of the particles \noriginating from the fluid elements at the freeze-out stage \n\\cite{Mohanty:2003va}. \nThe values of square sound velocity are extracted from the widths of rapidity \ndistributions \\cite{Mohanty:2003va,Gao:2015sdb,Adam:2016ddh}. For example, from \nthe measured data on the widths of the pion rapidity spectra, $v_s^2$ in the \ndense stage of the reactions has been extracted \\cite{Steinheimer:2012bp}.\n\nIn Fig. \\ref{fig:9}, we show $v_s^2$ along $s\/\\rho_B=10$ (top) and \n$s\/\\rho_B=1$ (bottom) for $G_V=0$ (blue) and $G_V=0.72G_s$ (red) models.\nAs the isentropic trajectories follow specific paths, $(T,\\mu_B)$, on the phase \ndiagram (see Figs. \\ref{fig:2} and \\ref{fig:3}), we show the $v_s^2$ dependence \non temperature (left) and baryon chemical potential (right), for both \nisentropic trajectories.\nFor each isentropic, we give the values of $v_s^2(T,\\mu_B)$ at its minimum and \nat the chiral and deconfinement boundaries in Table \\ref{Table1}. \n\nConsidering in first place the $s\/\\rho_B=10$ trajectory as a function of \n$T$ (Fig. \\ref{fig:9}, left upper panel), we see that the minimum of \n$v_s^2(T,\\mu_B)$ occurs closer to the deconfinement pseudocritical boundary \n(circles) than to the chiral pseudocritical boundary (squares).\nWhile the temperature dependence of the $s\/\\rho_B=10$ isentrope is a \nsingle-valued function, the same does not hold for its $\\mu_B$ dependence \n(Fig. \\ref{fig:9}, right upper panel).\nThe loop behavior for the $G_V=0$ model (blue curve in upper right panel of \nFig. \\ref{fig:9}) \nrises from the bending effect towards the CEP that the $s\/\\rho_B=10$ isentrope \nundergoes when crossing into the chiral broken region (solid red line in Fig. \n\\ref{fig:5}). \nThis effect occurring in $v_s^2$ can then be seen as a signal for the vicinity \nof a CEP (if some kind of bending effect into the CEP exists), once this effect \nis not seen for the $G_V=0.72G_s$ model.\nFor $s\/\\rho_B=1$ (lower panels of Fig. \\ref{fig:9}), the $v_s^2$ shows negative \nvalues for $G_V=0$ model (blue curve), reflecting the first-order phase\ntransition that occurs at lower $T$.\nIt is interesting to note that, for small values of $s\/\\rho_B$, the local \nminimum of $v_s^2$ at $\\mu_B\\approx1480$ MeV is associated  with the crossover\nof the strange quark. \n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe have analyzed the net-baryon number fluctuations for three-flavor quark \nmatter within the Polyakov extended Nambu--Jona-Lasinio model.\nFor a strong enough vector interaction intensity, the model predicts no CEP\nin the phase diagram.\nFrom  the net-baryon number fluctuations one concludes that, even in the \nabsence of a CEP, the nonmonotonic behavior persists.\nTherefore, the existence of a CEP cannot be taken solely from the existence of \nnonmonotonic behavior on the net-baryon number susceptibilities.\n\nWe have analyzed further other possible properties that may\ndistinguish the two scenarios: for the no CEP model ($G_V=0.72G_s$),\nlarge fluctuations in the susceptibility ratios occur only at small $T$, and \nthe values of $v_s^2$ are almost unchanged at moderates $s\/\\rho_B$ values.\nThe values of the susceptibility ratios along two or three isentropic lines \nwould possibly allow us to distinguish both cases. Also, the value of the sound \nvelocity at the chiral transition for two or three isentropes would give some \nuseful information. \nFor the $G_V=0$ model, by going from $s\/\\rho_B=9$ to $14$, the value of $v_s^2$ \nincreases at least $50\\%$ for each step $\\Delta(s\/\\rho_B)=2$. \nInstead, for the no CEP model, the change is of the order of $10\\%$.\nIt should be noticed, however, that in the present work we have discussed \ninfinite size matter. For a finite system it is expected that the signals we \nhave discussed are less intense but still might allow to distinguish both \nscenarios. \n\nWe have shown that, for high chemical potentials and low temperatures,\na signature of the strange quark chiral symmetry restoration is observed \nin a decrease of the sound velocity and in a region with negative \n$\\chi^4_B\/\\chi^2_B$. \n\n\\vspace{0.25cm}\n{\\bf Acknowledgments}\nThis work was supported by ``Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia,'' Portugal, \nunder Projects No. UID\/FIS\/04564\/2016 and No. POCI-01-0145-FEDER-029912 with\nfinancial support from POCI, in its FEDER component, and by the FCT\/MCTES \nbudget through national funds (OE), and under Grants No. SFRH\/BPD\/102273\/2014 \n(P.C.) and No. CENTRO-01-0145-FEDER-000014 (M.F.) through the CENTRO2020 \nprogram. Partial support comes from ``THOR'' (COST Action CA15213) and \n``PHAROS'' (COST Action CA16214).\n\n\\vspace{-0.5cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}