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+{"text":"\\section{Introduction}\nIn the era of information explosion, the recommender system has\nbecome one of the most effective ways to help users to discover\nwhat they are interested in enormous data. Generally speaking, the recommender systems usually follow   two steps: learn  vectorized representations (aka. embeddings) of users and items and then model interactions among them (e.g., whether a user buy an item).  Collaborative filtering (CF) learns  node embedding based on the historical interactions on user-item bipartite  graph and performs item recommendation based on the parameters.\n\nAs a matter of fact, there exist diverse relations among various types of nodes (e.g., buy relation and social relation) in real-world recommendation scenario, also widely known as heterogeneous graph \\cite{17tkde_shi}.  Taking the dataset Movielens as an example,  it contains three types\nof nodes include movie, user and genre.  \nMeta-path \\cite{11vldb_pathsim}, a composite relation connecting two objects, is a\nwidely used structure to capture the semantics. \nThe semantics revealed by different meta-paths are able to describe the  characteristics of nodes from different aspects. For example, meta-path User-Movie (U-M)  describes the preference of user, while  meta-path User-User (U-U)  describes social influence among users. \nBesides basic meta-path, multi-hop meta-path (e.g., U-U-U) which captures high-order semantics and enrich the connections among users is able to improve the node embedding and alleviate the cold-start problem.\n\nBased on the above analysis, when designing heterogeneous graph neural network for recommendation, we need to address the following  requirements.\n\\begin{itemize}\n\\item \\textbf{Heterogeneity of graph.} The heterogeneity is an intrinsic property of heterogeneous graph, i.e., various types of nodes and edges.  How to handle such complex structural information for recommendation is an urgent problem that needs to be solved.\n\\item \\textbf{High-order semantic preservation. } High-order semantic information which captures diverse  long-term dependencies among nodes  plays the key role in improving node embedding and alleviating the cold-start problem in recommender system. How to inject high-order semantic into node embedding is a fundamental problem in recommender system.\n\\item \\textbf{Rich semantics fusion.}  Different meaningful and complex semantic information are involved in heterogeneous graph, which are usually reflected by diverse meta-paths. For example, meta-path U-M and U-U can describe the preference and social influence of user and then  comprehensively describe the characteristics of user  from different aspects.   How to select the most meaningful meta-paths and fuse rich semantics to improve node embedding is an open problem.\n\\end{itemize}\nIn this paper, we  propose  \\textbf{H}eterogeneous\n\\textbf{G}raph neural network for  \\textbf{Rec}ommendation, named HGRec, which mainly considers high-order semantic preservation and rich semantics fusion. Specifically, \nsemantic aggregation layer injects high-order semantic  into node embedding via multi-hop meta-path and semantic fusion layer fuse rich semantics revealed by multiple meta-paths. \nAfter that, the overall model can be optimized via back propagation in an end-to-end manner.\n\nThe contributions of our work are summarized as follows:\n\\begin{itemize}\n\\item We  highlight the critical importance of rich high-order semantics in improving node embedding for recommendation system.\n\\item We propose a heterogeneous graph neural network based recommendation system, which explicitly injects high-order semantic into node embedding via multi-hops meta-path  and fuses rich semantics via multiple meta-paths for comprehensive node embedding.\n\\item  Empirical studies on real-world heterogeneous graphs \n demonstrate the state-of-the-art performance of HGRec and potentially\ngood interpretability for the recommendation results.\n\n\\end{itemize}\n\\section{METHODOLOGY}\nIn this section, we present the proposed model \\textbf{H}eterogeneous\n\\textbf{G}raph neural network for  \\textbf{Rec}ommendation (HGRec). The basic idea of HGRec is to learn representative node\nembedding of  users and items by injecting  and fusing high-order semantics. The proposed HGRec first adopts embedding layer to initialize node embedding. Then, semantic aggregation layer and semantic fusion layer will inject high-order semantic into node embedding via multi-hops meta-path and fuses rich semantics via multiple meta-paths, respectively. Lastly, we  leverage the fused embedding of user and item for recommendation.\n\\subsection{Embedding Initialization}\nFollowing the previous works \\cite{he2017neural,wang2019neural}, we random initialize node embedding matrix and use look-up to get the initial embedding of  user $u$ and item $i$, denoted as $\\mathbf{e}_u\\in \\mathbb{R}^d$ and  $\\mathbf{e}_i \\in \\mathbb{R}^d $, respectively. Here $d$ is the dimension of node embedding.\n\n\\subsection{Semantic Aggregation  Layer}\nAfter obtain initial node embedding, we  propose semantic aggregating layer to aggregate multi-hops meta-path based neighbors and update node embedding, so the high-order semantic information is well preserved. For clearly,  we first introduce the first-order aggregation in semantic aggregation layer and then generalize it to multiple\nsuccessive layers (aka. high-order semantic aggregation).\n\n\\textbf{First-order Semantic Aggregation}\nTaking one user $u$ and one user-related meta-path $\\Phi^U$ as an example, \nwe propose semantic aggregation layer $\\mathcal{A}$  to aggregate  meta-path based neighbors $\\mathcal{N}^{\\Phi^U}_u$ and get the first-order user embedding $\\mathbf{e}_{u }^{\\Phi^U, 1}$, shown as follows:\n\\begin{equation}\n    \\mathbf{e}_{u }^{\\Phi^U, 1}=\\mathcal{A}(u, \\Phi^U).\n\\end{equation}\nRather than simple neighbor combination, we consider the complex interaction between node and its neighbors in  aggregating process.  Specifically, we encode the interaction between node $u$ and its neighbor $k$  into aggregating process via $\\mathbf{e}_{k} \\odot \\mathbf{e}_{u}$, where $\\odot$ denotes the element-wise product. The overall aggregating process is shown as follows:\n\n\\begin{equation}\n\\mathbf{e}_{u }^{\\Phi^U, 1}= \\mathbf{W}_{1}^{\\Phi^U} \\mathbf{e}_{u}  + \\sum_{k\\in \\mathcal{N}^{\\Phi^U}_u}\\left(\\mathbf{W}_{1}^{\\Phi^U} \\mathbf{e}_{k}+\\mathbf{W}_{2}^{\\Phi^U}\\left(\\mathbf{e}_{k} \\odot \\mathbf{e}_{u}\\right)\\right) ,\n\\end{equation}\nwhere \n$\\mathbf{W}_{1}^{\\Phi^U},\\mathbf{W}_{2}^{\\Phi^U}$ are weight matrixes.\nThe first-order semantic aggregation only aggregates one-hop meta-path based neighbors  into node embedding, while high-order semantic revealed by multi-hops meta-path plays a crucial role in improving node embedding.\n\n\\textbf{High-order Semantic Aggregation} Considering the high-order semantic revealed by multi-hops meta-path, we  stack first-order  semantic aggregation for multiple layers and recurrently aggregate  corresponding meta-path based neighbors, \nso the  high-order semantic is injected into node embedding, shown as follows:\n\\begin{equation}\n\\mathbf{e}_{u }^{\\Phi^U, L}=\\mathcal{A}^L(\\cdots \\mathcal{A}^2(\\mathcal{A}^1(u, \\Phi^U))),\n\\end{equation}\nwhere $\\mathbf{e}_{u }^{\\Phi^U, L}$ denotes the $L$-order user embedding. \nThen, we concatenate different order user embedding and get the semantic-specific embedding of user $u$, shown as follows:\n\\begin{equation}\n    \\mathbf{e}_{u }^{\\Phi^U}=\\mathbf{e}_{u }^{\\Phi^U, 1}||\\mathbf{e}_{u }^{\\Phi^U, 2}||,\\cdots, ||\\mathbf{e}_{u }^{\\Phi^U, L},\n\\end{equation}\nwhere $||$ is the concatenation operation. However, one meta-path cannot comprehensively describe the characteristics of node from different aspects.  Considering a set of user-related meta-paths $\\{ \\Phi_1^U, \\Phi_2^U, \\cdots, \\Phi_{K_1}^U\\}$, we can get $K_1$ groups of user embeddings $ \\{\\mathbf{E}_{u }^{\\Phi_1^U},\\mathbf{E}_{u }^{\\Phi_2^U},\\cdots,\\mathbf{E}_{u }^{\\Phi_{K_1}^U} \\}$.\n\nSimilar to user embedding, given a set of item-related meta-paths $\\{ \\Phi_1^I, \\Phi_2^I, \\cdots, \\Phi_{K_2}^I\\}$, we can get $K_2$ groups of item embeddings $\\{\\mathbf{E}_{i }^{\\Phi_1^I},\\mathbf{E}_{i }^{\\Phi_2^I},\\cdots,\\mathbf{E}_{i }^{\\Phi_{K_2}^I} \\}$.\n\n\\subsection{Semantic Fusion Layer}\nAfter obtaining multiple higher-order node embedding, we need to learn the importance of different meta-paths and fuse them properly for better recommendation. Given $K_1$ groups of user embeddings $ \\{\\mathbf{E}_{u }^{\\Phi_1^U},\\mathbf{E}_{u }^{\\Phi_2^U},\\cdots,\\mathbf{E}_{u }^{\\Phi_{K_1}^U} \\}$, we propose semantic fusion layer $\\mathcal{F}$ to learn the weights of different meta-paths (e.g., $w^{\\Phi_1^U}, w^{\\Phi_2^U}, \\cdots, w^{\\Phi_{K_1}^U}$), shown as follows:\n\n\\begin{equation}\n    (w^{\\Phi_1^U}, w^{\\Phi_2^U}, \\cdots, w^{\\Phi_{K_1}^U}) = \\mathcal{F}(\\mathbf{E}_{u }^{\\Phi_1^U},\\mathbf{E}_{u }^{\\Phi_2^U},\\cdots,\\mathbf{E}_{u }^{\\Phi_{K_1}^U}).\n\\end{equation}\nTo learn the  importance of each meta-path (e.g., $\\alpha_{\\Phi _{k}^U}$), we first project node embedding into the  attention space and then use a semantic attention vector  $\\mathbf{q}_{U}$ to measure the importance  of  meta-path specific embedding, \n\\begin{equation}\n    \\alpha_{\\Phi _{k}^U}=\\frac{1}{\\left |V \\right |}\\sum_{i\\in V }\\mathbf{q}_U^{\\top}\\cdot \\tanh\\left (\\mathbf{W}_U\\cdot \\mathbf{e}_{u}^{\\Phi^U_k }+\\mathbf{b}_U\\right ),\n\\end{equation}\nwhere $\\mathbf{W}_U$ and $\\mathbf{b}_U$ are weight and bias, respectively.\nThen, we normalize them via softmax function and  get meta-path weights $w_{\\Phi _{k}^U}$, shown as follows:\n\\begin{equation}\n   w_{\\Phi _{k}^U}=\\frac{\\exp\\left ( w_{\\Phi _{k}^U} \\right )}{\\sum_{k=1}^{K_1}\\exp\\left ( w_{\\Phi _{k}^U} \\right )}.\n\\end{equation}\nWith the learned weights as coefficients, we\ncan fuse  multiple user embeddings to obtain the final\nembedding $\\mathbf{E}_u$ as follows:\n\n\\begin{equation}\n    \\mathbf{E}_u=\\sum_{k=1}^{K_1} w_{\\Phi _{k}^U}\\cdot \\mathbf{E}_u^{\\Phi _{k}^U}.\n\\end{equation}\n\nSimilar to user embedding,   we can fuse $K_2$ groups of item embeddings $\\{\\mathbf{E}_{i }^{\\Phi_1^U},\\mathbf{E}_{i }^{\\Phi_2^U},\\cdots,\\mathbf{E}_{i }^{\\Phi_{K_2}^U} \\}$ and  obtain the final\nembedding  of item $\\mathbf{E}_i$.\n\n\\subsection{Model Prediction}\nThe final part of the model is to recommend items for users based on their embedding. Here  we calculate the inner product  of user and item for recommendation, as follows:\n\n\\begin{equation}\n    \\hat{y}_{ui}=(\\mathbf{E}_{u })^\\top \\mathbf{E}_{i}.\n\\end{equation}\nThen, we calculate BPR loss \\cite{wang2019neural} and optimize the parameters, as follows:\n\\begin{equation}\n    L=\\sum_{\\left ( u,i,j \\right )\\in \\mathcal{O}}-\\ln\\sigma \\left ( \\hat{y}_{ui}- \\hat{y}_{uj} \\right )+\\lambda \\left \\| \\Theta  \\right \\|_{2}^{2},\n\\end{equation}\nwhere $\\mathcal{O}=\\left\\{(u, i, j) |(u, i) \\in \\mathcal{R}^{+},(u, j) \\in \\mathcal{R}^{-}\\right\\}$ denotes the pairwise\ntraining data,$ \\mathcal{R}^{+}$ indicates the observed interactions,  $\\mathcal{R}^{-}$ is\nthe unobserved interactions, $\\Theta $ denotes all trainable model parameters, and $\\lambda $\ncontrols the L2 regularization strength to prevent overfitting.\n\n\\section{EXPERIMENTS}\n\n\nWe conduct  experiments on three heterogeneous graphs: Amazon, Yelp and Movielens (details are shown in Table \\ref{dataset}).\n\\begin{table}[]\n\\caption{Statistics of the datasets.} \n\\label{dataset}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\nDatasets                   & Relation(A-B)     & \\#A   & \\#B   & \\#A-B   \\\\ \\hline\n\\multirow{3}{*}{Movielens} & User - Movie    & 943   & 1,682 & 100,000 \\\\ \\cline{2-5} \n                           & User - User    & 943   & 943   & 47,150  \\\\ \\cline{2-5} \n                           & Movie - Genre     & 1,682 & 18    & 2,861   \\\\ \\hline\n\\multirow{4}{*}{Amazon}    & User - Item     & 6,170 & 2,753 & 195,791 \\\\ \\cline{2-5} \n                           & Item - Cate.   & 2,753 & 22    & 5,508   \\\\ \\cline{2-5} \n                           & Item - Brand      & 2,753 & 334   & 2,753   \\\\ \\hline\n\\multirow{4}{*}{Yelp}    & User - Item     & 16,239 & 14,284 & 198,397 \\\\ \\cline{2-5} \n                           &User - User    & 16,239   & 16,239   & 158,590  \\\\ \\cline{2-5} \n                           & Item - City      & 14,284 & 47   & 14,267   \\\\ \\hline                        \n\\end{tabular}\n\\end{table}\nWe  compare with some state-of-art baselines, include BPRMF \\cite{bpr}, NMF \\cite{he2017neural}, GAT \\cite{18iclr_gat}, MCRec \\cite{hu2018leveraging}, NGCF \\cite{wang2019neural}, to verify the effectiveness of the proposed model. We also test a variant of HGRec, denotes as HGRec-, which assigns the same importance to each meta-path.\n\n\nFor evaluation, we split  datasets into training set and test set with 8:2 ratio and employ Pre@10, Recall@10, HR@10 and NDCG@10 as evaluation metrics. \n\nWe randomly initialize parameters and optimize models with Adam. For the proposed HGRec, we set the L2 regularization  to 1e-2, the dimension of the semantic attention vector $\\mathbf{q}$ to 64,  the dropout to 0.8,  and \nthe learning rate to 5e-4, 1e-3 and 5e-3 on Movielens  Amazon and  Yelp, respectively.\nWe also use early stopping with a patience of 100 to aviod overfitting.\n\n\\subsection{Overall Performance Analysis}\nThe experiment results   are shown in Table \\ref{perform} and  we have the following   observations:.\n\n\n\\begin{table}[]\n\\caption{Overall Performance Comparison }\n\\label{perform}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{Movielens}              \\\\ \\cline{2-5} \n                       & Pre@10  & Rec@10 & NDCG@10 & HR@10 \\\\ \\hline\nBMF                    & 0.3251 & 0.2096   & 0.4081 & 0.8928     \\\\ \\hline\nNMF                    & 0.1704 & 0.1163   & 0.2336 & 0.7739     \\\\ \\hline\nGAT                    & 0.2068 & 0.1210   & 0.2556  & 0.7548     \\\\ \\hline\nMCRec                  & 0.3310  & 0.2129    & 0.2624  & 0.9025      \\\\ \\hline\nNGCF                   & 0.3369 & 0.2179   & 0.4178 & 0.9045     \\\\ \\hline\nHGRec-            & \\textbf{0.3670} & \\textbf{0.2412}   & \\textbf{0.4551} & 0.9172     \\\\ \\hline\nHGRec                  & 0.3667 & 0.2405   & 0.4547 & \\textbf{0.9193}    \\\\ \\hline\nImprov.                    & 6.70$\\%$ &6.80$\\%$   & 8.19$\\%$ & 1.36$\\%$  \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{Amazon}               \\\\ \\cline{2-5} \n                       & Pre@10  & Rec@10 & NDCG@10 & HR@10 \\\\ \\hline\nBMF                    & 0.0490 & 0.0881   & 0.1176 & 0.3232     \\\\ \\hline\nNMF                    & 0.0168 & 0.0264   & 0.0463 & 0.1371     \\\\ \\hline\nGAT                    & 0.0410 & 0.0810  & 0.1096  & 0.2998     \\\\ \\hline\nMCRec                    & 0.0309 & 0.0697  & 0.1131  & 0.3027     \\\\ \\hline\nNGCF                   & 0.0495 & 0.0870  & 0.1150 &0.3224     \\\\ \\hline\nHGRec-                 & 0.0553 & 0.0988   & 0.1313 & 0.3503      \\\\ \\hline\nHGRec                  & \\textbf{0.0588} & \\textbf{0.1054}   & \\textbf{0.1384} & \\textbf{0.3746}   \\\\ \\hline\nImprov.                   &5.95$\\%$ &6.26$\\%$   &5.13$\\%$  & 6.48$\\%$   \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{Yelp}               \\\\ \\cline{2-5} \n                       & Pre@10  & Rec@10 & NDCG@10 & HR@10 \\\\ \\hline\nBMF                    & 0.0039 & 0.0287   & 0.0150 & 0.0291     \\\\ \\hline\nNMF                    &0.0012 & 0.0265   & 0.0233 & 0.0398     \\\\ \\hline\nGAT                    &0.0038  & 0.0240  & 0.0171  & 0.0363     \\\\ \\hline\nMCRec                    & 0.0031 & 0.0531  & 0.0201  & 0.0432     \\\\ \\hline\nNGCF                   & 0.0073 & 0.0410  & 0.0271 &0.0667     \\\\ \\hline\nHGRec-                 & 0.0076 & 0.0433   & 0.0237 & 0.0506      \\\\ \\hline\nHGRec                  & \\textbf{0.0078} & \\textbf{0.0447}   & \\textbf{0.0310} & \\textbf{0.0671}   \\\\ \\hline\nImprov.                   &6.41$\\%$ &3.13$\\%$   &12.6$\\%$  &1.03$\\%$   \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\textbullet ~The proposed  HGRec consistently performances  better than  baselines with significant gap on all the datasets. In particular, HGRec improves over the strongest baseline NGCF  w.r.t.\nRecall@10 by 6.80\\%, 17.49\\%,3.13\\%  in Movielens\nAmazon and Yelp, respectively. The results demonstrate that   injecting   rich high-order semantics  into the node embedding  indeed improves the recommendation performance.\n\n\\textbullet ~Compare HGRec with HGRec- , we can observe that HGRec outperforms HGRec-  on all  datasets. This proves that the semantic fusion layer is able to identify the importance of meta-paths and then  enhance the performance of HGRec.\n\n\\textbullet ~Graph neural network based recommendation models show their superiorities over traditional MF based models,  demonstrating the importance of nonlinear structural\ninteractions among nodes.\n\\begin{table}[]\n\\caption{Effectiveness of Layer Number}\n\\label{movie_depth}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{L} & \\multicolumn{2}{c|}{Movielens} & \\multicolumn{2}{c|}{Amazon} \\\\\\cline{2-5} \n                  & Rec@10      & NDCG@10      & Rec@10     & NDCG@10   \\\\\\hline\n1           & 0.2390        & 0.4506      & 0.0947       & 0.1251    \\\\\\hline\n2           & 0.2391        & 0.4526        & 0.0864       & 0.1151    \\\\\\hline\n3           & \\textbf{0.2405}        & \\textbf{0.4547}      & \\textbf{0.1054}       & \\textbf{0.1384}    \\\\\\hline\n 4          & 0.2391        & 0.4513      & 0.0743       & 0.1064  \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\useunder{\\uline}{\\ul}{}\n\n\n\\textbf{Effect of Layer Numbers.} \nTo investigate  the whether high-order semantic  improves node embedding, we vary the model depth (e.g., $L=1,2,3,4$) and show the results on Table \\ref{movie_depth}. We can find that with the growth of model depth, the performance of HGRec  are sustainable growth and achieves the best performance\nwhen   $L$ is set to 3,  indicating the effectiveness of high-order semantic. After that, the performance of\nHGRec starts to degenerate which may because of overfitting.\n\n\n\n\n\n\n\n\\section{Conclusion and Future Work}\nIn this work, we highlight the critical importance of rich  high-order semantics in improving node embedding for better recommendation. Specifically, we design a semantic aggregation layer which aggregates multi-hop meta-path neighbors so as to inject high-order semantic into node embedding.  To describe the characteristics of node comprehensively, we leverage a semantic fusion layer to  fuse rich semantic revealed by multiple meta-paths. Experimental results demonstrates the superiority of the proposed model and show the potentially good interpretability for the recommendation results.\n\n\n\n\n\n\\nocite{langley00}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\nThermally driven turbulent flows widely occur in geophysical flows and industrial processes. Examples are thermal convection in the atmospheric and mantle convection \\cite{mck1974,Wyngaard1992}, in the ocean \\cite{cheng2019fast}, and in many industrial processes \\cite{Bejan}. Rayleigh--B\\'enard convection (RBC), a fluid layer heated from below and cooled from above, is an ideal model for the study of thermally driven turbulent flows \\cite{ahl09,loh10,chi12}. \nThe main challenge for thermal turbulence studies is to explore the flow dynamics and heat transfer of the system in a wide range of the control parameters. The main attention has been on how the heat transfer depends on the Rayleigh number,  which is the dimensionless temperature difference and measures the intensity of the thermal driving of the system. The Rayleigh number is defined as $\\text{Ra}=\\frac{\\beta g \\Delta L^3 }{\\nu \\kappa}$, where $g$ is the gravitational acceleration, $\\beta$ is the isobaric thermal expansion coefficient, $\\nu$ the kinematic viscosity, $\\kappa$ is the thermal diffusivity, $\\Delta$ is the temperature difference between the hot plate and the cold plate, and $L$ is the thickness of the fluid layer between the aforementioned plates.  \n\nTo achieve high Rayleigh numbers, numerous strategies have been proposed in the past decades. Looking at the definition of $\\text{Ra}$, one can see that large $\\text{Ra}$ can be reached in several ways. \nOne approach is to use a working fluid for which the parameter $\\frac{\\beta}{{\\nu}\\kappa}$ has a large value. This approach has been widely used in the community, such as using helium gas at cryogenic temperatures \\cite{cas89,cha97,nie00,Urban2010} or mercury \\cite{san89}. Recently, it was found that pure gas, particularly gasses with high molecular weight and under high pressure (up to 19-bar), is also an effective way to push $\\text{Ra}$ to high values at a constant $\\text{Pr}$ number \\cite{fun09,ahl09b}. Another commonly used approach in the field of turbulent thermal convection is to increase the system thickness $L$ \\cite{nie00,fun05,nik05,sun05e,pui07,fun09,ahl09b,Urban2010}. \n\nThe fact that hitherto not much attention was on the changing  the gravitational acceleration provides the motivation of the present work. In this study, we propose a novel system, annular centrifugal RRC (ACRBC), which is a cylindrical annulus with cooled inner and heated outer walls under a solid body rotation. \nIn this system, the buoyancy force can be efficiently enhanced by replacing the gravitational acceleration ($g$) by the centrifugal acceleration, and  consequently, $\\text{Ra}$ can be increased by increasing the rotation rate for a given fluid and temperature difference. \nAnother advantage of this system is that the thermal convection in rapidly rotating cylindrical annulus has been recognized as a good model for the study of the flows in planetary cores and stellar interiors \\cite{hid58,bus74,hid75,bus76,aue95,kan19} and also in engineering applications under strong rotation, \\textit{i.e.}~flow in gas turbines \\cite{mar05,mic15,pit17}. The flow dynamics in this system can give insights into flows in geophysical context and flows in industrial processes. In this work, we aim to study the heat transfer and the turbulent structures in ACRBC.\n\n\n\n\n\\section{Governing equations, simulations, and experiments}\n\\subsection{Governing equations and numerical simulations}\n\nUsing the parameter definitions as shown in Fig.~\\ref{fig:setup}(b), the governing dimensionless Boussinesq equations in a rotating reference frame \\cite{lop13} can be expressed as\n\\begin{small}\n\\begin{align}\n \\nabla \\cdot {\\bf u} &= 0 \\\\\n\\frac{\\partial \\theta}{\\partial t} +{\\bf u}\\cdot \\nabla \\theta &= \\frac{1}{\\sqrt{RaPr}}\\nabla^2 \\theta \\\\\n\\frac{\\partial \\bf{u}}{\\partial t} +{\\bf u}\\cdot \\nabla {\\bf u} &= -\\nabla p+\\text{Ro}^{-1} \\hat{\\bm{\\omega}} \\times {\\bf u} \\nonumber \\\\ \n&\\phantom{=} +\\sqrt{\\frac{Pr}{Ra}}\\nabla^2 {\\bf u} -\\theta \\frac{2(1-\\eta)}{(1+\\eta)}\\bf r\n\\end{align}\n\\end{small}\n\\noindent where $\\bm{\\hat{\\omega}}$ is the unit vector pointing in the direction of the angular velocity, $\\bf u$ is the velocity vector normalized by the free fall velocity $\\sqrt{\\omega ^2\\frac{R_o+R_i}{2}\\beta \\Delta L}$, $t$ is the dimensionless time normalized by $\\sqrt{L\/(\\omega ^2\\frac{R_o+R_i}{2}\\beta\\Delta)}$, and $\\theta$ is the temperature normalized by $\\Delta = T_h - T_c$. Here, $\\omega$, $R_i$, $R_o$, and $\\Delta$ are the rotational speed, the thickness of the fluid layer between the two cylinders, the outer radius of the inner cylinder, the inner radius of the outer cylinder, and the temperature difference between the hot ($T_h$) and cold ($T_c$) cylinders, respectively.\n\n\nFrom the above governing equations, the relevant control parameters in ACRBC are the Rayleigh number (characterizing the thermal driving strength) \n$Ra=\\frac{1}{2}\\omega^2(R_o+R_i)\\beta \\Delta  L^3\/(\\nu \\kappa)$,\nthe inverse Rossby number (measuring Coriolis effects)\n$Ro^{-1}=2(\\beta \\Delta (R_o+R_i)\/(2L))^{-1\/2}$,\nthe Prandtl number (fluid property) $\\text{Pr}=\\nu\/\\kappa$, and the radius and aspect ratio (geometric properties) $\\eta=R_i\/R_o$ and $\\Gamma=H\/L$. The key response parameter of the system is the Nusselt number (measuring the ratio of the total heat transport over the conductive one) $\\text{Nu}=-Q\\ln(\\eta)\/(\\alpha \\Delta 2 \\pi H)$. Here, $Q$ is the measured heat input through the outer cylinder into the system per unit of time; $\\alpha=\\kappa\\rho c_p$ is the thermal conductivity of the working fluid with $\\rho$ and $c_p$ being the density and the specific heat capacity of the fluid, respectively; and $H$ is the height of the gap between two cylinders. It should be noted that the definition of $\\text{Nu}$ in ACRBC is slightly different from that in classical RBC because of the cylindrical geometry with a heat flux in the radial direction, and the detailed derivations of the $\\text{Nu}$ for ACRBC are documented in the Supplementary Materials.\n\n\nDirect numerical simulations (DNS) are performed using an energy conserving second-order finite-difference code \\cite{ver96,poe15cf,zhu18afid}. The code has been extensively validated and used in previous studies in both the Cartesian and cylindrical coordinates for convective systems \\cite{poe15cf,zhu18afid,zhu18prl,zhu18np}. In all numerical simulations and experiments, the radius ratio is fixed at $\\eta=0.5$. No-slip boundary conditions were used for the velocity and constant temperature boundary conditions for the inner and outer cylinders. Periodic boundary conditions were used for the top and bottom surfaces. For three-dimensional simulations, the aspect ratio was chosen the same as in the experiments $\\Gamma=H\/L=1$ (the only two exceptions are for the cases at $\\text{Ra}=1.16\\times10^9$ and $\\text{Ra}=2.2\\times10^9$, where the height of the domain is $L\/4$ and $L\/8$, respectively, as the flows here are quasi two-dimensional). For the cases of which the flows are quasi two-dimensional at large  $\\text{Ra}$, we use two-dimensional simulations. Pr was fixed at 4.3 for all the simulations. The adequate resolution was ensured for all cases, for example, at $\\text{Ra}=4.7 \\times 10^8$, $4608\\times384\\times384$ grid points were used and at  $\\text{Ra}=4.7 \\times 10^{10}$ with grid points $18432\\times1536$. To obtain sufficient statistics, each simulation was run at least 200 free fall time units. As reported in \\citep{kun10,kun11}, the boundary layers in rotation system are expected to be thiner compared with classical RBC. Hence, the verification of boundary layer grid resolution is necessary. A posteriori check of grid resolution shows that at $\\text{Ra}=4.7\\times10^{10}$, there are 48 grid points inside thermal boundary layers and 64 grid points inside viscous boundary layers, which guarantees to resolve the boundary layer adequately. Besides, we have conducted a set of grid independence studies and checked the spatial resolution in bulk region to resolve all relevant scales (Kolmogorov scale and Batchelor scale). The parameter space that was explored can be found in Fig. \\ref{fig:setup}(c). More details about the DNS are documented in the Supplementary Materials.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{fig1_mod.pdf}\n\\caption{{\\bf System configuration and parameter space.} (a) Three-dimensional render of the experimental set-up. A fluid is confined in a rotating cylindrical annulus having its inner(blue) and outer(red) cylindrical surfaces made out of copper, which is known for its excellent thermal conductivity. The bottom plate is made of teflon for thermal insulation, and top plate is made of plexiglass allowing flow visualization. The coolant pipes and electric cables go through the hollow stainless steel shaft and connect to a slip ring and a rotary union. The toothed pulley is driven by a servo motor. (b) Schematical diagram of the set-up, which defines the geometric parameters. (c) Explored parameter space of $\\text{Ra}$ and $\\text{Ro}^{-1}$ for our study of supergravitational thermal convection. The red discs, the blue circles and the blue triangles correspond to the parameters used in the experiments (EXP), in the three-dimensional (3D) numerical simulations and in the two-dimensional (2D) numerical simulations, respectively. The parameter space is divided into three regimes according to the influence of Coriolis force. In addition, for comparison we have also performed the simulations for $\\text{Ro}^{-1}=10^{-5}$ (not shown here), for which the Coriolis force is negligible. For more details about the experimental apparatus, we refer to the Supplementary Materials and movie S1.}\n\\label{fig:setup}\n\\end{figure*}\n\n\n\\bigskip\n\n\\subsection{Experiments}\nExperiments are performed in a cylindrical annulus with solid-body rotation as sketched in Fig.~\\ref{fig:setup} (a). The two concentric cylinders are machined from a solid piece of copper to control the symmetry of the system, and their surfaces are electroplated with a thin layer of nickel to prevent oxidation. The inner cylinder has an outer radius of $R_i=\\unit{120}{\\milli\\meter}$, and the outer cylinder has an inner radius of $R_o=\\unit{240}{\\milli\\meter}$, resulting in a gap of L=$\\unit{120}{\\milli\\meter}$, and a radius ratio of $\\eta = R_i\/R_o = 0.5$. The gap with a height $H=\\unit{120}{\\milli\\meter}$ is sandwiched by a top plexiglass plate and a bottom teflon plate, resulting in an aspect ratio of $\\Gamma = H\/L = 1$. Plexiglass with an excellent transparency is used as the material of the top lid, allowing us to visualize the flow field, whereas teflon with an excellent corrosion resistance and a good strength is machined as the bottom base. These end plates and cylinders are fixed together and leveled on a rotating aluminum frame with a rotation rate up to \\unit{546}{rpm}. Four silicone rubber film heaters are attached to the outside of the outer cylinder, and are supplied by a DC power supply (Ametek, \\texttt{XG 300-5}) with 0.005\\% long-term stability. We use water as our working fluid, which has $\\text{Pr} \\approx 4.3$. \n\n\nInside the cold inner cylinder, 16 channels are machined for the coolant fluid to pass through, and the inner cylinder is regulated at constant temperature using a temperature-controlled circulating bath (PolyScience, \\texttt{AP45R-20-A12E}). The coolant pipes and electric cables go through the hollow stainless steel shaft and connect with a slip ring and a rotary union (Moflon, \\texttt{MEPH200}), which has 2 channels for liquids, 6 channels for powers (\\unit{2220}{\\watt}), and 48 channels for electrical signals. The shaft is driven by a toothed belt and the pulley has a gear ratio of $60:32\\approx1.88:1$. The whole system is powered by a servomotor (Yaskawa, \\texttt{SGM7G-1AA}), which has a rated power of \\unit{11}{\\kilo\\watt}.\n\n\nFor high-precision temperature and heat flux measurements, it is essential to minimize the heat leakage from the experimental apparatus to the surroundings. Various thermal shields that are regulated at appropriate temperatures are installed in the system to prevent heat losses.  Furthermore, the whole system is placed in a big toughened glass box where the temperature is controlled by a Proportional-Integral-Derivative (PID) controller to match the mean temperature of the bulk fluid. For more detailed descriptions on the experimental set-up, we refer to Supplementary Materials. The measurement techniques are documented in Methods. \n\n\n\n\n\\begin{figure*} [ht]\n\\centering\n\\includegraphics[width=\\linewidth]{fig2_mod.pdf}\n\\caption{{\\bf Effects of the Coriolis force on the heat transfer and flow structures.} (a) Nusselt number ($\\text{Nu}$) as a function of $\\text{Ro}^{-1}$ for $\\text{Ra}=10^6$, $2.2\\times10^6$, $4.7\\times10^6$, $10^7$, $2.2\\times10^7$, $4.7\\times10^7$, $10^8$, $2.2\\times10^8$ and $4.7\\times10^8$ for $\\text{Pr} = 4.3$. (d) Root mean square axial velocity fluctuation $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$ versus $Ro^{-1}$ for $\\text{Ra}=10^7$ and $\\text{Ra}=10^8$. (b,c,e,f) Instantaneous temperature fields from DNS at $\\text{Ra}=10^8$ for $\\text{Ro}^{-1}=0.1$, $0.5$, $1$, and $25$ ($\\text{Pr} = 4.3$). The inner and outer surfaces locate 0.02 $L$ away from the cold and hot cylinder correspondingly.}\n\\label{fig:Ro}\n\\end{figure*}\n\n\n\\section {Results} \n\n\n\n\\subsection{Effects of the Coriolis force on the heat transport and flow structures}\nA series of numerical simulations are performed to explore the influence of the Coriolis force on the heat transfer and flow structures. In numerical experiments, $\\text{Ro}^{-1}$ varies from $10^{-2}$ to $4\\times 10^2$ with several fixed Rayleigh numbers $\\text{Ra}=10^{6}$, $2.2\\times10^6$, $4.7\\times10^6$, $10^7$, $2.2\\times10^7$, $4.7\\times10^7$, $10^8$, $2.2\\times10^8$, and $4.7\\times10^8$. As shown in Fig.~\\ref{fig:Ro} (a), for these $\\text{Ra}$ studied, the influence of Coriolis force can be divided into three regimes. In regime \\uppercase\\expandafter{\\romannumeral1} ($\\text{Ro}^{-1}<\\text{Ro}^{-1}_{c1}$), the Coriolis force is small and negligible as compared to the buoyancy so that the $\\text{Nu}$ does not depend on $\\text{Ro}^{-1}$. In regime  \\uppercase\\expandafter{\\romannumeral3} ($\\text{Ro}^{-1}>\\text{Ro}^{-1}_{c2}$), the rotation is so strong that the flow is nearly constrained by the effect of the Taylor-Proudman theorem~\\cite{pro16,tay22}, resulting in a quasi-two-dimensional flow state and thus a reduction of $\\text{Nu}$ compared with regime \\uppercase\\expandafter{\\romannumeral1}. For regime \\uppercase\\expandafter{\\romannumeral2} ($\\text{Ro}^{-1}_{c1}<\\text{Ro}^{-1}<\\text{Ro}^{-1}_{c2}$), the flow is governed by the combination of $\\text{Ra}$ and $\\text{Ro}^{-1}$ with rich flow states. In this regime, the general trend is that Nu decreases with $\\text{Ro}^{-1}$. The instantaneous temperature fields for several $\\text{Ro}^{-1}$ at $\\text{Ra}=10^8$ (see Fig.~\\ref{fig:Ro}(b,c,e,f)) from DNS support the explanation of this $\\text{Ro}$ dependence on the heat transport. As illustrated in Fig.~\\ref{fig:Ro}(b), the flow is three-dimensional at $\\text{Ro}^{-1}=0.1$ because the Coriolis force is too small to influence the flow effectively. However, with $\\text{Ro}^{-1}$ increasing, the enhanced Coriolis force tends to suppress vertical variation of the convection flow, and the flow gradually becomes two-dimensional, which is a manifestation of the Taylor-Proudman theorem. The two-dimensionalization of the flow field should be responsible for the reduction of heat transport at $\\text{Ro}^{-1} > \\text{Ro}^{-1}_{c1}$, which was also reported in \\cite{poe13}. As is evident in Fig.~\\ref{fig:Ro} (f), the flow is nearly two-dimensional at $\\text{Ro}^{-1} \\gtrsim \\text{Ro}^{-1}_{c2}$. So, further increased $\\text{Ro}^{-1}$ saturates the influence of the Coriolis force, which explains the nearly constant $\\text{Nu}$ when $\\text{Ro}^{-1} \\gtrsim \\text{Ro}^{-1}_{c2}$. Moreover, we use root mean square axial velocity fluctuation $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$ to measure the influence of the Taylor-Proudman theorem, as shown in Fig.~\\ref{fig:Ro}(d). It shows that $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$ has nearly a constant value when $\\text{Ro}^{-1}<\\text{Ro}^{-1}_{c1}$, then gradually decreases with $\\text{Ro}^{-1}$, and finally approaches to around zero after  $\\text{Ro}^{-1}>\\text{Ro}^{-1}_{c2}$. The overall trend of $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$ versus $\\text{Ro}^{-1}$ is consistent with the dependence of $\\text{Nu}$ on $\\text{Ro}^{-1}$ in general.\n\nWe have noticed that the critical $\\text{Ro}^{-1}_{c1}$ and $\\text{Ro}^{-1}_{c2}$ depend on the $\\text{Ra}$. As illustrated in Fig.~\\ref{fig:Ro} (d), for $\\text{Ra}=10^7$, we have $\\text{Ro}^{-1}_{c1}\\approx0.1$ and $\\text{Ro}^{-1}_{c2}\\approx10$, while for $\\text{Ra}=10^8$, we have $\\text{Ro}^{-1}_{c1}\\approx0.15$ and $\\text{Ro}^{-1}_{c2}\\approx15$.  According to the numerical data explored, we can roughly determine the boundaries of the three regimes, which is plotted in Fig.~\\ref{fig:Ro} (a) with green lines. In addition, we have also included the boundaries in Fig.~\\ref{fig:setup} (c), and extrapolated the boundaries to the parameter space of experiments. (More detailed discussions of the flow structures, $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$, and $\\text{Ro}^{-1}_{c}$ for these $\\text{Ra}$ cases are available in Supplementary Materials.) \n\n\n\n\n\\begin{figure*} [ht]\n\\centering\n\\includegraphics[width=\\linewidth]{fig3_mod.pdf}\n\\caption{{\\bf Global heat transport.} (a) Nusselt number ($\\text{Nu}$) as a function of $\\text{Ra}$ from experiments (the solid symbols), DNS (the open symbols) in ACRBC and the prediction from Grossmann-Lohse (G--L) theory\\cite{gro00} in classical RBC (dashed line). (b) The same plots as (a), but the vertical axis is compensated. Inset: an enlarged portion of the compensated plot at the large $\\text{Ra}$ regime, which shows the transition of the effective scaling exponent ($\\text{Nu} \\propto \\text{Ta}^{\\gamma}$) to $\\gamma > 1\/3$.}\n\\label{fig:heat}\n\\end{figure*}\n\n\n\\subsection{Global turbulent heat transport}\nAs shown in the section above, $\\text{Nu}$ does not depend much on $\\text{Ro}^{-1}$ for high $Ro^{-1}$ regime, in which we will study how heat transfer depends on $\\text{Ra}$ in this section. We have performed 48 experiments and more than 130 numerical simulations. Fig.~\\ref{fig:setup}(c) shows the parameter space of the experimental and numerical studies. In the experimental studies, we note that the existence of Earth's gravity and lids is unavoidable. Several numerical simulations have been performed to study the influences of Earth's gravity and lids, which show that their effects on $\\text{Nu}$ are small. \n In addition, we note that the centrifugal force increases linearly with radial location. To study the influence of nonuniform driving force, we have performed two sets of simulations with the radial-dependent centrifugal acceleration $\\omega^2 r$ and with a constant artificial acceleration $\\omega^2 (R_o+R_i)\/2$. The results show that the effect of the radial-dependent gravity has a similar role as the non-Oberbeck-Boussinesq effect \\citep{gro00,ahl06,sug09}, which does not have much influence on the heat transport and flow structures in the current parameter regime (see the Supplementary Materials for details). \n\n\n\n\nFig.~\\ref{fig:heat} (a) shows the measured $\\text{Nu}$ as a function of $\\text{Ra}$ for different $\\text{Ro}$. Each measurement lasts at least 4 hours after the system has reached a thermally steady situation to ensure a statistically stationary state (for the detailed measurement procedure, see the Supplementary Materials). As being evident in Fig.~\\ref{fig:heat}(a), the experiments and numerical simulations are in excellent agreement. Combining experiments and simulations, this study covers more than four decades of $\\text{Ra}$, \\textit{i.e.} from $10^6$ to $6.5\\times 10^{10}$. The range of $\\text{Ro}^{-1}$ is from 18 to 58 in experiments, and the data series show a consistent dependence of $\\text{Nu}$ on $\\text{Ra}$. For comparison, we also plot the data calculated from Grossmann--Lohse (G--L) theory \\cite{gro00} in the corresponding $\\text{Ra}$ range. \nTo better reveal the local exponent, we plot the compensated data in Fig.~\\ref{fig:heat}(b). The experimental and numerical data have a lower amplitude as compared to the classical RBC (G--L line) due to the different flow geometry. However, the scaling dependence of $\\text{Nu}$ versus $\\text{Ra}$ shows a good agreement between the data and G-L theory at $\\text{Ra} \\lessapprox 10^{10}$ with a scaling exponent $\\gamma=0.27\\pm0.01$, which is close to the typical value found in two-dimensional RBC. \n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{fig4_mod.pdf}\n\\caption{{\\bf Revolution of convective rolls.} (a, b) Experiments at $\\text{Ra}=1.8\\times 10^9$ and $\\text{Ro}^{-1}=13.8$. (a) Snapshots of streak images revealing the flow patterns. Seeing from the top, the whole system rotates clockwise in experiments. The corresponding movie is available in movie S2. (b) Time series of local temperature fluctuations. The measured point locates \\unit{30}{\\milli\\meter} away from the cold inner cylinder and at mid-height. (c, d) Simulations at $\\text{Ra}=10^7$, and $\\text{Ro}^{-1}=1$: (c) Snapshots (Top view) of instantaneous temperature fields. (d) The averaged azimuthal velocity profile along the radial direction. The reference of the frame is on the clockwise direction, and the corresponding movie is available in the Supplementary Movie.}\n\\label{fig:flowpattern}\n\\end{figure*}\n\nIt is very unexpected that the local effective exponent $\\gamma$ of $\\text{Nu} \\propto \\text{Ra}^{\\gamma}$ even exceeds one-third at $\\text{Ra} \\gtrsim 4\\times 10^{10} $ in experiments. \nIs this scaling regime connected to the appearance of ultimate regime of Rayleigh--B\\'enard turbulence \\cite{kra62,gro11,he12a}? If it is not in the ultimate regime, what causes this steep scaling exponent?  \nAs our current $\\text{Ra}$ is much lower than the transition $\\text{Ra}$ for ultimate turbulence observed in the classical RBC, we cannot provide a concrete answer on this question. The possible answers are that (i) because of the different way of the driving and the geometry, the transition Ra to the ultimate regime at the current system is lower than that in the classical RBC. (ii) Another possibility is that the enhanced local slope is due to the emergence of new flow states. As demonstrated in Fig.~\\ref{fig:setup} (c), with $\\text{Ra}$ increasing, the location of ($\\text{Ra},\\, \\text{Ro}^{-1}$) where the enhanced local effective scaling appears seems to move from regime III to regime II. Thus the flow may change from a two-dimensional flow state to a three-dimensional flow state due to the strong thermal driving, resulting in a higher $\\text{Nu}$ in this $\\text{Ra}$ range. \nThe two-dimensional simulations for high $\\text{Ra}$ coincide well with the experiments, indicating that within the $\\text{Ra}$ range explored in two-dimensional simulations the flow is still quasi-two-dimensional. Unfortunately, we cannot push the simulations to the Ra regime with the effective exponent larger than one-third, in which the flow may not be in the quasi-two-dimensional state. We incline to anticipate that the changing from a two-dimensional flow state to a three-dimensional flow state may be responsible for the enhancement of local effective exponent at $Ra \\gtrsim 4\\times10^{10}$. \nWe notice that the scaling range with $\\gamma > 1\/3$ is very narrow in the current work, therefore further work at even higher Ra is needed to clarify this issue. \n\n\n\\subsection{Zonal flow}\n\n\nWe now study the dynamics of zonal flow experimentally and numerically. The \"zonal flow\" phenomenon has been investigated in experiments of geophysical and astrophysical flows \\cite{hei05,har15}. Fig.~\\ref{fig:flowpattern}(b) shows the time series of local temperature fluctuations in water at $\\text{Ra}=6.6\\times 10^9$, $\\text{Pr}=4.3$, and $\\text{Ro}^{-1}=18$, which unexpectedly shows a noticeable periodicity. To prove that this novel phenomenon is connected to the azimuthal movement of the coherent flow, we perform flow visualization with aqueous glycerol solution as mentioned in Methods to demonstrate the flow pattern.\n\n\nFig.~\\ref{fig:flowpattern}(a) shows some typical streak images from flow visualization at three different time (for video, see movie S2), where there are four pairs of rolls parallel to the rotating axis. For reference, we highlight one of the convective rolls using a yellow ellipse. From Fig.~\\ref{fig:flowpattern} (aI--aIII), unexpectedly the convection rolls move clockwise around the center with a faster rotation rate than the background rotation of the experimental system, which we name \"zonal flow\". The flow visualization further justifies that the revolution of the cold plume arms triggers the periodic temperature signals measured by the thermistors. \n\n\nNext to the experiments, we also find this phenomenon in the simulations (see Fig.~\\ref{fig:flowpattern}(c) and movie S3).\nWe use a black ellipse to highlight a selected cold plume. As shown in Fig.~\\ref{fig:flowpattern}(c),\non the reference of the (clockwise) rotating frame, the plume arms still evolve in a clockwise direction, indicating a net rotation of the coherent flow (zonal flow).\nThis zonal flow is further quantified through the averaged (axially and azimuthally in space and over time) azimuthal velocity profile versus the radial distance from the inner cylinder wall $r$, as shown in Fig.~\\ref{fig:flowpattern}(d).\n\n\nWhat causes this net rotation of the convection rolls? \nThe dashed lines in Fig. \\ref{fig:flowpattern}(c\\uppercase\\expandafter{\\romannumeral1}) mark the direction of hot\/cold plumes without Coriolis force. But because of the effects of Coriolis force, the plumes deflect to their left when the system rotates clockwise. As is evident in the Fig. \\ref{fig:flowpattern}(c\\uppercase\\expandafter{\\romannumeral1}) where the plumes are just formed in the beginning stage, the deflected angle of hot plumes ($\\angle a$) is approximately equal to the deflected angle of cold plumes ($\\angle b$). However, because of the different curvature of the cylinders, the similar deflected angles of the hot and cold plumes induce the different effects. The hot plumes directly affect the position (A) where the cold plumes are ejected, resulting in the clockwise rotation of the cold plumes and flow, whereas the impact of the cold plumes does not directly affect the motion of the hot plumes due to a relatively large distance the impact position of the cold plumes (B) and the ejecting position of the hot plumes. Thus, the hot plumes win and push the overall flow to move in the same direction of the background rotation. \nWe also perform the experiments and simulations with the opposite direction of the background rotation, and the results are consistent. In addition, several numerical simulations have been performed to test the dependence of zonal flow on the radius ratio $\\eta$, which shows that the zonal flows become weaker and weaker with $\\eta$ increasing from 0.4 to 0.9. The difference in the curvature of the two cylinders plays the key role for the net rotation of convection rolls. One may expect to observe different types of zonal flows at different $\\eta$, such as two large scale winds with the opposite directions near the plates \\cite{har15}. This remarkable net rotation of the coherent flow structures along the same direction of the background rotation could be connected to many relevant flow phenomena in nature, which deserves systematical studies in the future.\n\n \n\n\\section{Summary}\nIn summary, we have experimentally and numerically studied the global heat transport and turbulent flow structures in a rotating annulus with hot outer cylinder and cold inner cylinder, \\textit{i.e.}, ACRBC. In experiments, the mean centrifugal acceleration covers the range from 5 to 60 times gravitational acceleration. We show that the effective scaling exponent of global heat transport transitions to $\\gamma=1\/3$ at $\\text{Ra} \\approx 10^{10}$ and finally exceeds $\\gamma > 1\/3$ for $\\text{Ra} \\gtrapprox 4\\times 10^{10}$. \nUnexpectedly, we observed the faster azimuthal motion of the coherent overall flow structure faster than the background rotation of the system, which we call zonal flow, and provided a physical understanding on this remarkable net rotation motion of the coherent structures. This novel experimental approach sheds new lights on how to efficiently extend the parameter regimes for the study of buoyancy driven turbulent flows. Furthermore, the findings in the current system driven by the centrifugal acceleration can help us understand phenomena in geophysical and astrophysical flows. \n\n\n\\section{Methods}\n{\\it Measurement techniques.} In our experiments, all the temperature measurements are based on negative temperature coefficient thermistors. We use $6\\frac{1}{2}$-digit multimeter (Keithley, \\texttt{2701}) to measure the resistances of the thermistors, and the resistance can be converted to temperature using the Steinhart-Hart equation\\cite{lav97}. Two types of thermistors are used: One with a head diameter of \\unit{2.5}{\\milli\\meter} (Omega, \\texttt{44131}) is used to measure the temperature of the inner and outer cylinders, and the other with a head diameter of \\unit{300}{\\micro\\meter} (Measurement Specialties, \\texttt{GAG22K7MCD419}) is inserted into the convection cell to resolve fast temperature fluctuations, which is located \\unit{30}{\\milli\\meter} away from the cold cylinder and at mid-height in the vertical direction. \n\nTo visualize the flow field, we use nylon fibers with length $l = \\unit{3}{\\milli\\meter}$, and diameter $d =  \\unit{0.5}{\\milli\\meter}$. Nylon has a density of $\\rho_{p}\\approx \\unit{1.14}{\\gram\\per\\centi\\meter\\cubed}$ that is a little heavier than water. In the flow visualization experiments, we use aqueous glycerol solution with 54\\% glycerol by volume to match the density of nylon. The properties of the aqueous glycerol solution are listed in the Supplementary Materials for reference. Four light-emitting diods are used as light source, and a charge-coupled device camera (Ximea, \\texttt{MD028MU-SY}) is put on the top of the cell to record images. As the system rotates rapidly, it is difficult to make the camera rotate synchronously. So we keep the camera fixed, and then set the frame rate of the camera the same as the rotational speed. Processing the images taken by camera through a MATLAB code, we can get streak images to visualize the flow field. More processing details can be found in the Supplementary Materials.\n\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsubsection{Irreducible algebraic representations}\n\nThe simplest irreducible algebraic representations are the weighted Specht modules $S^{\\lambda,k}$ where as an $S_n$ representation this is just the usual Specht module $S^\\lambda$, and each copy of $\\mathbb{C}^\\times$ just acts by scalars via the character $z \\to z^k$.\n\nIf $\\lambda^1, \\lambda^2, \\dots, \\lambda^m$ are partitions of total size $n$, and $k_1 < k_2 < \\dots < k_m$ are distinct integers then the induced representation \n\\begin{equation}\\label{algirr}\n\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^m|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{\\lambda^1,k_1} \\otimes S^{\\lambda^2,k_2} \\otimes \\dots \\otimes S^{\\lambda^m,k_m})\n\\end{equation}\nis irreducible and moreover every irreducible algebraic representation of $T \\rtimes S_n$ is obtained this way. Hence we see the irreducible algebraic representations of $T \\rtimes S_n$ can be naturally labeled by collections of partitions of total size $n$ indexed by the integers .\n\n\\subsubsection{Induction of representations}\n\nMoreover, it is easy to describe the induction of two weighted Specht modules of the same $\\mathbb{C}^\\times$ weight. We just get a direct sum of weighted Specht modules of same $\\mathbb{C}^\\times$  weight with multiplicities coming from the usual Littlewood-Richardson rule. More precisely: \n$$\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\mu|})} ^ {\\mathbb{C}^\\times \\wr S_{|\\lambda| + |\\mu|} }(S^{\\lambda,k} \\otimes S^{\\mu,k}) = \\bigoplus_\\nu  c_{\\lambda, \\mu}^\\nu S^{\\nu, k}$$\nIt's then clear how to decompose the induction of two arbitrary irreducibles, you just run the Littlewood-Richardson rule separately on each $\\mathbb{C}^\\times$-weight. In other words the Grothendieck ring of $\\bigoplus_n \\text{Rep}(\\mathbb{C}^\\times \\wr S_{n})$ with monoidal structure coming from induction is just a direct sum indexed by $\\mathbb{Z}$ of copies of the ring of symmetric functions $\\Lambda$.\n\n\\subsubsection{Restriction to $S_n$}\n\nFinally, we will close out this section by noting that because irreducible representations of $T \\rtimes S_n$ are just induced up from products of weighted Specht modules, we can easily describe the restriction of arbitrary irreducible $T \\rtimes S_n$ representations to $S_n$. The irreducible representation \n\n$$\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^m|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{\\lambda^1,k_1} \\otimes S^{\\lambda^2,k_2} \\otimes \\dots \\otimes S^{\\lambda^m,k_m})$$\njust becomes\n$$\\text{Ind}_{S_{|\\lambda^1|} \\times S_{|\\lambda^2|} \\times \\dots S_{|\\lambda^m|}}^{S_{n} }(S^{\\lambda^1} \\otimes S^{\\lambda^2} \\otimes \\dots \\otimes S^{\\lambda^m})$$\nwhich of course can be decomposed using the Littlewood-Richardson rule.  Explicitly in terms of symmetric functions we have the following corollary.\n\n\\begin{corollary}\\label{restmult}\nThe multiplicity of the Specht module $S^\\mu$ inside the restriction to $S_n$ of $$\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^m|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{\\lambda^1,k_1} \\otimes S^{\\lambda^2,k_2} \\otimes \\dots \\otimes S^{\\lambda^m,k_m})$$ is equal to the multiplicity of the Schur function $s_\\mu$ in the product $s_{\\lambda^1}s_{\\lambda^2}\\dots s_{\\lambda^m}$\n\n\\end{corollary}\n\nAs suggested in the introduction we see that the difficulty in understanding the $S_n$ action on $\\overline{W(\\lambda)_\\nu}$ lies in understanding it as a $T \\rtimes S_n$ representation, and that then restricting to $S_n$ is straightforward.\n\n\\end{subsection}\n\n\\begin{subsection}{Polynomial representations of $T \\rtimes S_n$}\n\nWe say that an algebraic representation of $T \\rtimes S_n$ is \\emph{weakly polynomial} if the $T$-weights that occur are all polynomial weights. The weighted Specht module $S^{\\lambda, k}$ is weakly polynomial if and only if $k \\ge 0$, and more generally an irreducible representation of the form (\\ref{algirr}) is weakly polynomial if and only if $k_i \\ge 0$ for all $i$.  The class of weakly polynomial representations is preserved by induction and the corresponding Grothendieck ring of $\\bigoplus_n \\text{Rep}^{wp}(\\mathbb{C}^\\times \\wr S_{n})$ is again a direct sum of copies of $\\Lambda$, this time indexed by $\\mathbb{Z}_{\\ge 0}$.\n\nLet $V$ denote the standard $n$-dimensional representation of $T \\rtimes S_n$ obtained by restricting the defining representation of $GL_n$ to $T \\rtimes S^n$. As a representation of $S_n$ this is just the standard $n$-dimensional permutation representation, and $T$ acts on the permutation basis with weights $(1,0,0,\\dots, 0)$, $(0,1,0,0,\\dots 0)$, $\\dots$, and $(0,0,\\dots, 0, 1)$. In the notation of (\\ref{algirr}) we have that $$V \\cong \\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{n-1}) \\times (\\mathbb{C}^\\times \\wr S_{1})} ^ {\\mathbb{C}^\\times \\wr S_{n} }(S^{(n-1),0} \\otimes S^{(1),1}).$$\n\nWe say that a representation of $T \\rtimes S_n$ is \\emph{polynomial of degree $d$} if it is a direct summand of copies of $V^{\\otimes d}$, and say that a representation is \\emph{polynomial} if it is a direct sum of polynomial representations of various degrees.  In particular, a representation of $T \\rtimes S_n$ is polynomial if and only if it is a direct summand of the restriction of a polynomial representation of $GL_n$. \n\n\\medskip\n\n\\noindent \\textbf{Warning:} We'll note that unlike the classes of algebraic and weakly polynomial representations, induction from $(\\mathbb{C}^\\times \\wr S_{n}) \\times (\\mathbb{C}^\\times \\wr S_{m})$ to $\\mathbb{C}^\\times \\wr S_{n+m}$ does not preserve the class of polynomial representations.  However it is obvious by the definition that polynomial representations are preserved by the internal tensor product of $T \\rtimes S_n$ representations.\n\n\n\\subsubsection{Irreducible polynomial representations}\n\nThe following proposition characterizes which irreducible weakly polynomial representations are polynomial.\n\n\\begin{proposition}\\label{polirr}\nGiven partitions $\\lambda^1, \\lambda^2, \\dots, \\lambda^m$ of total size $n$, and distinct nonnegative integers $0 \\le k_1 < k_2 < \\dots < k_m$ then the irreducible weakly polynomial representation\n\n$$\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^m|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{\\lambda^1,k_1} \\otimes S^{\\lambda^2,k_2} \\otimes \\dots \\otimes S^{\\lambda^m,k_m})$$\nis polynomial (of degree $k_1+k_2+\\dots k_m$) if and only if either $k_1 \\ne 0$ or $k_1 = 0$ and $\\lambda^1$ is a one-row partition.\n\n\\end{proposition}\n\nBefore proving Proposition \\ref{polirr} it will be convenient to introduce a new labeling for these representations.  If $\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell$ is a sequence of (possibly empty) partitions of total size $m \\le n$ then define $$V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell} := \\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{n-m}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^\\ell|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{(n-m),0} \\otimes S^{\\lambda^1,1} \\otimes \\dots \\otimes S^{\\lambda^\\ell,\\ell})$$\n\nFor example in this notation the standard $n$-dimensional representation $V$ from the previous section is now denoted by $V^{(1)}$, and if instead we took the $n$-dimensional permutation representation for $S_n$ and had $T$ act on the permutation basis with weights $(2,0,0,\\dots, 0)$, $(0,2,0,0,\\dots 0)$, $\\dots$, and $(0,0,\\dots, 0, 2)$ this representation would be labeled $V^{\\emptyset, (1)}$.  Under this labeling the degree of $V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}$ is given by $|\\lambda^1| + 2|\\lambda^2| + 3|\\lambda^3| + \\dots + \\ell | \\lambda^\\ell|.$\n\nWe'll note that adding empty partitions to the end of the sequence $\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell$ doesn't change the corresponding representation.  It will occasionally be useful to think of this as being a labeling by infinite sequences of partitions where all but finitely many are the empty partition.  In particular, the trivial representation just corresponds to the sequence where all the partitions are empty.\n\nAt times we will want to use this notation to describe representations of $\\mathbb{C}^\\times \\wr S_{n}$ for different values of $n$, in which case we will add subscripts $V_{n}^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}$ to avoid ambiguity.\n\n\\medskip\n\nSince by definition all irreducible polynomial representations are direct summands of tensor powers of $V^{(1)}$ it will be useful to see how tensoring with $V^{(1)}$ looks under this labeling.  The following lemma gives such a description:\n\n\\begin{lemma}\\label{tensorV} \\hspace{1cm} \\\\\n\n \\noindent If $|\\lambda^1| + |\\lambda^2| + \\dots + |\\lambda^\\ell| = n$ then  $$V^{(1)} \\otimes V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell} = \\bigoplus_{\\substack{{\\lambda^i}' = \\lambda^i - \\square \\\\ {\\lambda^{i+1}}' = \\lambda^{i+1} + \\square}} V^{\\lambda^1, \\dots ,{\\lambda^i}' , {\\lambda^{i+1}}',  \\dots, \\lambda^\\ell}$$\nand if $|\\lambda^1| + |\\lambda^2| + \\dots + |\\lambda^\\ell| < n$\n\n$$V^{(1)} \\otimes V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell} = \\bigoplus_{\\substack{{\\lambda^i}' = \\lambda^i - \\square \\\\ {\\lambda^{i+1}}' = \\lambda^{i+1} + \\square}} V^{\\lambda^1, \\dots ,{\\lambda^i}' , {\\lambda^{i+1}}',  \\dots, \\lambda^\\ell} \\oplus \\bigoplus_{{\\lambda^1}' = \\lambda^1 + \\square} V^{{\\lambda^1}', \\lambda^2,\\dots \\lambda^\\ell}$$\nwhere the sums are over all ways of removing\/adding a box to the corresponding Young diagrams.\n\\end{lemma}\n\nIn terms of our labeling by sequences of partitions this just says that tensoring with $V^{(1)}$ corresponds to the following process:  If we start with one sequence of partitions we create new ones in all possible ways by removing one box from one partition and adding it to the next partition in the sequence, or if there aren't already $n$ total boxes we can add one box to the first partition.\n\nWe'll mostly be interested in the case where $n$ is strictly larger than the degree so we are always allowed to add a box to the first partition, but we've included the general case here for completion. In terms of the Young diagrams involved here is a picture of the first few rows of the Bratteli diagram for tensoring with $V^{(1)}$ when $n$ is at least $3$ (when $n$ is less than $3$ just delete all sequences of Young diagrams with more than $n$ boxes):\n\n\\medskip\n\n\\begin{tikzpicture}[scale=.7]\n  \\node (empty) at (0,0) {\\scalebox{1.5}{$\\emptyset$}};\n  \\node (1) at (0,-2) {$\\ydiagram{1}$};\n  \\node (a) at (0,-5) {$\\ydiagram{1,1}$};\n  \\node (b) at (3,-5) {$\\ydiagram{2}$};\n  \\node (c) at (6,-5) {$\\Big{(} \\scalebox{1.5}{$\\emptyset$},  \\ydiagram{1}$ \\ \\Big{  )}};\n  \\node (d) at (0,-9) {$\\ydiagram{1,1,1}$};\n   \\node (e) at (3,-9) {$\\ydiagram{2,1}$};\n \\node (f) at (6,-9) {$\\ydiagram{3}$};\n \\node (g) at (10,-9) {$\\Big{(} \\ \\ydiagram{1},  \\ydiagram{1}$ \\ \\Big{  )}};\n  \\node (h) at (14,-9) {$\\Big{(}$ \\scalebox{1.5}{$\\emptyset$}, \\scalebox{1.5}{$\\emptyset$},  $\\ydiagram{1}$ \\ \\Big{  )}};\n  \n  \\draw (empty)--(1)--(a)--(d)\n  (1)--(b)--(e)\n  (1) -- (c)--(g)\n  (a)--(e)\n  (b)--(f)\n  (a)--(g)--(b) \n (c)--(h);\n\\end{tikzpicture}\n\n\\medskip\n\n\\noindent \\textbf{Proof of Lemma \\ref{tensorV}:} $V^{(1)}$ is induced from $(\\mathbb{C}^\\times \\wr S_{n-1}) \\times (\\mathbb{C}^\\times \\wr S_{1})$ so in order to tensor with it we may use the push-pull formula:\n\n$$\\text{Ind}(U) \\otimes W \\cong \\text{Ind}(U \\otimes \\text{Res}(W))$$\n \nSo to calulate this first we need to restrict $V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}$ to $(\\mathbb{C}^\\times \\wr S_{n-1}) \\times (\\mathbb{C}^\\times \\wr S_{1})$. We already saw that induction between algebraic representations was controlled by the Littlewood-Richardson rule, hence by Frobenius reciprocity it follows that if $|\\lambda^1| + |\\lambda^2| + \\dots + |\\lambda^\\ell| = n$ then\n\n$$\\text{Res} (V_n^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}) = \\bigoplus_{{\\lambda^i}' = \\lambda^i - \\square} V_{n-1}^{\\lambda^1, \\dots {\\lambda^i}' \\dots, \\lambda^\\ell} \\otimes S^{(1),i}$$\n\nand if $|\\lambda^1| + |\\lambda^2| + \\dots + |\\lambda^\\ell| < n$ then \n\n$$\\text{Res}  (V_n^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}) = \\bigoplus_{{\\lambda^i}' = \\lambda^i - \\square} V_{n-1}^{\\lambda^1, \\dots {\\lambda^i}' \\dots, \\lambda^\\ell} \\otimes S^{(1),i} \\hspace{.4cm} \\oplus \\hspace{.4cm}  V_{n-1}^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell} \\otimes S^{(1),0}$$\n\nNext we need to tensor these factors with the $(\\mathbb{C}^\\times \\wr S_{n-1}) \\times (\\mathbb{C}^\\times \\wr S_{1})$ representation that is the trivial representation of $(\\mathbb{C}^\\times \\wr S_{n-1})$ tensored with $S^{(1),1}$. Of course tensoring with the trivial representation does nothing, and the second factor is just a character of $\\mathbb{C}^\\times$ so we see this just sends $V_{n-1}^{\\mu^1, \\dots, \\mu^\\ell} \\otimes S^{(1),i}$ to $V_{n-1}^{\\mu^1, \\dots, \\mu^\\ell} \\otimes S^{(1),i+1}$.\n\nFinally, we need to induce back up to $(\\mathbb{C}^\\times \\wr S_{n-1})$.  But we know by the Littlewood-Richardson rule that $$\\text{Ind} V_{n-1}^{\\mu^1, \\dots, \\mu^\\ell} \\otimes S^{(1),i+1} = \\bigoplus_{{\\mu^{i+1}}' = \\mu^{i+1} + \\square} V_n^{\\mu^1, \\dots {\\mu^{i+1}}', \\dots, \\mu^\\ell}$$\nPutting this all together gives the desired formula. \\hfill $\\square$\n\n\\medskip\n\nFrom here Proposition \\ref{polirr} follows immediately: By induction every irreducible subrepresentation of $(V^{(1)})^{\\otimes d}$ is of the desired form, and conversely it's clear that any sequence of partitions can be obtained from the empty sequence by the process described in Lemma \\ref{tensorV} of adding boxes to the first partition and moving them down the sequence arbitrarily (moreover the multiplicity is given by the number of downward walks on the Bratteli diagram from the empty sequence to the desired sequence of partitions).\n\n\\medskip\n\n\\noindent \\textbf{Example:} If $n$ is at least $2$ Lemma \\ref{tensorV} gives us the decomposition of the tensor square of the defining representation:\n\n$$V^{(1)} \\otimes V^{(1)} \\cong V^{(1,1)} \\oplus V^{(2)} \\oplus V^{\\emptyset, (1)}$$\nIn terms of $GL_n$ representations this is familiar, if we take the usual decomposition of $GL_n$ representations\n$$V \\otimes V = \\Lambda^2(V) \\oplus \\text{Sym}^2(V)$$\nthen $\\Lambda^2(V)$ restricts to $V^{(1,1)}$, and if we think of $\\text{Sym}^2(V)$ as degree $2$ polynomials of standard basis vectors $x_1, x_2, \\dots x_n$ then $V^{\\emptyset, (1)}$ corresponds to the span of the polynomials $x_i^2$ and $V^{(2)}$ corresponds to the span of the polynomials $x_ix_j$ with $i \\ne j$.\n\n\n\\end{subsection}\n\n\\end{section}\n\n\\begin{section}{Permutation modules, symmetric powers, and multiset combinatorics}\\label{combo}\n\n\\begin{subsection}{Weighted permutation modules}\n\nRecall for the symmetric group $S_n$ and a partition (or composition) $\\lambda =(\\lambda_1,\\lambda_2, \\dots, \\lambda_\\ell)$, the permutation module $M(\\lambda)$ is defined as $$M(\\lambda) := \\text{Ind}_{S_{\\lambda_1} \\times  S_{\\lambda_2} \\times \\dots  S_{\\lambda_\\ell}}^{S_n}(\\mathbf{1}) = \\text{Ind}_{S_{\\lambda_1} \\times  S_{\\lambda_2} \\times \\dots  S_{\\lambda_\\ell}}^{S_n}(S^{(\\lambda_1)} \\otimes S^{(\\lambda_2)}\\otimes \\dots \\otimes S^{(\\lambda_\\ell)}  )$$\nwhich combinatorially just means $M(\\lambda)$ is the linearized representation of the action of $S_n$ on ordered set partitions of $\\{1,2,\\dots,n\\}$ into sets $A_1, A_2, \\dots A_\\ell$ with $|A_i| = \\lambda_i$ for each $i$.\n\nFor algebraic representations of $T \\rtimes S_n$ there are natural analogs of these modules where we allow the copies of $\\mathbb{C}^\\times$ to act by characters and then induce up. Explicitly we have the \\emph{weighted permutation modules} $M(\\lambda, \\mathbf{k})$ where $\\lambda = (\\lambda_1,\\lambda_2, \\dots, \\lambda_\\ell)$ is a composition of $n$ and $\\mathbf{k} = (k_1, k_2, \\dots, k_\\ell)$ is a list of $\\mathbb{C}^\\times$ weights of the same length\n$$M(\\lambda, \\mathbf{k}) := \\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda_1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda_2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda_\\ell|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{(\\lambda_1),k_1} \\otimes S^{(\\lambda_2),k_2} \\otimes \\dots \\otimes S^{(\\lambda_\\ell),k_\\ell})$$\nNote that we can rearrange the parts of the composition or the order of the $k_i$'s, so long as we permute the other accordingly. In particular, at times it will be convenient to reorder things so that $\\lambda$ is a partition and other times it will be convenient to order the $k_i$'s in increasing order. Just note you can't in general do both simultaneously.\n\nIt's then easy to see that a weighted permutation module is polynomial if and only if each $k_i$ is non-negative and at most one of them is equal to zero.  In this case it will be convenient to reindex as before by lists of partitions $\\lambda^1, \\lambda^2, \\dots \\lambda^j$ of total size at most $n$. If we denote $\\lambda^i = (\\lambda^i_1, \\lambda^i_2,\\dots, \\lambda^i_{\\ell_i})$ and $|\\lambda^1|+ \\dots + |\\lambda^j| = m$ then define\n$$\\tilde{M}(\\lambda^1, \\lambda^2,\\dots, \\lambda^k) := M((n-m, \\lambda^1_1, \\lambda^1_2, \\dots, \\lambda^j_{\\ell_j}), (0, 1,1,\\dots, 2,2,\\dots, j,j))$$\nwhere there are $\\ell_1$ $1$'s, $\\ell_2$ $2$'s, and so on. \n\n In other words, $\\tilde{M}(\\lambda^1, \\lambda^2,\\dots, \\lambda^k)$ is just the representation induced from the representation of $(\\mathbb{C}^\\times \\wr S_{n-m})\\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^j|})$ where the first factor acts trivially and in the $\\mathbb{C}^\\times \\wr S_{|\\lambda^i|}$ factor we take a copy of the $S_n$ permutation module $M(\\lambda^i)$ with the copies of $\\mathbb{C}^\\times$ scaling by the character $z \\to z^i$.\n \n\\medskip\n\n\\noindent \\textbf{Examples:} The defining representation $V^{(1)}$ is isomorphic to the permutation module $\\tilde{M}((1))$.  If $n \\ge 2$ the tensor square $V^{(1)} \\otimes V^{(1)}$ decomposes as $\\tilde{M}((1,1)) \\oplus \\tilde{M}(\\emptyset, (1))$. In terms of the standard basis vectors $x_1, x_2, \\dots x_n$ of $V^{(1)}$, $\\tilde{M}((1,1))$ is the span of all vectors of the form $x_i \\otimes x_j$ with $i \\ne j$, and $\\tilde{M}(\\emptyset, (1))$ is spanned by the vectors $x_i \\otimes x_i$.\n\n\\medskip\n\nFor the rest of the section we will recall some facts about permutation modules for $S_n$ and describe appropriate analogs for weighted permutation modules.\n\n\\subsubsection{Decomposition into irreducibles}\n\nIn the unweighted case, the decomposition of a permutation module into irreducibles given by the Kostka numbers which combinatorially count semistandard Young tableaux.  Explicitly we have $$M(\\lambda) \\cong \\bigoplus_\\mu K_{\\lambda,\\mu} S^\\mu$$\nwe'll note however that this is just a special case of the Littlewood-Richardson rule (or even the Pieri rule) applied to the induction of the trivial representation of the Young subgroup corresponding to $\\lambda$.\n\nSo unsurprisingly, the decomposition of weighted permutation modules will also be governed by Kostka numbers.  Our notation for polynomial representations will be more convenient for stating this result (and ultimately will be the case we care about), but we'll note that it holds for algebraic representations as well.\n\n\\begin{lemma}\\label{multikostka}\n$$\\tilde{M}(\\lambda^1, \\lambda^2, \\dots, \\lambda^m) = \\bigoplus_{\\mu^1, \\mu^2, \\dots, \\mu^m} K_{\\mu^1,\\lambda^1}K_{\\mu^2,\\lambda^2}\\dots K_{\\mu^m,\\lambda^m} V^{\\mu^1,\\mu^2,\\dots, \\mu^m}$$\n\\end{lemma}\n\n\\noindent \\textbf{Proof:} We've already seen that for algebraic $T \\rtimes S_n$ modules induction is governed by the Littewood-Richardson rule on each $\\mathbb{C}^\\times$-weight separately.  As such, the decomposition of a weighted permutation modules is given by a product of Kostka numbers for each $\\mathbb{C}^\\times$ weight. \\hfill $\\square$\n\n\n\\subsubsection{$S_n$-invariants in weighted permutation modules}\n\nThe usual permutation module $M(\\lambda)$ contains a one dimensional space of $S_n$-invariants. Indeed this is true anytime a representation of a finite group is constructed as the linearization of a transitive group action, and the space of invariants is spanned by the sum of the elements of the set being acted upon (or alternatively one can see this using Frobenius reciprocity as we are inducing up the trivial representation from a subgroup).\n\nFor weighted permutation modules, asking for a $T \\rtimes S_n$ invariant vector is asking too much. Indeed this only occurs when $T$ acts trivially (which in the $\\tilde{M}(\\lambda, \\mathbf{k})$ notation is when $\\mathbf{k}$ is the zero vector), which just factors through the unweighted case. Nevertheless, we still have the following:\n\n\\begin{lemma}\\label{perminv}\nThe permutation module $\\tilde{M}(\\lambda, \\mathbf{k})$ has a one dimensional space of $S_n$-invariants.\n\\end{lemma}\n\n\\noindent \\textbf{Proof:} By construction $M(\\lambda, \\mathbf{k})$ restricted to $S_n$ is isomorphic to $M(\\lambda)$, which as we just said has a one dimensional space of invariants. \\hfill $\\square$\n\n\\subsubsection{Tensor products of weighted permutation modules}\n\nAnother key fact about permutation modules for $S_n$ is that the tensor product of two permutation modules is isomorphic to a direct sum of permutation modules.  More precisely, if $\\lambda$ and $\\mu$ are partitions of $n$ then $M(\\lambda) \\otimes M(\\mu)$ decomposes into permutation modules as follows:\n\nDefine a \\emph{tabloid} of type $(\\lambda, \\mu)$ to be a matrix of non-negative integers such that the $i$th row sums to $\\lambda_i$ and the $j$th column sums to $\\mu_j$ for all $i$ and $j$, and let $T(\\lambda, \\mu)$ be the set of all tabloids of type $(\\lambda, \\mu)$. Note that tabloids are just a convenient way of indexing the double cosets of the corresponding Young subgroups. For every tabloid $T \\in T(\\lambda, \\mu)$, let $M(T)$ denote the corresponding permutation module where we think of the nonzero entries of $T$ as a composition of $n$. Then we have that\n$$M(\\lambda) \\otimes M(\\mu) = \\bigoplus_{T(\\lambda,\\mu)} M(T)$$\nand in fact this holds over arbitrary rings, although we'll stick to complex representations here (see \\cite{Stanley} chapter 7). In particular we'll note that Lemma \\ref{perminv} then says the space of $S_n$-invariants in $M(\\lambda) \\otimes M(\\mu)$ is therefore equal to $|T(\\lambda,\\mu)|$, which is equal to the number of double cosets of the corresponding Young subgroups.\n\nFor weighted permutation modules essentially nothing changes, we just need to keep track of the weights.  If we are decomposing the tensor product $M(\\lambda, \\mathbf{k}) \\otimes M(\\lambda, \\mathbf{k'})$, then define $T(\\lambda, \\mu)$ as before but this time define $M(T, \\mathbf{k + k'})$ to be the weighted permutation module where if there is a nonzero entry in position $(i,j)$ we weight it by $k_i + k_j'$. We have the following lemma:\n\n\\begin{lemma}\\label{permtensor}\n$$M(\\lambda, \\mathbf{k}) \\otimes M(\\lambda, \\mathbf{k'}) =  \\bigoplus_{T(\\lambda,\\mu)} M(T, \\mathbf{k + k'})$$\n\\end{lemma}\n\nThe proof is essentially identical to the unweighted case which is well known and a simple application of Mackey's theorem, so we'll omit it.\n\n\\medskip\n\n\\noindent \\textbf{Example}: Suppose $n = 5$ and we want to decompose $M(\\lambda,\\mathbf{k}) \\otimes M(\\mu, \\mathbf{k'})$ where $\\lambda =  \\mu = (3,2)$, $\\mathbf{k} = (0,2)$ and $\\mathbf{k'} = (1,4)$.  The relevant tabloids in $T(\\lambda, \\mu)$ are:\n\n\n$$\\begin{pmatrix}\n3 & 0 \\\\\n0 & 2\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n2 & 1 \\\\\n1 & 1\n\\end{pmatrix} \n\\hspace{1cm}\n\\begin{pmatrix}\n1 & 2 \\\\\n2 & 0\n\\end{pmatrix}\n$$\nwhich give us the decomposition $$M(\\lambda,\\mathbf{k}) \\otimes M(\\mu, \\mathbf{k'}) = M((3,2), (1,6)) \\oplus M((2,1,1,1), (1,3,4,6)) \\oplus M((1,2,2), (1,3,4))$$\n\n\\medskip\n\nNext, suppose we want to decompose a tensor product of three or more permutation modules \n$$M(\\lambda^1) \\otimes M(\\lambda^2) \\otimes \\dots \\otimes M(\\lambda^m)$$\nas a direct sum of permutation modules.  Of course one could just repeatedly use the rule from above for a product of two permutation modules in terms of tabloids, however it turns out that it's often simpler to just do it in one step.\n\nDefine a \\emph{multitabloid} of type $\\lambda^1, \\lambda^2, \\dots, \\lambda^m$ to be a $m$-dimensional array of nonnegative integers $b_{i_1,i_2,\\dots, i_m}$  such that the $\\ell$th generalized row sum in the $j$th direction is $\\lambda^j_\\ell$. That is,\n$$\\sum_{\\substack{(i_1,i_2,\\dots, i_m) \\\\ i_j = \\ell}} b_{i_1,i_2,\\dots, i_m} = \\lambda^j_\\ell$$\nif we then let $T(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)$ denote the set of all such multitabloids. Then by an easy induction on the number of terms $m$ we get the following corollary:\n\n\\begin{corollary} \\ \\\\\n\\begin{enumerate}\n\\item For unweighted permutation modules $M(\\lambda^1), M(\\lambda^2), \\dots , M(\\lambda^m)$ we have: $$M(\\lambda^1) \\otimes M(\\lambda^2) \\otimes \\dots \\otimes M(\\lambda^m) = \\bigoplus_{B \\in T(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)} M(B)$$\n\n\\item For weighted permutation modules $M(\\lambda^1, \\mathbf{k}^1), M(\\lambda^2, \\mathbf{k}^2), \\dots , M(\\lambda^m,\\mathbf{k}^m)$ we have $$M(\\lambda^1, \\mathbf{k}^1) \\otimes M(\\lambda^2, \\mathbf{k}^2) \\otimes \\dots \\otimes M(\\lambda^m, \\mathbf{k}^m) = \\bigoplus_{B \\in T(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)} M(B, \\mathbf{k}^1+ \\mathbf{k}^2 + \\dots + \\mathbf{k}^m)$$\n\nWhere we view $T \\in T(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)$ as a composition by taking the nonzero entries, and $\\mathbf{k}^1+ \\mathbf{k}^2 + \\dots + \\mathbf{k}^m$ assigns the entry $b_{i_1,i_2,\\dots, i_m}$ of $B$ the weight $k^1_{i_1} + k^2_{i_2}+ \\dots +k^m_{i_m}$.\n\n\\end{enumerate}\n\\end{corollary}\n \n\n\n\\subsubsection{Stable tensor products for polynomial permutation modules}\n\nWe'll note our description of the decomposition of a tensor product of weighted permutation modules immediately gives us a compatible description for the polynomial permutation representations.  Indeed one just needs to recall that $\\tilde{M}(\\lambda^1, \\lambda^2, \\dots, \\lambda^k)$ has an extra factor of size $(n-m)$ in degree zero when we write it as $M(\\lambda, \\mathbf{k})$. \n\n The benefit of this notation for polynomial permutation representations is that it is independent of $n$ provided $n$ is large enough, and it is often easier to work with. For example, when we look at the matrices involved in decomposing a tensor product of permutation modules only thing that changes when we vary $n$ is the upper left entry. \n \n For example recall from a previous example the decomposition\n \n $$V^{(1)} \\otimes V^{(1)} = \\tilde{M}(1) \\otimes \\tilde{M}(1)= \\tilde{M}((1,1)) \\oplus \\tilde{M}(\\emptyset, (1))$$\n in our other notation this is $$M((n-1,1), (0,1)) \\otimes M((n-1,1), (0,1))$$\n and if $n$ is at least $2$ the corresponding tabloids are\n \n $$\\begin{pmatrix}\nn-2 & 1 \\\\\n1 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\nn-1 & 0 \\\\\n0 & 1\n\\end{pmatrix} $$\nand we see that as $n$ changes the only thing that changes is the entry in the upper left corner.  Moreover we'll note that since the sum of the entries of these matrices sum to $n$, one can always recover that entry from knowing the rest of the matrix and $n$.\n\nThis motivates the notion of \\emph{stable tabloids}, introduced in \\cite{Harman} to study periodicity phenomena in the modular representation theory of symmetric groups, and independently in \\cite{OZ1} to study certain symmetric functions related to $S_n$-representations.  If $\\lambda = (\\lambda_1, \\lambda_2,\\dots \\lambda_m)$ and $\\mu = (\\mu_1, \\mu_2, \\dots ,\\mu_\\ell)$ are compositions of arbitrary sizes a stable tabloid of type $(\\lambda,\\mu)$ is a $(m+1)\\times(\\ell+1)$ matrix (with rows and columns indexed from zero to $m$ and $\\ell$ respectively) such that:\n\n\\begin{enumerate}\n\\item The $(0,0)$ entry is empty and all other entries are non-negative integers.\n\n\\item For $i \\ge 1$ the $i$th row sums to $\\lambda_i$ and the $i$th column sums to $\\mu_i$.\n\n\\end{enumerate}\n\nThe point being that for $n$ sufficiently large these are just obtained from deleting the upper left entries from the matrices in $T(\\lambda[n],\\mu[n])$ where $\\lambda[n] = (n-|\\lambda|, \\lambda_1, \\lambda_2,\\dots \\lambda_m)$ and $\\mu[n] = (n-|\\mu|, \\mu_1, \\mu_2, \\dots ,\\mu_\\ell)$ are the padded partitions.\n\nIf we let $\\tilde{T}(\\lambda,\\mu)$ denote the set of stable tabloids of type $(\\lambda, \\mu)$ then it follows (See \\cite{Harman} or \\cite{OZ1}) that the decomposition of the tensor product of (unweighted) permutation modules $M(\\lambda[n]) \\otimes M(\\mu[n])$ for $n$ sufficiently large is just:\n\n$$M(\\lambda[n]) \\otimes M(\\mu[n]) = \\bigoplus_{T \\in \\tilde{T}(\\lambda,\\mu)} M(T[n])$$\nWhere $T[n]$ is the the padded composition of $n$ obtained by taking the nonzero entries of $T$ along with an extra part of size $(n-|T|)$.\n\nIn the weighted case, Lemma \\ref{permtensor} says that if we decompose a tensor product of two polynomial weighted permutation modules of the form $\\tilde{M}(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)$, then the decomposition into weighted permutation modules will again be indexed by weighted tabloids. To get the weights for a stable tabloid one just adds the corresponding weights on the rows and columns, with the caveat that the zeroth row and column have weight zero, in particular the ``extra\" part of size $n-|T|$ always gets weight zero.\n\n\\medskip\n\n\\noindent \\textbf{Remark:} We'll note that if $n$ is below the stable range we can still recover $T(\\lambda[n],\\mu[n])$ from $\\tilde{T}(\\lambda,\\mu)$ just by throwing out those stable tabloids where the sum of the entries is larger than $n$ and then filling in the upper left corner appropriately.  Hence in this setting it will often be easy to deduce non-stable decompositions from the calculations in the stable range.\n\n\\medskip\n\nAs in the non-stable case, if we want to take a tensor product of three or more permutation modules\n$$M(\\lambda^1[n]) \\otimes M(\\lambda^2[n]) \\otimes \\dots \\otimes M(\\lambda^m[n])$$\n we can combine things into a single combinatorial object, a \\emph{stable multitabloid}. If $\\lambda_i$ has length $\\ell_i$ these will be $m$-dimensional arrays of dimensions $(\\ell_1 +1) \\times (\\ell_2 +1) \\times \\dots \\times (\\ell_m +1)$ such that:\n \n \\begin{enumerate}\n \\item The $(0,0,\\dots, 0)$ position is left empty, and all other entries $b_{i_1,i_2, \\dots i_m}$ are non-negative integers.\n \n \\item The $r$th generalized row sum in the $j$th direction is $\\lambda^j_r$. That is, $$\\sum_ {\\substack{(i_1,i_2, \\dots i_m) \\\\ i_j = r}} b_{i_1,i_2, \\dots i_m} = \\lambda^j_r$$\n \n \\end{enumerate}\n As before this can be seen by a simple inductive argument on the number of terms $m$, iterating the $m=2$ case at every step. Similarly to the case with two factors we can extend this to tensor products of weighted permutation modules by keeping track of the  weights accordingly.\n\n\\end{subsection}\n\n\\begin{subsection}{Symmetric powers}\nWe are now ready to start talking about symmetric powers of the defining representation $V$ and tensor products thereof.  The key observation is that as a representation of $T \\rtimes S_n$ the symmetric power $Sym^k(V)$ decomposes as a direct sum of weighted permutation modules.\n\nMore precisely, let $A_k$ be the set of all $k$-tuples of non-negative integers $(a_1,a_2, \\dots, a_k)$ such that \n$$a_1 + 2a_2 + 3a_3+\\dots+ka_k = k$$\nand let $A_k^n$ denote the subset of $A_k$ such that\n$$a_1 + a_2 + a_3+\\dots+a_k \\le n$$\nin particular note that if $n \\ge k$ then $A_k^n = A_k$. The following proposition describes $Sym^k(V)$ as a sum of weighted permutation modules.\n\n\\begin{proposition}\\label{symdec}\n $$Sym^k(V) \\cong \\bigoplus_{A_k^n} \\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_k))$$\n\n\\end{proposition}\n\n\\noindent \\textbf{Proof:} If $x_1,x_2,\\dots x_n$ is the standard basis of $V$ then $Sym^k(V)$ is the space of homogeneous degree $k$ polynomials in the $x_i$'s.  Then for $(a_1,a_2, \\dots, a_k) \\in A_k^n$ consider the space spanned by all monomials $x_1^{b_1}x_2^{b_2} \\dots x_n^{b_n}$ where $b_i = 1$ for $a_1$ values of $i$, $b_i = 2$ for $a_2$ values of $i$, and so on.  This space is isomorphic to the weighted permutation module $\\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_k))$, with the monomials forming the permutation basis.  $Sym^k(V)$ is spanned by monomials and each monomial lies in some such space for a unique $(a_1,a_2, \\dots, a_k) \\in A_k^n$. \\hfill $\\square$\n\n\\medskip\n\nWe'll note that in this case the weighted permutation modules $\\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_k))$ involved are all irreducible $T \\rtimes S_n$ representations so we could also have written:\n\n$$Sym^k(V) \\cong \\bigoplus_{A_k^n} V^{(a_1),(a_2),(a_3), \\dots, (a_k)}$$\n\n\n\\subsubsection{Combinatorics of $A_k$}\n\nIn preparation for the next section we'd like to give two interpretations for $A_k$, which will generalize nicely to tensor products of symmetric powers.\n\n\\medskip\n\n\\noindent \\textbf{Polynomial interpretation:} Given a sequence $(a_1,a_2, \\dots, a_k)$ consider the polynomial $$a(x) := a_1x+a_2x^2+a_3x^3+\\dots+a_kx^k$$\nthe condition that $(a_1,a_2, \\dots, a_k) \\in A_k$ just corresponds to the condition that $a'(1) = k$, and the condition that $(a_1,a_2, \\dots, a_k) \\in A_k^n$ adds the additional constraint that $a(1) \\le n$. Hence $A_k^n$ can be naturally identified with the set of polynomials $a(x) \\in \\mathbb{Z}_{\\ge 0}[x]$ such that $a(0)= 0$, $a(1) \\le n$, and $a'(1) = k$.\n\n\\medskip\n\n\\noindent \\textbf{Multiset partition interpretation:} Suppose $M$ is a multiset.  A \\emph{multiset partition} of $M$ is a collection (i.e a multiset)  $\\{ M_1, M_2, \\dots, M_m\\}$ of non-empty sub-multisets of $M$ such that the multiplicity of an element of $M$ is equal to the sum of the multiplicities of it in the $M_i$'s. \n\n$A_k$ can naturally be thought of as an indexing set for multiset partitions of the multiset $M = \\{1,1,1, \\dots, 1\\} = \\{1^k\\}$ consisting of a single element with multiplicity $k$. Indeed an element $(a_1,a_2, \\dots, a_k) \\in A_k$ just corresponds to the unique multiset partition with $a_1$ parts of size $1$, $a_2$ parts of size 2, and so on. Under this identification $A_k^n$ is just indexing those multiset partitions of $M$ into at most $n$ parts.\n\n\n\\vspace{.5cm}\n\nAs mentioned before, in this case the decomposition into weighted permutation modules also gives the decomposition into irreducible $T \\rtimes S_n$ modules. If we forget about the action of $T$ this gives us a combinatorial interpretation for the decomposition of $Sym^k(V)$ as a representation of $S_n$.\n\n  For $(a_1, a_2, \\dots, a_k) \\in A_k^n$ define the \\emph{associated composition of $n$} as $(n-(a_1 + a_2 + a_3+\\dots+a_k), a_1, a_2, \\dots, a_k)$, and define the associated composition of $n$ to a multiset partition of $M = \\{1,1,1, \\dots, 1\\}$  via the identification above. \n\n\\begin{corollary} If $\\lambda$ is a partition of $n$ then the multiplicity of $S^\\lambda$ in $Sym^k(V)$ is equal to the number of pairs $(P,T)$ where $P$ is a multiset partition of $\\{1^k\\}$ with at most $n$ parts and $T$ is a semistandard Young tableau of shape $\\lambda$ and content equal to the associated composition of $n$ to $P$.\n\n\n\\end{corollary}\n\n\\noindent \\textbf{Proof:} Proposition \\ref{symdec} and the above interpretation tells us that $Sym^k(V)$ decomposes into weighted permutation modules indexed by these multiset partitions.  Restricting a weighted permutation modules to $S_n$ just gives an ordinary permutation module corresponding to the associated composition of $n$, and permutation modules decompose into irreducibles with multiplicities given by Kostka numbers which count semistandard Young tableaux.  \\hfill $\\square$\n\n\\medskip\n\nIn particular, the case where $\\lambda = (n)$ and we are just looking at the space of symmetric group invariants is of particular interest so we'll state it as a separate corollary.\n\n\\begin{corollary}\nThe space of $S_n$-invariants in $Sym^k(V)$  is equal to the number of multiset partitions of $\\{1^k\\}$ with at most $n$ parts.\n\n\\end{corollary}\n\n\\end{subsection}\n\n\\begin{subsection}{Tensor products of symmetric powers}\n\nNow let's extend this analysis from a single symmetric power of $V$ to a product of symmetric powers $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V).$$\nWe already saw how to decompose a single symmetric power into weighted permutation modules, and we also saw how to decompose a tensor product of two weighted permutation modules, so all that remains is to put it all together and keep track of the terms to get a concise combinatorial description.\n\n\\subsubsection{The two factor case}\n\nFirst let's go through how one would do this in the case where there are just two factors $Sym^{k_1}(V) \\otimes Sym^{k_2}(V)$ just to demonstrate the idea.\n\n\\begin{enumerate}\n\\item First one would use Proposition \\ref{symdec} to write  $$Sym^{k_i}(V) \\cong \\bigoplus_{A_{k_i}^n} \\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_{k_i}))$$ for $i = 1,2$.\n\n\\item Then for each pair in $(a,a') \\in A_{k_1}^n \\times A_{k_2}^n$ we would decompose $$\\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_{k_1})) \\otimes \\tilde{M}((a'_1),(a'_2),(a'_3), \\dots, (a'_{k_2}))$$\naccording to Lemma \\ref{permtensor} into weighted permutation modules indexed by stable tabloids of type $(a,a')$ (where we think of $a = (a_1,\\dots, a_{k_1})$ and $a' = (a'_1,\\dots, a'_{k_2})$ as compositions) with appropriate weights.\n\n\\end{enumerate}\n\nCombining this into a single step by summing over all stable tabloids as $(a,a')$ varies over $A_{k_1}^n \\times A_{k_2}^n$ we are summing over all $(k_1+1) \\times (k_2+1)$ matrices with rows and columns indexed from $0$ to $k_1$ and $k_2$ respectively such that:\n\n\\begin{itemize}\n\n\\item The $(0,0)$ entry is blank, and all other entries are non-negative integers $b_{ij}$.\n\n\\item  The sum of the entries weighted by their row number is equal to $k_1$ and the sum of the entries weighted by their column number is equal to $k_2$. That is,\n\n$$\\sum_{i,j} i b_{ij} = k_1 \\hspace{1cm} \\sum_{i,j} j b_{ij} = k_2.$$\n\n\n\\item The sum of the entries of the matrix is at most $n$ (if we are in the stable range where $n > k_1 +k_2$ this condition is redundant).\n\n\\end{itemize}\n\nFor each such matrix we get a weighted permutation module $\\tilde{M}(\\lambda_1,\\lambda_2, \\dots, \\lambda_{k_1 + k_2})$ where $\\lambda_\\ell$ is the partition obtained by taking the entries $b_{ij}$ with $i+j = \\ell$.\n\n\\medskip\n\n  For example if we want to decompose $Sym^2(V) \\otimes Sym^2(V)$ there are $8$ such stable tabloids that appear:\n  \n  \n   $$\\begin{pmatrix}\n\\ & 2 & 0 \\\\\n2 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 1 & 0 \\\\\n1 & 1 & 0 \\\\\n0 & 0 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 0 & 0 \\\\\n0 & 2 & 0 \\\\\n0 & 0 & 0\n\\end{pmatrix}  \n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 1 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 1 & 0\n\\end{pmatrix}$$\n\n   $$\n\\begin{pmatrix}\n\\ & 0 & 0 \\\\\n1 & 0 & 1 \\\\\n0 & 0 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 2 & 0 \\\\\n0 & 0 & 0 \\\\\n1 & 0 & 0\n\\end{pmatrix} \n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 0 & 1 \\\\\n2 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix}  $$\n\nWe then read off the corresponding weighted permutation modules by looking at the diagonals that go from the lower left to the upper right to obtain the decomposition\n\n\\begin{equation}\n\\begin{split}\nSym^2(V) \\otimes Sym^2(V) = \\tilde{M}((2,2)) \\oplus \\tilde{M}((1,1),(1)) \\oplus \\tilde{M}(\\emptyset, (2)) \\\\ \\oplus \\ 2\\tilde{M}((1),\\emptyset, (1)) \\oplus 2\\tilde{M}((2),(1)) \\oplus \\tilde{M}(\\emptyset, \\emptyset, \\emptyset, (1))\n\\end{split}\n\\end{equation}\nwhich holds for all $n \\ge 4$. If $n$ is less than $4$ we obtain the decomposition by keeping just those terms where the sum of the sizes of the partitions involved is at most $n$.\n\n\\subsubsection{The general case}\n\nNow let's extend this to the general case of $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V).$$\nHere the program will be basically identical to the two factor case\n\n\\begin{enumerate}\n\\item First one would use Proposition \\ref{symdec} to write  $$Sym^{k_i}(V) \\cong \\bigoplus_{A_{k_i}^n} \\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_{k_i}))$$ for $i = 1,2,\\dots, m$.\n\n\\item Then for each $m$-tuple in $(a^1,a^2,\\dots, a^m) \\in A_{k_1}^n \\times A_{k_2}^n\\times \\dots A_{k_m}^n$ we would decompose $$ \\bigotimes_i \\tilde{M}((a^i_1),(a^i_2),(a^i_3), \\dots, (a^i_{k_i}))$$\naccording to Lemma \\ref{permtensor} into weighted permutation modules indexed by stable multitabloids of type $(a^1,a^2, \\dots, a^m)$ (where we think of $a^i = (a^i_1,\\dots, a^i_{k_i})$ as compositions) with appropriate weights.\n\n\\end{enumerate}\n\nAs before we can combine this into a single step by counting up the multitabloids  as we vary our choice of $(a^1,a^2,\\dots, a^m) \\in A_{k_1}^n \\times A_{k_2}^n\\times \\dots A_{k_m}^n$. Now the objects we care about are $(k_1+1)\\times (k_2 +1) \\times \\dots \\times (k_m+1)$ arrays with rows, columns, etc. indexed from $0$ to $k_1$, $k_2$, and so on respectively such that:\n\n\n\\begin{itemize}\n\n\\item The $(0,0, \\dots , 0)$ entry is blank, and all other entries are non-negative integers $b_{i_1,i_2,\\dots, i_m}$.\n\n\\item  The sum of the entries weighted by their $jth$ index value is equal to $k_j$  for all $j$. That is:\n\n$$\\sum_{i_1,i_2,\\dots, i_m} i_j b_{i_1,i_2,\\dots, i_m} = k_j$$\n\n\n\\item The sum of the entries of the array is at most $n$ (if we are in the stable range where $n > k_1 +k_2 + \\dots +k_m$ this condition is redundant).\n\n\\end{itemize}\n\n\\medskip\n\nIf we let $A^n_{k_1,k_2,\\dots,k_m}$ denote the set of such arrays then then putting everything together gives the following proposition describing the decomposition of a product of symmetric powers of $V$ into permutation modules.\n\n\\begin{proposition}\\label{symproddec}\n$$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V) = \\bigoplus_{B \\in A^n_{k_1,k_2,\\dots,k_m}} \\tilde{M}(\\lambda_1,\\lambda_2, \\dots, \\lambda_{k_1 + k_2+\\dots +k_m})$$\nwhere $\\lambda_\\ell = \\lambda_\\ell (B)$ is the partition obtained from the $B$ by taking the nonzero entries $b_{i_1,i_2,\\dots,i_m}$ with $i_1+i_2+\\dots+i_m = \\ell$.\n\n\\end{proposition}\n\n\n\\subsubsection{Combinatorics of $A^n_{k_1,k_2,\\dots,k_m}$}\n\nAs in the case of a single symmetric power, we will now give two interpretations for the set $A^n_{k_1,k_2,\\dots,k_m}$, one in terms of polynomials and one in terms of multiset partitions.\n\n\\medskip\n\n\\noindent \\textbf{Polynomial interpretation}:  Given an array $B$ of numbers $b_{i_1, i_2, \\dots, i_m}$ (for $i_j \\ge 0$) with finitely many nonzero entries we can construct the polynomial\n\n$$P_B(x_1,x_2, \\dots x_m) := \\sum_{i_1, i_2, \\dots, i_m} b_{i_1, i_2, \\dots, i_m}x_1^{i_1}x_2^{i_2}\\dots x_m^{i_m}$$\nthen the condition that $B \\in A^n_{k_1,k_2,\\dots,k_m}$ translates into\n\n\\begin{enumerate}\n\\item $P_B(x_1,x_2, \\dots x_m)$ has constant term zero and all other coefficients are non-negative integers.\n\n\\item For $i =1, 2, \\dots m$ $$ \\frac{d}{dx_i}P_B(x_1,x_2, \\dots x_m) |_{(x_1,x_2, \\dots x_m) = (1,1,\\dots 1)} = k_i$$\n\n\\item$ P_B(1,1,\\dots 1) \\le n$\n\n\\end{enumerate}\n\nHence we may identify $A^n_{k_1,k_2,\\dots,k_m}$ with the set of such polynomials satisfying these three conditions.\n\n\\medskip\n\n\\noindent \\textbf{Multiset partition interpretation:} $A^n_{k_1,k_2,\\dots,k_m}$ has a natural bijection with the set of multiset partitions of $\\{1^{k_1}, 2^{k_2}, \\dots m^{k_m} \\}$.  Explicitly, an array $B \\in A^n_{k_1,k_2,\\dots,k_m}$ with entries $b_{i_1, i_2, \\dots, i_m}$ corresponds to the multiset partition where $\\{1^{i_1}, 2^{i_2}, \\dots, m^{i_m}\\}$ appears $b_{i_1, i_2, \\dots, i_m}$ times.\n\n\\vspace{.5cm}\n\nWe find the multiset partition interpretation to be more conceptually satisfying and will mostly use it to state results, however we've included the polynomial interpretation as we feel it may be more amenable to computations.\n\n\\medskip\n\n The \\emph{type}, $\\text{Type}(P)$, of a multiset partition $P \\vDash M$ is the sequence $\\lambda^1, \\lambda^2, \\lambda^3, \\dots$ of partitions where $\\text{Type}(P)^i = \\lambda^i$ records the multiplicities of the parts of $P$ of size $i$. \n \n  For example, if $M = \\{1,1,2\\}$ and $P = \\{\\{1\\},\\{1\\},\\{2\\}\\}$ then $\\text{Type}(P)$ is the sequence $\\lambda^1, \\lambda^2, \\lambda^3, \\dots$ where $\\lambda^1 = (2,1)$ and all other partitions are the empty set. This is since $P$ has three parts of size $1$, two of which are the same as one another (i.e. $\\{1\\}$) and one that appears with multiplicity $1$ (i.e $\\{2\\}$). By convention we'll drop off the trailing empty sets and just write $\\text{Type}(P) = ((2,1))$.  If instead we took $P' = \\{1,1,2\\}$ then $\\text{Type}(P') = (\\emptyset, \\emptyset, (1))$.\n \nTranslating Proposition \\ref{symproddec} into this language we get the following corollary:\n\n\\begin{corollary}\\label{symtoperm}\n\n$$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V) = \\bigoplus_{\\substack{P \\vDash \\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\} \\\\ |P| \\le n}}\\tilde{M}(\\text{Type}(P))$$\n\n\\end{corollary}\n\n\\medskip\n\nWe are now ready to give our main result, a combinatorial interpretation for the decomposition of a tensor product of symmetric powers into irreducible $T \\rtimes S_n$ modules.\n\n\\begin{proposition}\\label{maindecomp}\nThe multiplicity of $V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^j}$  inside $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$\nis equal to the number of tuples $(P,T_1,T_2, \\dots T_j)$ such that $P$ is a multiset partition of $\\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\}$ with at most $n$ parts and $T_i$ is a semistandard Young tableau of shape $\\lambda^i$ and content $\\text{Type}(P)^i$.\n\n\\end{proposition}\n\n\\noindent \\textbf{Proof:} Corollary \\ref{symtoperm} gives us a decomposition of this tensor product of symmetric powers into permutation modules indexed by multiset partitions. Lemma \\ref{multikostka} then gives us the decomposition of a permutation module in terms of Kostka numbers.  Finally, the combinatorial interpretation of Kostka numbers as counting semistandard Young tableaux gives the result. \\hfill $\\square$\n\n\\medskip\n\n\\noindent \\textbf{Remark:} In the case where each $k_i$ is $1$ this refines Proposition $5.10$ in \\cite{BHH} to the case of weighted permutation modules, but the proof is morally very similar.\n\n\\medskip\n\nIf we forget about the action of $T$, Corollary \\ref{symtoperm} recovers an interpretation of the decomposition of $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$ into irreducible representations of $S_n$ due to Orellana and Zabrocki (\\cite{OZ1} Theorem 5). \n\n Define the \\emph{unweighted type} $\\text{UType}(P)$ of a multiset partition $P$ to be the partition recording the multiplicities of the multisets in $P$ (in other words just combine the partitions making up $\\text{Type}(P)$ into a single partition). And let $\\text{UType}(P)[n]$ be the corresponding partition of $n$ obtained by adding a part of size $(n-|P|)$ to $\\text{UType}(P)$.\n\n\\begin{proposition}\nThe multiplicity of the Specht module $S^\\lambda$  inside $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$\nis equal to the number of tuples $(P,T)$ such that $P$ is a multiset partition of $\\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\}$ with at most $n$ parts and $T$ is a semistandard Young tableau of shape $\\lambda$ and content $\\text{UType}(P)[n]$.\n\\end{proposition}\n\n\\noindent \\textbf{Proof:} Upon restricting to $S_n$  Corollary \\ref{symtoperm} tells us that $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V) = \\bigoplus_{\\substack{P \\vDash \\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\} \\\\ |P| \\le n}}M(\\text{UType}(P)[n])$$\nas a representation of $S_n$.  Then the decomposition of $M(\\text{UType}(P)[n])$ is just given by appropriate Kostka numbers, which again count semistandard Young tableaux of fixed shape and content. \\hfill $\\square$\n\n\\medskip\n\n\\noindent \\textbf{Remark:} In \\cite{OZ1} Orellana and Zabrocki combine a pair $(P,T)$ into a single combinatorial object: a multiset tableau.  However morally these are the same description, and we'll leave it as an exercise to any interested parties to make explicit the bijection between these pairs and appropriate multiset tableaux.  \n\nWe'll also note that they stated their results in terms of a new basis of the ring of symmetric functions they defined corresponding to irreducible symmetric group representations of large symmetric groups.  In this context we'll mention that one should not expect such a basis to exist for $T \\rtimes S_n$, as (among other reasons) it is possible for two elements of $T \\rtimes S_n$ to be $GL_n$-conjugate but not $T \\rtimes S_n$-conjugate.\n\n\\medskip\n\nFinally, we'll close out this section by stating as a corollary the important special cases of the above propositions where we just look at the space of $S_n$-invariants.\n\n\\begin{corollary}\n\nIf $\\mu = (\\mu_1, \\mu_2, \\dots, \\mu_\\ell)$ is a partition of $k_1+k_2+\\dots+k_m$ with at most $n$ parts the dimension of $S_n$-invariants in the symmetrized weight space\n\n$$(Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V))_{\\bar{\\mu}}$$\nis equal to the number of multiset partitions of $\\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\}$ with parts of sizes $\\mu_1, \\mu_2, \\dots, \\mu_\\ell$. In particular, the dimension of the space of $S_n$-invariants inside full space $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$\nis equal to the number multiset partitions of $\\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\}$ with at most $n$ parts.\n\n\\end{corollary}\n\nIn particular if all the $k_i$'s are equal to $1$ and $m > n$ this recovers the fact that the space of $S_n$-invariants in $V^{\\otimes m}$ is given by the Bell number $B_m$.  If we let the $k_i$'s be arbitrary but assume $n$ is sufficiently large this gives a representation theoretic context for the generalized Bell numbers studied in \\cite{Griffiths}.\n\n\n\\end{subsection}\n\\end{section}\n\n\\begin{section}{Future directions and computations}\\label{future}\n\nThe main motivating problem of finding a combinatorial interpretation for the restriction of an irreducible polynomial representation of $GL_n$ to $T \\rtimes S_n$ remains open. In light of Proposition \\ref{maindecomp} and the fact that weighted permutations satisfy an upper triangularity property with respect to the irreducibles it is reasonable to expect that such an interpretation could be formulated in terms of multiset partitions and Young tableaux, however for the time being such an interpretation remains elusive.\n\nFor the remainder of the paper we will outline some future directions of study beyond this motivating question. In particular we will outline a version of Schur-Weyl duality for $T \\rtimes S_n$, highlight a connection between the restriction problem and the Foulkes conjecture, and include some low-degree computations of restriction from $GL_n$ to $T \\rtimes S_n$.\n\n\\begin{subsection}{Schur-Weyl duality for $T \\rtimes S_n$}\n\nNow we'll breifly describe a version of Schur-Weyl duality for $T \\rtimes S_n$.  As before let $V$ denote the defining representation of $GL_n$.  We have a tower of subgroups \n$$S_n \\subset T\\rtimes S_n \\subset GL_n$$\nacting on $V$ and hence on $V^{\\otimes k}$.  Therefore we get a reverse inclusion of endomorphism algebras \n$$\\text{End}_{S_n}(V^{\\otimes k}) \\supset \\text{End}_{T \\rtimes S_n}(V^{\\otimes k}) \\supset \\text{End}_{GL_n}(V^{\\otimes k})$$\nwhere the outer two terms are familiar instances of Schur-Weyl duality.\n\nOn the right we have classical Schur-Weyl duality in which $S_k$ acts $GL_n$-equivariantly on $V^{\\otimes k}$ by permuting the factors.  For any $n$ and $k$ these maps span $ \\text{End}_{GL_n}(V^{\\otimes k})$, and if $n \\ge k$ this provides an isomorphism $\\mathbb{C}[S_k] \\cong  \\text{End}_{GL_n}(V^{\\otimes k})$.\n\nOn the left the situation is similar. The partition algebra $Par(n,k)$ acts $S_n$-equivariantly on $V^{\\otimes k}$ for all $n$ and $k$. The map $Par(n,k) \\to \\text{End}_{S_n}(V^{\\otimes k})$ is always surjective, moreover the kernel has an explicit combinatorial description. In particular, if $n \\ge 2k$ this map is an isomorphism (See \\cite{HR}, \\cite{BH1}, \\cite{BH2} for more details).\n\nAs such one may expect that we can explicitly describe the endomorphism algebra $\\text{End}_{T \\rtimes S_n}(V^{\\otimes k})$ at least for $n$ sufficiently large compared to $k$ and indeed this is the case.  Recall that $Par(n,k)$ has a basis indexed by set-partitions of the set $\\{1,2,\\dots,k\\} \\cup \\{1',2',\\dots k'\\}$, we say that such a set-partition is \\emph{balanced} if each part $P$ of the partition satisfies $|P \\cap \\{1,2,\\dots,k\\}| = |P \\cap \\{1',2',\\dots,k'\\}|$. \n\n\\medskip\n\nThe following proposition describes a version of Schur-Weyl duality for $T\\rtimes S_n$:\n \n\\begin{proposition} Let $Par^{bal}(k)$ denote the subspace of $Par(n,k)$ spanned by the balanced set partitions.  The following holds:\n\\begin{enumerate}\n\\item $Par^{bal}(k)$ is a subalgebra of $Par(n,k)$.\n\\item The obvious identification of these subspaces for $Par(n,k)$ and $Par(m,k)$ is an isomorphism of algebras (motivating the notation $Par^{bal}(k)$ not involving $n$).\n\\item  $Par^{bal}(k)$ acts $T\\rtimes S_n$-equivariantly on $V^{\\otimes k}$ giving an algebra homomorphism $Par^{bal}(k) \\to  \\text{End}_{T \\rtimes S_n}(V^{\\otimes k})$, this is surjective for all values of $n$ and $k$ and is an isomorphism whenever $n \\ge k$.\n\n\n\\end{enumerate}\n\\end{proposition}\n\n\\noindent \\textbf{Sketch of proof:} For parts $1$ and $3$ we use the description of Schur-Weyl duality for $S_n$ and check that $Par^{bal}(k)$ is exactly the subalgebra of $Par(n,k)$ that preserves $S_n$-orbits of $T$-weights. For part $2$ one just notes that in the partition algebra $Par(n,k)$, the dependence on $n$ arises when we stack two partition diagrams and there are isolated components in the middle, but for balanced partitions that can never happen. \\hfill $\\square$\n\n\\medskip\n\nWe didn't pursue this direction any further, but here are a few problems we think may be interesting:\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Describe $Par^{bal}(k)$ by generators and relations.\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Is there a double centralizer property? In other words, does the image of $T \\rtimes S_n$ span the algebra of $Par^{bal}(k)$-endomorphisms of $V^{\\otimes k}$? If so, use this to describe the representations of $Par^{bal}(k)$.\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Describe the kernel of the natural map $Par^{bal}(k) \\to  \\text{End}_{T \\rtimes S_n}(V^{\\otimes k})$ for $n < k$.\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Describe the $T \\rtimes S_n$ endomorphism algebras of $V^{\\otimes r} \\otimes V^{* \\otimes s}$. We'll note that this should be something in between a partition algebra and a walled Brauer algebra.\n\\end{subsection}\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Describe diagrammatically the endomorphism algebras of $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$ as $S_n$ and $T \\rtimes S_n$ representations.  We'll go ahead and coin the terms ``multiset partition algebras\" and ``balanced multiset partition algebras\".\n\n\n\\begin{subsection}{$S_n$-invariants, plethysm, and Foulkes conjecture}\n\nA particularly important subcase of general problem of restricting an irreducible representation $W(\\lambda)$ from $GL_n$ to $T \\rtimes S_n$ is to understand the the space of $S_n$-invariants in each symmetrized weight space $W(\\lambda)_{\\bar{\\mu}}$ as $\\mu$ varies over $S_n$-orbits of weights.\n\nIf $\\mu$ has $m_1$ parts of size $1$, $m_2$ parts of size $2$, and so on.  Let \n\n$$S_{\\mu} := S_1^{m_1} \\times S_2^{m_2} \\times \\dots \\times S_k^{m_k}$$\ndenote the Young subgroup of $S_{|\\mu|}$ corresponding to $\\mu$, and let $N(\\mu)$ denote its normalizer in $S_{|\\mu|}$.  Note that $N(\\mu)$ is just the product of the wreath products $S_i \\wr S_{m_i}$. \n\nThe following proposition, essentially due to Gay, relates the space of $S_n$-invariants in the symmetrized weight space $W(\\lambda)_{\\bar{\\mu}}$to the representation theory of $S_{|\\lambda|}$.\n\n\\begin{proposition} \\textbf{(\\cite{Gay} Theorem 2)} The dimension of the space of $S_n$-invariants in $W(\\lambda)_{\\bar{\\mu}}$is equal to the multiplicity of the Specht module $S^\\lambda$ in $\\text{Ind}_{N(\\mu)}^{S_{|\\mu|}}(\\mathbf{1})$.\n\\end{proposition}\n\n\\noindent \\textbf{Remark:} We'll note that Gay only considered the ``zero\" weight space case where $\\mu = (a,a,a,\\dots,a)$.  However his proof easily generalizes, and moreover one can reduce the general case to this by a straightforward application of the Littlewood-Richardson rule for restricting to the Levi subgroup $GL_{m_1} \\times GL_{m_2} \\times \\dots \\times GL_{m_k}$.\n\n\\medskip\n\nFinding a combinatorial interpretation of such modules induced from normalizers of Young subgroups is an important open problem in combinatorial representation theory which in the language of symmetric functions is equivalent to decomposing the plethysms $h_a[h_b]$ into Schur functions. \n\nIn particular one important conjecture in this area is Foulkes conjecture from 1950, about embedding one such induced module into another.  Translated into the language of this paper Foulkes conjecture is the following:\n\n\\begin{conjecture} \\textbf{(Foulkes conjecture \\cite{Foulkes})} If $a < b$ then the space of $S_n$-invariants in the symmetrized weight space of weight $(a,a,\\dots, a, 0,0,\\dots,0)$ is at least as large as the space of $S_n$-invariants in the symmetrized weight space of weight $(b,b,\\dots, b, 0,0,\\dots,0)$ for any polynomial representation of $GL_n$ of degree $ab$.\n\n\\end{conjecture}\n\nWe'll note that the usual representation theoretic formulations of this conjecture are either entirely about general linear group representations or entirely about symmetric group representations. As far as we can tell this ``mixed\"  formulation of the conjecture seems to be missing from much of the literature on Foulkes conjecture.\n\n  It suggests a possible approach to the conjecture by studying the representation theory of the spherical subalgebra $eAe$, where $$A = U(\\mathfrak{gl}_n) \\rtimes \\mathbb{C}(S_n) \\hspace{.5cm} \\text{ and }  \\hspace{.5cm} e = \\frac{1}{n!}\\sum_{\\sigma \\in S_n} \\sigma$$\nwhich naturally acts on the space of $S_n$-invariants inside a $GL_n$ representation. \n\n\\medskip\n\nUsing Proposition \\ref{maindecomp} we can give a purely combinatorial formulation of a weak version of Foulkes conjecture for tensor products of symmetric powers:\n\n\\begin{conjecture} \\textbf{(Weak Foulkes conjecture)} If $a < b$ and $M$ is a multiset of size $ab$ then the number of multiset partitions of $M$ into $b$ parts of size $a$ at least as large as the number of multiset partitions of $M$ into $a$ parts of size $b$.\n\\end{conjecture}\n\nNote that in terms of symmetric functions this is equivalent to the conjecture that $h_b[h_a]-h_b[h_a]$ has nonnegative coefficients when expressed in the basis of monomial symmetric functions (whereas the full Foulkes conjecture says it has non-negative coefficients in the basis of Schur functions).\n\n\n\n\\end{subsection}\n\n\n\n\\subsection{Low degree calculations}\nProposition \\ref{maindecomp} combined with the Jacobi-Trudi identity gives us a combinatorial method for computing the decomposition of an irreducible polynomial representation of $GL_n$ to $T \\rtimes S_n$ (although not a positive combinatorial formula). Here are some explicit calculations for polynomial representations of degree at most $4$, where we always assume we are in the stable range with $n$ larger than the degree.\n\n$$ W( \\emptyset) \\longrightarrow V^\\emptyset $$\n$$W(1) \\longrightarrow V^{(1)} $$\n$$W(2) \\longrightarrow V^{(2)} \\oplus V^{\\emptyset, (1)}$$\n$$ W(1,1) \\longrightarrow V^{(1,1)}$$\n$$W(3) \\longrightarrow V^{(3)} \\oplus V^{(1),(1)} \\oplus V^{\\emptyset,\\emptyset, (1)}$$\n$$W(2,1) \\longrightarrow V^{(2,1)} \\oplus V^{(1),(1)}$$\n$$W(1,1,1) \\longrightarrow V^{(1,1,1)}$$\n$$W(4) \\longrightarrow V^{(4)} \\oplus V^{(2),(1)} \\oplus V^{\\emptyset, (2)} \\oplus V^{(1), \\emptyset, (1)} \\oplus V^{\\emptyset,\\emptyset, \\emptyset, (1)}$$\n$$W(3,1) \\longrightarrow V^{(3,1)} \\oplus V^{(1,1),(1)} \\oplus V^{(2),(1)} \\oplus V^{\\emptyset, (1,1)} \\oplus V^{(1),\\emptyset,(1)} $$\n$$ W(2,2) \\longrightarrow V^{(2,2)} \\oplus V^{(2),(1)} \\oplus V^{\\emptyset, (2)}$$\n$$W(2,1,1) \\longrightarrow V^{(2,1,1)} \\oplus V^{(1,1),(1)}$$\n$$W(1,1,1,1) \\longrightarrow V^{(1,1,1,1)}$$\n\n\n\n\n\\end{section}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we study the 2D viscous\nshallow water equations with a more general diffusion\n\\begin{equation}\\label{1.1}\n\\left\\{\n\\begin{array}{ll}\nh_t+\\mbox{div}(hu)=0,\\\\\nh(u_t+u\\cdot\\nabla u)-\\nu\\nabla\\cdot(hD(u))-\\nu\\nabla(h \\textrm{div}(u))+h\\nabla h=0, \\\\\nu(0,\\cdot)=u_0,\\,\\,h(0,\\cdot)=h_0 ,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $h(t,x)$ is the height of fluid surface,\n$u(t,x)=(u_1(t,x),u_2(t,x))$ is the horizontal velocity vector\nfield, $D(u)=\\f12(\\nabla u+\\nabla u^t)$ is the deformation tensor, and\n$\\nu>0$ is the viscous coefficient. If the diffusion terms in (\\ref{1.1}) are\nreplaced by $-\\nu\\nabla\\cdot(h\\nabla u)$,\nthen (\\ref{1.1}) turns into the usual viscous shallow water equations.\n\nRecently, the viscous shallow water equations have been widely\nstudied by Mathematicians, see the review paper\\cite{BDM}.  Bui\n\\cite{Bui} proved the local existence and uniqueness of classical\nsolutions to the Cauchy-Dirichlet problem for the shallow water\nequations with initial data $h_0$, $u_0$ in H\\\"{o}lder spaces as\nwell as $h_0$ bounded away from vacuum. Kloeden\\cite{Klo} and\nSundbye\\cite{Su1} independently proved global existence and\nuniqueness of classical solutions to the Cauchy-Dirichlet problem in\nSobolev spaces. Later, Sundbye\\cite{Su2} also proved global\nexistence and uniqueness of classical solutions  to the Cauchy\nproblem. However, for all above results (except \\cite{Bui}), the\nauthors only consider the case when the initial data $h_0$ is a\nsmall perturbation of some positive constant $\\bar h_0$ and $u_0$ is\nsmall in some sense. Very recently, Wang and Xu\\cite{WX} proved the local\nwell-posedness of the Cauchy problem in Sobolev spaces\nfor the large data $u_0$ and $h_0$ closing to $\\bar h_0$. More\nprecisely, they obtained the following result.\n\\begin{Theorem}\\label{WX}\\cite{WX} Let $\\bar{h}_0$ be a strictly positive constant and $s>2$.\nAssume that\n\\begin{eqnarray*}&&(i)\\quad(u_0,h_0-\\bar{h}_0)\\in\nH^s(\\mathop{\\bf R\\kern 0pt}\\nolimits^2)\\otimes H^s(\\mathop{\\bf R\\kern 0pt}\\nolimits^2);\\hspace{7cm}\\\\\n&&(ii)\\quad\\|h_0-\\overline{h}_0\\|_{H^s}\\ll\\overline{h}_0.\\end{eqnarray*} Then\nthere exist a positive time $T$ and a unique solution $(u,h)$ of\n$(\\ref{1.1})$ such that\n\\begin{eqnarray}\\label{1.2} u,\\,\\, h-\\bar h_0\\in\nL^\\infty([0,T],H^s),\\quad \\nabla u\\in L^2([0,T]; H^s).\n\\end{eqnarray}\nMoreover, there exists a strictly positive constant $c$ such that if\n\\begin{eqnarray}\\label{1.3}\n\\|u_0\\|_{H^s}+ \\|h_0-\\bar{h}_0\\|_{H^s}\\le c,\n\\end{eqnarray}\nthen we can choose $T=+\\infty$.\n\\end{Theorem}\n\nOne purpose of this paper is to study the well-posedness of\n$(\\ref{1.1})$ for the initial data with the minimal regularity.\nFor the incompressible Navier-Stokes equations, such research has been initiated by\nFujita and Kato\\cite{FK}, see also \\cite{Can1, Can2, Mey2} for other relevant results. They proved local well-posedness for the incompressible Navier-Stokes equations\nin the scaling invariant space. The scaling invariance means that\nif $(u,p)$ is a solution of the incompressible Navier-Stokes equations with  initial data $u_0(x)$, then\n\\begin{eqnarray}\n\\label{1.4}u_\\lambda(t,x)\\triangleq\\lambda u(\\lambda^2 t,\\lambda x),\\quad\np_\\lambda(t,x)\\triangleq \\lambda^2 p(\\lambda^2 t,\\lambda x)\n\\end{eqnarray}\n is also a solution of the incompressible Navier-Stokes equations with\n$u_{0,\\lambda}\\triangleq \\lambda u_0(\\lambda x)$.\nObviously, $\\dot H^{\\frac d 2-1}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$\nis a scaling invariant space under the scaling of (\\ref{1.4}), i.e.\n$$\\|u_\\lambda\\|_{\\dot{H}^{\\frac d 2-1}}=\\|u\\|_{\\dot{H}^{\\frac d 2-1}}.$$\nThe equations $(\\ref{1.1})$ have no scaling invariance like the incompressible Navier-Stokes equations.\nHowever, due to the similarity of the structure between (\\ref{1.1}) and the incompressible Navier-Stokes equations,\nwe still solve (\\ref{1.1}) for initial data whose regularity fits with the scaling of (\\ref{1.4}).\nIt should be pointed out that R. Danchin was the first to consider the similar problem for the compressible\nNavier-Stokes equations, and some ideas of this paper is motivated by \\cite{D1}.\n\nThe second purpose of this paper is to prove the local well-posedness of (\\ref{1.1}) under  more natural assumption that\nthe initial height is bounded away from zero.\nFor the initial data with slightly higher regularity,\nthis can be easily obtained by modifying the argument of Danchin\\cite{D5}.\nHowever, for the initial data with low regularity, his method is not applicable any more, since\nthe proof of \\cite{D5} relies on the fact that some profits can be gained from the inclusion map\n$B^s\\hookrightarrow L^\\infty$ in the case of $s>\\frac d2$.\nFor this reason, we have to introduce some kind of weighted Besov space $E^s_T$(see Section 3) which is crucial to\nget rid of the condition that the initial height $h_0$ is close to $\\bar{h}_0$.\nOne important observation is that the $E^s_T$ norm of the solution is small for small time $T$.\n\n\n\nBefore stating our main result, let us first introduce some notations and definitions.\nChoose a radial function $\\varphi\\in{\\cal S}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ such that\n\n$${\\rm supp}\\,\\varphi\\subset\\{\\xi\\in \\mathop{\\bf R\\kern 0pt}\\nolimits^d;\n\\frac{5}{6}\\le|\\xi|\\le \\frac{12}{5}\\},\\quad \\sum_{k\\in{\\mathbf Z}}\\varphi(2^{-k}\\xi)=1,\n\\quad\\xi\\in\\mathop{\\bf R\\kern 0pt}\\nolimits^d\\setminus \\{0\\}.$$\nHere $\\varphi_k(\\xi)=\\varphi(2^{-k}\\xi)$,\n$k\\in {\\mathbf Z}$.\n\n\n\\begin{Def}\\label{Def1.1}\nLet $k\\in {\\mathbf Z}$, the Littlewood-Paley projection operators $\\Delta_k$ and $S_k$ are defined as follows\n$$\\Delta_kf=\\varphi(2^{-k}D)f,\\quad\nS_kf=\\sum_{j\\le k-1}\\Delta_jf, \\quad \\textrm{for}\\quad f\\in {\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d).$$\n\\end{Def}\n\n\nWe denote the space ${\\cal Z'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ by the dual space of ${\\cal\nZ }(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)=\\{f\\in {\\cal S}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d);\\,D^\\alpha \\hat{f}(0)=0;\n\\forall\\alpha\\in\\mathop{\\bf N\\kern 0pt}\\nolimits^d \\,\\mbox{multi-index}\\}$, it also can be\nidentified by the quotient space of ${\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)\/{\\cal P}$ with the\npolynomials space ${\\cal P}$. The formal equality $$f=\\sum_{k\\in{\\mathbf Z}}\\Delta_kf$$ holds true for $f\\in {\\cal Z'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$\nand is called the homogeneous Littlewood-Paley decomposition. It\nhas nice properties of quasi-orthogonality: with our choice of\n$\\varphi$,\n\\begin{eqnarray}\\label{1.5}\n\\Delta_j\\Delta_kf=0\\quad i\\!f\\quad|j-k|\\ge 2\\quad \\textrm{and}\n\\quad \\Delta_j(S_{k-1}\\Delta_k\nf)=0\\quad i\\!f\\quad|j-k|\\ge 4.\n\\end{eqnarray}\n\n\\begin{Def}\\label{Def1.2}\nLet $s\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$, $1\\le p, r\\le+\\infty$. The homogeneous Besov space $\\dot{B}^{s}_{p,r}$\nis  defined by\n$$\\dot{B}^{s}_{p,r}=\\{f\\in {\\cal Z'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d):\\,\\|f\\|_{\\dot{B}^{s}_{p,r}}<+\\infty\\},$$\nwhere\n\\begin{eqnarray*}\n&&\\|f\\|_{\\dot{B}^{s}_{p,r}}=\\left\\{\\begin{array}{l}\\displaystyle\\bigg(\\sum_{k\\in{\\mathbf Z}} 2^{ksr}\n\\|\\Delta_kf\\|^r_{p}\\bigg)^{\\frac{1}{r}},\\quad\\mbox{for}\\quad r<+\\infty,\\\\\n\\displaystyle\\sup_{k\\in{\\mathbf Z}}2^{ks}\\|\\Delta_kf\\|_{p},\\quad\\mbox{for}\\quad r=+\\infty.\\end{array}\\right.\\end{eqnarray*}\n\\end{Def}\nIf\\, $p=r=2$, $\\dot{B}^s_{2,2}=\\dot{H}^s$,\nand if\\, $d=2$, we have $\\dot{B}^{1}_{2,1}\\hookrightarrow L^\\infty$ and\n$$\\|f\\|_{\\infty}\\le C\\|f\\|_{\\dot{B}^{1}_{2,1}}.$$\nWe refer to \\cite{Ch1,Tr} for more details.\n\nIn addition to the general time-space space such as $L^\\rho(0,T; \\dot{B}^{s}_{p,r})$,\nwe introduce a useful mixed time-space homogeneous Besov space\n$\\widetilde{L}^\\rho_T(\\dot{B}^{s}_{p,r})$\nwhich is initiated in \\cite{CL} and is used in the proof of the uniqueness.\n\\begin{Def}\\label{Def1.3}\nLet $s\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$, $1\\le p,r,\\rho\\le+\\infty$, $0<T\\le+\\infty$. The  mixed time-space homogeneous\nBesov space $\\widetilde{L}^r_T(\\dot{B}^s_{p,q})$ is defined by\n$$\\widetilde{L}^\\rho_T(\\dot{B}^{s}_{p,r})=\n\\{f\\in {\\cal Z'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^{d+1}):\\,\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^{s}_{p,r})}<+\\infty\\},$$\\\nwhere$$\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^{s}_{p,r})}=\n\\bigg\\|2^{ks}\n\\bigg(\\int_0^T\\|\\Delta_kf(t)\\|^\\rho_{p}dt\\bigg)\n^{\\frac{1}{\\rho}}\\bigg\\|_{\\ell^r}.$$\n\\end{Def}\nUsing the Minkowski inequality, it is easy to verify that\n$$L^\\rho_T(\\dot{B}^{s}_{p,r})\\subseteq\\widetilde{L}^\\rho_T(\\dot{B}^{s}_{p,r})\n\\quad\\hbox{if}\\quad \\rho\\le r\\quad \\textrm{and}\\quad\n\\widetilde{L}^\\rho_T(\\dot{B}^{s}_{p,r})\\subseteq{L}^\\rho_T(\\dot{B}^{s}_{p,r})\n\\quad\\hbox{if}\\quad \\rho\\ge r.$$\n\nNext, we introduce a hybrid-index Besov space\nwhich plays an important role in the study of compressible\nfluids and is initiated in \\cite{D1,D2}.\n\\begin{Def}\\label{Def1.4}\nLet $s$, $\\sigma\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$, and set\n$$\\|f\\|_{\\widetilde{B}^{s,\\sigma}_2}\\triangleq\\sum_{k\\le0}2^{ks}\\|\\Delta_k f\\|_{2}\n+\\sum_{k>0}2^{k\\sigma}\\|\\Delta_k f\\|_{2}.$$\nLet $m=-[\\frac{d}{2}+1-s]$, we define\n\\begin{eqnarray*}\n&&\\widetilde{B}^{s,\\sigma}_2(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)=\\{f\\in{\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d):\n\\|f\\|_{\\widetilde{B}^{s,\\sigma}_2}<+\\infty\\}\\quad\n\\hbox{if}\\quad m<0,\\\\\n&&\\widetilde{B}^{s,\\sigma}_2(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)=\\{f\\in{\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)\/{\\cal P}_m:\n\\|f\\|_{\\widetilde{B}^{s,\\sigma}_2}<+\\infty\\}\\quad\n\\hbox{if}\\quad m\\ge0,\\end{eqnarray*}\nwhere ${\\cal P}_m$ denotes the set of polynomials of degree $\\le m$.\n\\end{Def}\n\nThroughout this paper, we will denote\n$\\dot{B}^{s}_{2,1}$ by $B^s$, and $\\widetilde{B}^{s,\\sigma}_2$ by\n$\\widetilde{B}^{s,\\sigma}$. The following facts can be easily verified by using the definition of $\\widetilde{B}^{s,\\sigma}$:\\vspace{0.1cm}\n\n(i) $\\widetilde{B}^{s,s}=\\dot{B}^s_{2,1}$;\n\n(ii)\\,If $s\\le \\sigma$, then $\\widetilde{B}^{s,\\sigma}=\\dot{B}^s_{2,1}\\cap\\dot{B}^\\sigma_{2,1}$.\nOtherwise, $\\widetilde{B}^{s,\\sigma}=\\dot{B}^s_{2,1}+\\dot{B}^\\sigma_{2,1}$.\n\\vspace{0.2cm}\n\n\nNow we state our main result as follows.\n\\begin{Theorem}\\label{Them1.2} Let $\\bar{h}_0$ be a positive constant.\nAssume that\n\n(i)\\, $(u_0,h_0-\\bar{h}_0)\\in B^0(\\mathop{\\bf R\\kern 0pt}\\nolimits^2)\\otimes\n\\widetilde{B}^{0,1}(\\mathop{\\bf R\\kern 0pt}\\nolimits^2);$\\vspace{0.1cm}\n\n(ii)\\,\\,$h_0\\ge \\bar{h}_0$.\\\\ \\vspace{0.1cm}\nThen there exist a positive time $T$ and a unique   solution $(u,h)$ of $(\\ref{1.1})$ such that\n\\begin{eqnarray}\\label{1.6}\nu\\in C([0,T]; B^0)\n\\cap L^1(0,T; B^2),\\,\\,\nh-\\bar{h}_0\\in C([0,T]; \\widetilde{B}^{0,1})\\cap\nL^1(0,T; \\widetilde{B}^{2,1}),\\quad h\\ge \\f12 \\bar{h}_0.\n\\end{eqnarray}\nMoreover, there exists a strictly positive constant $c$ such that if\n\\begin{eqnarray}\\label{1.7}\n\\|u_0\\|_{B^0}+\n\\|h_0-\\bar{h}_0\\|_{\\widetilde{B}^{0,1}}\\le c,\\end{eqnarray}\nthen we can choose $T=+\\infty$.\n\\end{Theorem}\n\nThe structure of this paper is as follows.\n\nIn Section 2, we recall some useful multilinear estimates in the Besov spaces .\nIn Section 3, we prove the existence of solution. In Section 4,\nwe prove the uniqueness of the solution.\nFinally, in the Appendix, we prove some multilinear estimates in the weighted Besov spaces.\n\nThroughout the paper, $C$ denotes various ``harmless'' large\nfinite constants, and $c$ denotes various ``harmless\" small\nconstants. We shall sometimes use $X\\lesssim Y$ to\ndenote the estimate $X\\le CY$ for some constant $C$.\nWe denote $\\|\\cdot\\|_p$ by the $L^p$ norm of a function.\n\\vspace{.3cm}\n\n\\setcounter{equation}{0}\n\\section{Multilinear estimates in the Besov spaces}\n\nLet us first recall the Bony's paraproduct decomposition.\n\n\\begin{Def}\\label{Def2.1}\nWe shall use the following  Bony's paraproduct decomposition(see \\cite{BC,B})\n\\begin{eqnarray}\\label{2.1}\nfg=T_fg+T_gf+R(f,g),\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\\label{2.2}\nT_fg=\\sum_{k\\in{\\mathbf Z}}S_{k-1}f\\Delta_kg\\quad \\textrm{and} \\quad\nR(f,g)=\\sum_{k\\in{\\mathbf Z}}\\sum_{|k'-k|\\le 1}\\Delta_{k} f\\Delta_{k'}g.\n\\end{eqnarray}\n\\end{Def}\n\nNext, let us recall some useful lemmas and multilinear estimates in the Besov spaces.\n\n\\begin{Lemma}(Bernstein's inequality)\\label{Lem2.1}\nLet $1\\le p\\le q\\le+\\infty$. Assume that $f\\in {\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$,\nthen for any $\\gamma\\in{\\mathbf Z}^d$,\nthere exist\nconstants $C_1$, $C_2$ independent of $f$, $j$ such that\n\\begin{eqnarray*}\n&&{\\rm supp}\\hat f\\subseteq \\{|\\xi|\\le\nA_02^{j}\\}\\Rightarrow\n\\|\\partial^\\gamma f\\|_q\\le C_12^{j{|\\gamma|}+j d(\\frac{1}{p}-\\frac{1}{q})}\\|f\\|_{p},\n\\\\\n&&{\\rm supp}\\hat f\\subseteq \\{A_12^{j}\\le|\\xi|\\le A_22^{j}\\}\\Rightarrow\n\\|f\\|_{p}\\le C_22^{-j|\\gamma|}\\sup_{|\\beta|=|\\gamma|}\\|\\partial^\\beta f\\|_p.\n\\end{eqnarray*}\n\\end{Lemma}\nThe proof can be found in \\cite{Ch1}.\n\n\\begin{Proposition}\\label{Prop2.2}\nIf $s>0$, $f,g\\in B^s\\cap L^\\infty$.\nThen $fg\\in B^s\\cap L^\\infty$ and\n\\begin{equation}\\label{2.3}\n\\|fg\\|_{B^s}\\le C(\\|f\\|_{\\infty}\\|g\\|_{B^s}+\\|g\\|_{\\infty}\\|f\\|_{B^s}).\n\\end{equation}\nIf $s_1$, $s_2\\le\\frac{d}{2}$ such that $s_1+s_2>0$, $f\\in B^{s_1}$, and\n$g\\in B^{s_2}$. Then $fg\\in B^{s_1+s_2-\\frac{d}{2}}$ and\n\\begin{eqnarray}\\label{2.4}\n\\|fg\\|_{B^{s_1+s_2-\\frac{d}{2}}}\\le C\\|f\\|_{B^{s_1}}\\|g\\|_{B^{s_2}}.\n\\end{eqnarray}\nIf $|s|<\\frac{d}{2}$, $1\\le r\\le+\\infty$, $f\\in \\dot{B}^s_{2,r}$ ,and\n$g\\in {B}^{\\frac{d}{2}}$. Then $fg\\in \\dot{B}^{s}_{2,r}$ and\n\\begin{equation}\\label{2.5}\n\\|fg\\|_{\\dot{B}^{s}_{2,r}}\\le C\\|f\\|_{\\dot{B}^{s}_{2,r}}\n\\|g\\|_{{B}^{\\frac{d}{2}}}.\n\\end{equation}\nIf $s\\in(-\\frac{d}{2},\\frac{d}{2}]$, $f\\in B^s$, and\n$g\\in \\dot{B}^{-s}_{2,\\infty}$. Then $fg\\in \\dot{B}^{-\\frac{d}{2}}_{2,\\infty}$ and\n\\begin{eqnarray}\\label{2.6}\n\\|fg\\|_{\\dot{B}^{-\\frac{d}{2}}_{2,\\infty}}\\le C\\|f\\|_{{B}^{s}}\n\\|g\\|_{\\dot{B}^{-s}_{2,\\infty}}.\\end{eqnarray}\nIf $1\\le \\rho_1, \\rho_2, \\rho\\le \\infty, s\\in(-\\frac{d}{2},\\frac{d}{2}]$,\n$f\\in \\widetilde{L}^{\\rho_1}_T(B^s)$, and\n$g\\in \\widetilde{L}^{\\rho_2}_T(\\dot{B}^{-s}_{2,\\infty})$. Then there holds\n\\begin{eqnarray}\\label{2.7}\n\\|fg\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^{-\\frac{d}{2}}_{2,\\infty})}\\le\nC\\|f\\|_{\\widetilde{L}^{\\rho_1}_T({B}^{s})}\n\\|g\\|_{\\widetilde{L}^{\\rho_2}_T(\\dot{B}^{-s}_{2,\\infty})},\n\\end{eqnarray}\nwhere $\\frac{1}{\\rho_1}+\\frac{1}{\\rho_2}=\\frac{1}{\\rho}$.\n\\end{Proposition}\n{\\it Proof.}\\quad\nFor the sake of simplicity, we only present the proof of (\\ref{2.4}) below, the others can be\ndeduced in the same way (see also \\cite{D3,RS}).\nBy the  Bony's paraproduct decomposition  and the\nproperty of quasi-orthogonality (\\ref{1.5}), for fixed $j\\in{\\mathbf Z}$, we write\n\\begin{align}\n\\Delta_j(fg)&=\\sum_{|k-j|\\le3}\\Delta_j(S_{k-1}f\\Delta_kg)+\n\\sum_{|k-j|\\le3}\\Delta_j(S_{k-1}g\\Delta_kf)\n+\\sum_{k\\ge j-2}\\sum_{|k-k'|\\le 1}\\Delta_j(\\Delta_{k}f\\Delta_{k'}g)\\nonumber\\\\\n&\\triangleq {I+II+III}.\\nonumber\\end{align}\nThanks to the definition of Besov space $B^s$, we have\n\\begin{align}\\label{2.8}\n\\|fg\\|_{B^{s_1+s_2-\\frac{d}{2}}}&\\le\\bigg(\\sum_{j\\in\n{\\mathbf Z}}2^{(s_1+s_2-\\frac{d}{2})j}\\|I\\|_{2}\\bigg)\n+\\cdots+\\bigg(\\sum_{j\\in\n{\\mathbf Z}}2^{(s_1+s_2-\\frac{d}{2})j}\\|III\\|_{2}\\bigg)\\nonumber\\\\\n&\\triangleq {I'+II'+III'}.\\end{align}\nIt suffices to  estimate the\nabove three terms separately.\nUsing the Young's inequality and lemma \\ref{Lem2.1}, we have\n\\begin{align}\n\\|\\Delta_j(S_{k-1}f\\Delta_kg)\\|_{2}&\\lesssim\n\\|S_{k-1}f\\|_{\\infty}\\|\\Delta_kg\\|_{2}\n\\lesssim\n\\sum_{k'\\le k-2}\\|\\Delta_{k'} f\\|_{\\infty}\\|\\Delta_kg\\|_{2}\\nonumber\\\\\n&\\lesssim \\sum_{k'\\le k-2}2^{k' s_1}\n\\|\\Delta_{k'} f\\|_{2}2^{k'(\\frac{d}{2}-s_1)}\\|\\Delta_kg\\|_{2}\n\\nonumber\\\\\n&\\lesssim \\|f\\|_{B^{s_1}}\n\\|\\Delta_kg\\|_{2}2^{k(\\frac{d}{2}-s_1)},\\nonumber\n\\end{align}\nwhere  we have used the fact $s_1\\le \\frac{d}{2}$ in the last inequality.\nHence, we get\n\\begin{align}\\label{2.9}\n{I'}&\\lesssim \\|f\\|_{B^{s_1}}\\sum_{j\\in{\\mathbf Z}}2^{(s_1+s_2-\\frac{d}{2})j}\n\\sum_{|k-j|\\le3}2^{k(\\frac{d}{2}-s_1)}\n\\|\\Delta_kg\\|_{2}\n\\nonumber\\\\&\\lesssim \\|f\\|_{B^{s_1}}\n\\sum_{|\\ell|\\le3}2^{-(s_1+s_2-\\frac{d}{2})\\ell}\n\\sum_{j\\in{\\mathbf Z}}2^{s_2(j+\\ell)}\n\\|\\Delta_{j+\\ell}g\\|_{2}\n\\lesssim \\|f\\|_{B^{s_1}}\\|g\\|_{B^{s_2}}.\n\\end{align}\nSimilarly, using the fact $s_2\\le\\frac{d}{2} $, we can obtain\n\\begin{align}\\label{2.10}\n{II'}&\\lesssim \\|f\\|_{B^{s_1}}\\|g\\|_{B^{s_2}}.\n\\end{align}\nNow we turn to estimate $III'$. From Lemma \\ref{Lem2.1} and H\\\"{o}lder inequality\n, it follows that\n\\begin{eqnarray*}\n\\|\\Delta_j(\\Delta_{k}f\\Delta_{k'}g)\\|_{2}\\lesssim\n2^{j\\frac{d}{2}}\\|\\Delta_{k}f\\Delta_{k'}g\\|_{1}\n\\lesssim 2^{j\\frac{d}{2}}\\|\\Delta_{k}f\\|_{2}\\|\\Delta_{k'}g\\|_{2}.\n\\end{eqnarray*}\nSo, we get by Minkowski inequality that for $s_1+s_2>0$\n\\begin{align}\\label{2.11}\n{III'}&\\lesssim \\sum_{j\\in{\\mathbf Z}}2^{(s_1+s_2-\\frac{d}{2})j}2^{j\\frac{d}{2}}\n\\bigg(\\sum_{k\\ge j-2}\\sum_{|k-k'|\\le1}\n\\|\\Delta_{k}f\\|_{2}\n\\|\\Delta_{k'}g\\|_{2}\\bigg)\n\\nonumber\\\\&\\lesssim\n\\sum_{\\ell\\ge-2}2^{-(s_1+s_2)\\ell}\n\\sum_{j\\in{\\mathbf Z}}2^{s_1(j+\\ell)}\n\\|\\Delta_{j+\\ell}f\\|_{2}\\|g\\|_{B^{s_2}}\n\\lesssim \\|f\\|_{B^{s_1}}\\|g\\|_{B^{s_2}}.\n\\end{align}\nSumming up (\\ref{2.8})-(\\ref{2.11}), we get the desired inequality\n(\\ref{2.4}).  \\hfill $ \\blacksquare $ \\vskip 3mm\n\n\\begin{Proposition}\\label{Prop2.3}\n\\textrm{(1)} Let $s>0$. Assume that $F\\in\nW^{[s]+2,\\infty}_{loc}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ such that $F(0)=0$. Then there\nexists a constant $C(s,d,F)$ such that if $u\\in B^s\\cap L^\\infty$,\nthere holds \\begin{eqnarray}\\label{2.12} &&\\|F(u)\\|_{B^s}\\le\nC(1+\\|u\\|_{\\infty})^{[s]+1}\\|u\\|_{B^s} ;\\end{eqnarray} and if $u\\in\n\\dot{B}^s_{2,\\infty}\\cap L^\\infty$, there holds \\begin{eqnarray}\\label{2.13}\n&&\\|F(u)\\|_{\\dot{B}^s_{2,\\infty}}\\le\nC(1+\\|u\\|_{\\infty})^{[s]+1}\\|u\\|_{\\dot{B}^s_{2,\\infty}} . \\end{eqnarray}\n(2)Assume that  $G\\in W^{[\\frac{d}{2}]+3,\\infty}_{loc}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ such\nthat $G'(0)=0$. Then there exists a functions $C(s,d,G)$ such that\nif  $-\\frac{d}{2}<s\\le\\frac{d}{2}$, $u, v\\in B^{\\frac{d}{2}}\\cap\nL^\\infty$ and $u-v\\in B^s$, there holds\n\\begin{align}\\label{2.14}\n\\|G(u)-G(v)\\|_{B^s}\\le C(\\|u\\|_{\\infty},\\|v\\|_{\\infty})\n(\\|u\\|_{B^{\\frac{d}{2}}}+\\|v\\|_{B^{\\frac{d}{2}}})\n\\|u-v\\|_{B^s};\\end{align}\nand if $|s|<\\frac{d}{2}$, $u, v\\in B^\\frac{d}{2}\\cap L^\\infty$ and $u-v\\in\\dot{B}^s_{2,\\infty}$, there holds\n\\begin{eqnarray}\\label{2.15}\n\\|G(u)-G(v)\\|_{\\dot{B}^s_{2,\\infty}}\\le C(\\|u\\|_{\\infty},\\|v\\|_{\\infty})\n(\\|u\\|_{B^{\\frac{d}{2}}\n}+\\|v\\|_{B^{\\frac{d}{2}}})\n\\|u-v\\|_{\\dot{B}^s_{2,\\infty}}\n.\\quad\n\\end{eqnarray}\n\\end{Proposition}\n{\\it Proof.}\\quad  We can refer to \\cite{BC1,RS} for the proof of (1).\nFor (2), we refer to \\cite{D1,D4}. For example, we write\n$$G(u)-G(v)=(u-v)\\int_0^1G'(v+\\tau(u-v))d\\tau,$$\nthen it follows from (\\ref{2.5}) that for $|s|<\\frac{d}{2}$\n\\begin{eqnarray*}\n\\|G(u)-G(v)\\|_{\\dot{B}^s_{2,\\infty}}\\le C\\|u-v\\|_{\\dot{B}^s_{2,\\infty}}\n\\|G'(v+\\tau(u-v))\\|_{B^\\frac{d}{2}},\n\\end{eqnarray*}\nwhich together with (\\ref{2.12}) implies (\\ref{2.15}). \\hfill $ \\blacksquare $ \\vskip 3mm\n\n\\begin{Proposition}\\label{Prop2.4}\nLet $A$ be a homogeneous smooth function of degree $m$. Assume that\n$-\\frac{d}{2}< s_1, t_1, s_2, t_2\\le1+\\frac{d}{2}$, Then there hold:\nif\\, $k\\ge1$,\n\\begin{align}\\label{2.16}\n|&(A(D)\\Delta_k(v\\cdot\\nabla f),A(D)\\Delta_kf)|\\nonumber\\\\ &\n\\lesssim\n\\alpha_k2^{-k(s_2-m)}\\|v\\|_{B^{\\frac{d}{2}+1}}\\|f\\|_{\\widetilde{B}^{s_1, s_2}}\n\\|A(D)\\Delta_kf\\|_2,\\qquad\\end{align}\nand if\\, $k\\le0$,\n\\begin{align}\\label{2.17}\n|&(A(D)\\Delta_k(v\\cdot\\nabla f),A(D)\\Delta_kf)|\\nonumber\\\\& \\lesssim\n\\alpha_k2^{-k(s_1-m)}\\|v\\|_{B^{\\frac{d}{2}+1}}\\|f\\|_{\\widetilde{B}^{s_1, s_2}}\n\\|A(D)\\Delta_kf\\|_2,\\qquad\\end{align}\nand if\\, $k\\ge1$,\n\\begin{align}\\label{2.18}\n|&(A(D)\\Delta_k(v\\cdot\\nabla f),\\Delta_kg)+(\\Delta_k(v\\cdot\\nabla g),A(D)\\Delta_kf)|\n\\nonumber\\\\&\\lesssim\n\\alpha_k\\|v\\|_{B^{\\frac{d}{2}+1}}(2^{-kt_2}\\|g\\|_{\\widetilde{B}^{t_1, t_2}}\\|A(D)\\Delta_kf\\|_2\n+2^{-k(s_2-m)}\\|f\\|_{\\widetilde{B}^{s_1, s_2}}\\|\\Delta_kg\\|_2),\\qquad\n\\end{align}\nand if\\, $k\\le0$,\n\\begin{align}\\label{2.19}\n|&(A(D)\\Delta_k(v\\cdot\\nabla f),\\Delta_kg)+(\\Delta_k(v\\cdot\\nabla g),A(D)\\Delta_kf)|\n\\nonumber\\\\&\\lesssim\n\\alpha_k\\|v\\|_{B^{\\frac{d}{2}+1}}(2^{-kt_1}\\|g\\|_{\\widetilde{B}^{t_1, t_2}}\\|A(D)\\Delta_kf\\|_2\n+2^{-k(s_1-m)}\\|f\\|_{\\widetilde{B}^{s_1, s_2}}\\|\\Delta_kg\\|_2),\\qquad\n\\end{align}\nwhere $\\displaystyle\\sum_{k\\in{\\mathbf Z}}\\alpha_k\\le1$.\n\\end{Proposition}\nFor the proof we refer to \\cite{D2}.\n\n\n\n\\setcounter{equation}{0}\n\\section{Existence}\n\nIn this section, we prove the existence of the solution for the 2D viscous\nshallow water equations. Without loss of generality, we assume that $\\bar{h}_0=1$ and $\\nu=1$. Replacing\n$h$ by $h+1$ in (\\ref{1.1}), we  rewrite (\\ref{1.1}) as\n\\begin{equation}\\label{3.1}\n\\left\\{\n\\begin{array}{ll}\nh_t+\\mbox{div} u+\\mbox{div}(hu)=0,\\\\\nu_t-(\\nabla\\cdot D(u)+\\nabla \\textrm{div}\\,u)+u\\cdot\\nabla u-\\frac{\\nabla h}{1+h}(D(u)+\\textrm{div}\\, u)+\\nabla h=0, \\\\\nu(0,\\cdot)=u_0,  \\quad   h(0,\\cdot)=h_0.\n\\end{array}\n\\right.\n\\end{equation}\n\n\\subsection{The linearized system}\n\nIn this subsection, we consider the linearized system of (\\ref{3.1}):\n\\begin{equation}\\label{3.2}\n\\left\\{\n\\begin{array}{ll}\nh_t+v\\cdot\\nabla h+\\mbox{div} u={\\cal H},\\\\\nu_t-(\\nabla\\cdot D(u)+\\nabla \\textrm{div}\\,u)+v\\cdot\\nabla u+\\nabla h={\\cal G}, \\\\\nu(0,\\cdot)=u_0,  \\quad   h(0,\\cdot)=h_0.\n\\end{array}\n\\right.\n\\end{equation}\n\nLet us first introduce some definitions. Set\n$$e_k^r(t)\\triangleq(1-e^{-cr2^{2k}t})^\\f1r,\\quad\n\\omega_k(t)=\\sum_{\\widetilde{k}\\ge k}\n2^{-(\\widetilde{k}-k)}(e_{\\widetilde{k}}^1(t)+e_{\\widetilde{k}}^2(t)),\n$$\nwhere $c$ is a positive constant which will be determined later. We remark that\n$$\n\\omega_k(t)\\le C, \\quad \\textrm{for}\\,\\, \\textrm{any}\\,\\, k\\in {\\mathbf Z},\n$$\nwhich will be constantly used in the following.\n\n\\begin{Def}\\label{Def3.1}\nLet $s\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$ and $T>0$. The function space $E^{s}_T$\nis  defined by\n$$E^{s}_T=\\{f\\in {\\cal Z'}((0,T)\\times\\mathop{\\bf R\\kern 0pt}\\nolimits^d):\\,\\|f\\|_{E^{s}_T}<+\\infty\\},$$\nwhere\n\\begin{eqnarray*}\n\\|f\\|_{E^{s}_T}\\triangleq\\sum_{k\\in Z}2^{ks}\\omega_k(T)\\|\\Delta_k f\\|_{L^\\infty_T(L^2)}.\n\\end{eqnarray*}\n\\end{Def}\n\n\\begin{Def}{\\label{Def3.2}}\nLet $s_1, s_2\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$ and $T>0$. The function space $\\widetilde{E}^{s_1,s_2}_T$\nis  defined by\n$$\\widetilde{E}^{s_1,s_2}_T=\\{f\\in {\\cal Z'}((0,T)\\times\\mathop{\\bf R\\kern 0pt}\\nolimits^d):\\,\\|f\\|_{\\widetilde{E}^{s_1,s_2}_T}<+\\infty\\},$$\nwhere\n$$\\|f\\|_{\\widetilde{E}^{s_1,s_2}_T}\\triangleq \\sum_{k\\le0}2^{ks_1}\\omega_k(T)\\|\\Delta_k f\\|_{L^\\infty_T(L^2)}\n+\\sum_{k\\ge 1}2^{ks_2}\\omega_k(T)\\|\\Delta_k f\\|_{L^\\infty_T(L^2)}.$$\n\\end{Def}\n\\begin{rmk}\nIf $s_1\\le s_2$, then $\\widetilde{E}^{s_1,s_2}_T=E^{s_1}_T\\cap E^{s_2}_T$.\nOtherwise, $\\widetilde{E}^{s_1,s_2}_T=E^{s_1}_T+E^{s_2}_T$.\n\\end{rmk}\n\nLet $(u,h)$ be a smooth solution of (\\ref{3.2}). We want to establish the following {\\it a-priori} estimates for $(h,u)$:\n\\begin{eqnarray}\\label{3.3}\n&&\\|u\\|_{L^1_T(B^2)}+\\|u\\|_{L^2_T(B^1)}+\\|h\\|_{\\widetilde{E}^{0,1}_T}\\nonumber\\\\\n&&\\quad\\le C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)E_k(0)\n+C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\|\\Delta_k{\\cal G}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\sum_{k\\ge 1}\\omega_k(T)\\|\\nabla \\Delta_k{\\cal H}(t)\\|_{L^1_T(L^2)}\n+C\\sum_{k<1}\\omega_k(T)\\|\\Delta_k{\\cal H}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+C\\|h\\|_{\\widetilde{E}^{0,1}_T}\\|v\\|_{L^1_T(B^2)},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{3.4}\n&&\\|u\\|_{L^\\infty_T(B^0)}+\\|h\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h\\|_{L^1_T(\\widetilde{B}^{2,1})}\\nonumber\\\\\n&&\\quad\\le E_0+C\\Bigl(\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0,1})}\n+\\|{\\cal G}\\|_{L^1_T(B^0)}+\\int_0^TV'(t)(\\|u(t)\\|_{B^0}+\\|h(t)\\|_{\\widetilde{B}^{0,1}})dt\\Bigr),\n\\end{eqnarray}\nwhere $V(t)=\\|v(t')\\|_{L^1_t(B^2)}$ and\n$$\nE_0=\\sum_{k\\in{\\mathbf Z}}E_k(0),\\quad E_{k}(t)=\\bigg\\{\n\\begin{array}{ll}E_{hk}(t)\n&\\quad k\\ge 1,\\\\\nE_{lk}(t)&\\quad k< 1,\n\\end{array}\\bigg.\n$$\nwith\n\\begin{eqnarray*}\n&& E_{hk}^2(t)=\\f12\\|u_k(t)\\|^2_{2}+\\|\n\\nabla h_k(t)\\|^2_{2}+(u_k(t), \\nabla h_k(t)), \\quad \\textrm{and}\\\\\n&& E_{lk}^2(t)= \\f12\\|u_k(t)\\|^2_{2}+\\f12\\|\nh_k(t)\\|^2_{2}+\\f18(u_k(t),\\nabla h_k(t)).\n\\end{eqnarray*}\n\nLet us begin with the proof of (\\ref{3.3}) and (\\ref{3.4}). Set\n$$\nu_k=\\Delta_k u,\\quad h_k=\\Delta_k h,\\quad {\\cal H}^k=\\Delta_k {\\cal\nH},\\quad {\\cal G}^k=\\Delta_k{\\cal G}.\n$$\nThen we get by applying the operator $\\Delta_k$ to (\\ref{3.2}) that\n\\begin{equation}\\label{3.5}\n\\left\\{\n\\begin{array}{ll}\n\\partial_th_k+\\Delta_k(v\\cdot\\nabla h)+\\mbox{div} u_k={\\cal H}_k,\\\\\n\\partial_tu_k-(\\nabla\\cdot D(u_k)+\\nabla \\textrm{div}\\,u_k)+\\Delta_k(v\\cdot\\nabla u)+\\nabla h_k={\\cal G}_k, \\\\\nu_k(0,\\cdot)=\\Delta_ku_0,  \\quad   h_k(0,\\cdot)=\\Delta_kh_0.\n\\end{array}\n\\right.\n\\end{equation}\nMultiplying the second equation of (\\ref{3.5}) by $u_k$, and\nintegrating the resulting equation over $\\mathop{\\bf R\\kern 0pt}\\nolimits^2$, we obtain\n\\begin{eqnarray}\\label{3.6} \\frac{1}{2}\\frac{d}{dt}\\|u_k\\|^2_{2} +\\f12\\|\\nabla\nu_k\\|^2_{2}+\\f32\\|\\mbox{div} u_k\\|_2^2+(\\nabla h_k,u_k)=({\\cal G}_k, u_k)\n-(\\Delta_k(v\\cdot\\nabla u), u_k). \\end{eqnarray}\n\nIn the following, we will deal with the high frequency and the low frequency of $h$ in a different manner.\n\n\n\n\\vspace{.15cm}\n\n{\\bf High  frequencies}:\\, $k\\ge 1$.\\vspace{.15cm}\n\nFirstly, applying $\\nabla$ to the first equation of $(\\ref{3.5})$,\nand  multiplying it by $\\nabla h_k$, then integrating the resulting\nequation over $\\mathop{\\bf R\\kern 0pt}\\nolimits^2$, we obtain \\begin{eqnarray}\\label{3.7}\n\\frac{1}{2}\\frac{d}{dt}\\|\\nabla h_k\\|^2_{2}+ (\\nabla\\mbox{div} u_k, \\nabla\nh_k)=(\\nabla {\\cal H}_{k}, \\nabla h_k)-(\\nabla\\Delta_k(v\\cdot \\nabla\nh), \\nabla h_k).\\quad \\end{eqnarray} Secondly, applying the operator $\\nabla$\nto the first equation of $(\\ref{3.5})$ and taking the $L^2$ product\nof the resulting equation with $u_k$; then taking the $L^2$ product\nof  second equation of $(\\ref{3.5})$ with $\\nabla h_k$, we get by\nsumming them up that\n\\begin{eqnarray}\\label{3.8} &&\\frac{d}{dt}(u_k, \\nabla\nh_k)-\\|\\mbox{div} u_k\\|^2_{2} -2(\\nabla \\mbox{div} u_k,\\nabla h_k)+\\|\\nabla\nh_k\\|^2_{2}\\nonumber\\\\&& \\quad=(\\nabla{\\cal H}_{k}, u_k)+({\\cal\nG}_{k}, \\nabla h_k) -(\\nabla\\Delta_k(v\\cdot \\nabla h),\nu_k)-(\\Delta_k(v\\cdot\\nabla u), \\nabla h_k), \\end{eqnarray} where we used the\nfact that\n$$\n(\\nabla\\cdot D(u_k)+\\nabla \\textrm{div}\\,u_k, \\nabla h_k)=2(\\nabla \\mbox{div} u_k,\\nabla h_k).\n$$\nThen we get by summing up (\\ref{3.6}), (\\ref{3.7})$\\times 2$, and (3.8) that\n\\begin{eqnarray}\\label{3.9}\n&&\\frac{d}{dt}\\bigl[\\f12\\|u_k\\|^2_{2}+\\|\n\\nabla h_k\\|^2_{2}+(u_k, \\nabla h_k)\\bigr]\\nonumber\\\\\n&&\\qquad\\qquad+\\bigl[\\|\\nabla h_k\\|^2_{2}+\\f12\\|\\nabla u_k\\|^2_{2}+\\f12\\|\\mbox{div} u_k\\|^2_2\n+(\\nabla h_k, u_k)\\bigr]\\nonumber\\\\\n&&=\\Bigl[(\\nabla {\\cal H}_{k}, u_k)+2(\\nabla {\\cal H}_{k},\\nabla h_k)\n+({\\cal G}_{k}, u_k)+({\\cal G}_{k}, \\nabla h_k)\\Bigr]\\nonumber\\\\\n&&\\quad-(\\Delta_k(v\\cdot\\nabla u), u_k)\n-2(\\nabla\\Delta_k(v\\cdot\\nabla h), \\nabla h_k)\\nonumber\\\\\n&&\\quad-\\Bigl[(\\nabla\\Delta_k(v\\cdot\\nabla h), u_k)\n+(\\Delta_k(v\\cdot\\nabla u), \\nabla h_k)\\Bigr]\\nonumber\\\\\n&&\\triangleq I+II+III+IV.\n\\end{eqnarray}\nNote that\n\\begin{eqnarray*}\n(u_k,\\nabla h_k)\\le \\f13\\|u_k\\|^2_{2}+\n\\f34\\|\\nabla h_k\\|^2_{2},\n\\end{eqnarray*}\nhence, we get by the definition of $E_{hk}$ that\n\\begin{eqnarray}\\label{3.10}\n\\f16(\\|u_k\\|^2_{2}+\\|\\nabla h_k\\|^2_{2})\\le E_{hk}^2\\le 2(\\|u_k\\|^2_{2}+\\|\\nabla h_k\\|^2_{2}).\n\\end{eqnarray}\nSimilarly, using the fact that $\\f562^k\\ge \\f53$ and (\\ref{3.10}), we have\n\\begin{eqnarray}\\label{3.11}\n\\|\\nabla h_k\\|^2_{2}+\\f12\\|\\nabla u_k\\|^2_{2}+\\f12\\|\\mbox{div} u_k\\|^2_{2}\n+(\\nabla h_k, u_k)\\ge\\frac{1}{8}E_{hk}^2.\n\\end{eqnarray}\n\nBy summing up (\\ref{3.9})-(\\ref{3.11}), we obtain\n\\begin{eqnarray}\\label{3.12}\n\\frac{d}{dt}E_{hk}^2+cE_{hk}^2\\le C|I+II+III+IV|.\n\\end{eqnarray}\n\nIn order to obtain (\\ref{3.3}), we use Lemma \\ref{Lem5.1} to deal with the right hand terms of (\\ref{3.12}).\nFirstly, we get by using the Cauchy-Schwartz inequality and (\\ref{3.10}) that\n\\begin{eqnarray}\\label{3.13}\n|I|\\le C(\\|\\nabla {\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2})E_{hk}.\n\\end{eqnarray}\nFrom Lemma \\ref{Lem5.1} and (\\ref{3.10}), it follows that\n\\begin{eqnarray}\\label{3.14}\n|II+III+IV|\\le C(\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2})E_{hk}.\n\\end{eqnarray}\n\nBy summing up (\\ref{3.12}), and (\\ref{3.13})-(\\ref{3.14}), we obtain\n\\begin{eqnarray}\\label{3.15}\n\\frac{d}{dt}E_{hk}+cE_{hk}\\le C\\Bigl(\\|\\nabla {\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2}\\Bigr),\n\\end{eqnarray}\nwhich implies that\n\\begin{align}\\label{3.16}\n\\|E_{hk}(t)\\|_{L^\\infty_T}\\le & E_{hk}(0)+C\\Bigl(\\|\\nabla {\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&+\\|{\\cal F}_k^1(t)\\|_{L^1_T(L^2)}+\\|{\\cal \\widetilde{F}}_k^0(t)\\|_{L^1_T(L^2)}\\Bigr).\n\\end{align}\nFurthermore, by (\\ref{5.1}) and (\\ref{5.2}), there holds\n\\begin{eqnarray*}\n\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\Big(\\|{\\cal F}_k^1(t)\\|_{L^1_T(L^2)}+\\|{\\cal \\widetilde{F}}_k^0(t)\\|_{L^1_T(L^2)}\\Big)\\le C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}\n+\\|h\\|_{E^{1}_T}\\|v\\|_{L^1_T(B^2)}\\Big).\n\\end{eqnarray*}\nMultiplying $\\omega_k(T)$ on both sides of \\eqref{3.16}, then\nsumming up  the resulting equation over $k\\ge 1$, we obtain\n\\begin{align}\\label{3.17}\n\\sum_{k\\ge 1}\\omega_k(T)\\|E_{hk}(t)\\|_{L^\\infty_T}\\le &\\sum_{k\\ge 1}\\omega_k(T)E_{hk}(0)\\nonumber\\\\\n&+C\\sum_{k\\ge 1}\\omega_k(T)\\Big(\\|\\nabla {\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)}\\Big)\\nonumber\\\\\n&+C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{E^{1}_T}\\|v\\|_{L^1_T(B^2)}\\Big).\n\\end{align}\n\nNext, we use the decay effect of the parabolic operators to  estimate  $\\|u\\|_{L^2_T(B^1)\\cap L^1_T(B^2)}$. It follows from (\\ref{3.6}) and Lemma 5.1 that\n\\begin{eqnarray*}\n\\frac{d}{dt}\\|u_k\\|_2+c2^{2k}\\|u_k\\|_2\\le C(\\|\\nabla h_k(t)\\|_2+\\|{\\cal G}_{k}(t)\\|_{L^2}\n+\\|{\\cal \\widetilde{F}}_{k}^0(t)\\|_{L^2}),\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray*}\n\\|u_k\\|_2\\le e^{-ct2^{2k}}\\|u_k(0)\\|_{2}+Ce^{-ct2^{2k}}\\ast_t\\Bigl(\\|\\nabla h_k(t)\\|_2\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal \\widetilde{F}}_{k}^0(t)\\|_{2}\\Bigr),\n\\end{eqnarray*}\nwhere the sign $\\ast$ denotes the convolution of functions defined in $\\mathop{\\bf R\\kern 0pt}\\nolimits^+$, more precisely,\n$$e^{-ct2^{2k}}\\ast_t f\\triangleq \\int_0^t\ne^{-c(t-\\tau)2^{2k}}f(\\tau)d\\tau.$$\nTaking the $L^r$ norm for $r=1,2$ with respect to $t$, we get by using the Young's inequality that\n\\begin{eqnarray*}\n\\|u_k\\|_{L^r_T(L^2)}\\le C2^{-2k\/r}e^r_k(T)\\Big(\\|u_k(0)\\|_{2}+\n\\|\\nabla h_k\\|_{L^1_T(L^2)}+\\|{\\cal G}_{k}\\|_{L^1_T(L^2)}+\\|{\\cal \\widetilde{F}}_{k}^0\\|_{L^1_T(L^2)}\\Big),\n\\end{eqnarray*}\nwhich together with (\\ref{5.2}) implies that\n\\begin{align}\\label{3.18}\n&\\sum_{k\\ge1}\\Bigl(2^{2k}\\|u_k\\|_{L^1_T(L^2)}+2^{k}\\|u_k\\|_{L^2_T(L^2)}\\Bigr)\\le C\\sum_{k\\ge1}\\omega_k(T)\\|u_k(0)\\|_2\n\\nonumber\\\\&\\qquad+C\\sum_{k\\ge1}\\omega_k(T)\\Bigl(\\|\\nabla h_k\\|_{L^1_T(L^2)}+\\|{\\cal G}_{k}\\|_{L^1_T(L^2)}\\Bigr)\n+C\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)},\n\\end{align}\nwhere we used the fact that\n\\begin{eqnarray*}\ne_k^1(T)+e_k^2(T)\\le \\omega_k(T).\n\\end{eqnarray*}\nOn the other hand, it follows from (\\ref{3.15}) that\n\\begin{eqnarray*}\n\\|E_{hk}\\|_2\\le e^{-ct}E_{hk}(0)+Ce^{-ct}\\ast_t\\Bigl(\\|\\nabla {\\cal H}_k(t)\\|_2\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal{F}}_{k}^1(t)\\|_{2}+\\|{\\cal \\widetilde{F}}_{k}^0(t)\\|_{2}\\Bigr).\n\\end{eqnarray*}\nTaking the $L^1$ norm with respect to $t$, we get by using the Young's inequality that\n\\begin{align}\\label{3.19}\n\\|E_{hk}\\|_{L_T^1}&\\le C(1-e^{-cT})E_{hk}(0)+C(1-e^{-cT})\\Bigl(\\|\\nabla {\\cal H}_k(t)\\|_{L_T^1(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L_T^1(L^2)}\\nonumber\\\\\n&\\quad+\\|{\\cal{F}}_{k}^1(t)\\|_{L_T^1(L^2)}+\\|{\\cal \\widetilde{F}}_{k}^0(t)\\|_{L_T^1(L^2)}\\Bigr).\n\\end{align}\nNote that for $k\\ge 1$\n$$\n1-e^{-ct}\\le 1-e^{-ct2^{2k}}\\le \\omega_k(t),\n$$\nwhich together with (\\ref{3.19}) and Lemma \\ref{Lem5.1} gives\n\\begin{align}\\label{3.20}\n\\sum_{k\\ge 1}\\|E_{hk}\\|_{L_T^1}\\le& C\\sum_{k\\ge 1}\\omega_k(T)E_{hk}(0)\n+C\\sum_{k\\ge 1}\\omega_k(T)\\Bigl(\\|\\nabla {\\cal H}_k(t)\\|_{L_T^1(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L_T^1(L^2)}\\Bigr)\\nonumber\\\\\n&+C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{E^1_T}\\|v\\|_{L^1_T(B^2)}\\Big).\n\\end{align}\nPlugging \\eqref{3.20} into (\\ref{3.18}), we obtain\n\\begin{align}\\label{3.21}\n&\\sum_{k\\ge 1}\\Bigl(2^{2k}\\|u_k\\|_{L^1_T(L^2)}+2^{k}\\|u_k\\|_{L^2_T(L^2)}\\Bigr)\\nonumber\\\\&\\quad\\le C\\sum_{k\\ge1}\n\\omega_k(T)E_{hk}(0)\n+C\\sum_{k\\ge 1}\\omega_k(T)\\Bigl(\\|\\nabla {\\cal H}_k(t)\\|_{L_T^1(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L_T^1(L^2)}\\Bigr)\\nonumber\\\\\n&\\qquad+C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{E^{1}_T}\\|v\\|_{L^1_T(B^2)}\\Big).\n\\end{align}\n\nOn the other hand, in order to obtain (\\ref{3.4}), we use Proposition \\ref{Prop2.4} to deal with the right hand terms of (\\ref{3.12}).\nApplying (\\ref{2.16}) with $s_1=s_2=0$ to $II$, (\\ref{2.16}) with $s_1=0, s_2=1$ to $III$, (\\ref{2.18})\nwith $t_1=t_2=0, s_1=0, s_2=1$ to $IV$, we obtain\n\\begin{eqnarray}\\label{3.22}\n|II+III+IV|\\le CE_{hk}\\alpha_kV'(t)(\\|u\\|_{B^0}+\\|h\\|_{\\widetilde{B}^{0,1}}),\n\\end{eqnarray}\nwith $\\displaystyle\\sum_{k\\in {\\mathbf Z}}\\alpha_k\\le 1$ and $V(t)=\\|v(t')\\|_{L^1_t(B^2)}$. From (\\ref{3.13}) and (\\ref{3.22}), it follows that\n\\begin{eqnarray*}\n\\frac{d}{dt}E_{hk}+cE_{hk}\\le C\\Bigl(\\|\\nabla {\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\alpha_kV'(t)(\\|u\\|_{B^0}+\\|h\\|_{\\widetilde{B}^{0,1}})\\Bigr),\n\\end{eqnarray*}\nfrom which, a similar proof of (\\ref{3.21}) ensures that\n\\begin{eqnarray}\\label{3.23}\n&&\\sum_{k\\ge 1}\\Bigl(\\|E_{hk}\\|_{L_T^1}+\\|E_{hk}\\|_{L_T^\\infty}\\Bigr)\\le C\\sum_{k\\ge 1}E_{hk}(0)\\nonumber\\\\\n&&\\qquad+C\\Bigl(\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0,1})}\n+\\|{\\cal G}\\|_{L^1_T(B^0)}+\\int_0^TV'(t)(\\|u(t)\\|_{B^0}+\\|h(t)\\|_{\\widetilde{B}^{0,1}})dt\\Bigr).\n\\end{eqnarray}\n\n\\vspace{.15cm}\n\n{\\bf Low frequencies}:\\, $k<1$.\\vspace{.15cm}\n\nMultiplying the first equation of $(\\ref{3.5})$ by $h_k$, we get by integrating\nthe resulting equation over $\\mathop{\\bf R\\kern 0pt}\\nolimits^2$ that\n\\begin{eqnarray}\\label{3.24}\n\\frac{1}{2}\\frac{d}{dt}\\|h_k\\|^2_{2}+\n(\\mbox{div} u_k, h_k)=({\\cal H}_{k}, h_k)-(\\Delta_k(v\\cdot \\nabla h),h_k).\n\\end{eqnarray}\nSumming up (\\ref{3.6}), $(\\ref{3.8})\\times\\frac{1}{8}$, and (\\ref{3.24}), we obtain\n\\begin{align}\\label{3.25}\n&\\frac{d}{dt}\\Bigl[\\f12\\|u_k\\|^2_{2}+\\f12\\|\nh_k\\|^2_{2}+\\frac{1}{8}(u_k,\\nabla h_k)\\Bigr]\n\\nonumber\\\\\n&\\qquad+\\Bigr[\\frac{1}{8}\\|\\nabla h_k\\|^2_{2}\n+\\f12\\|\\nabla u_k\\|^2_{2}+\\frac{11}{8}\\|\\mbox{div} u_k\\|^2_{2}\n-\\frac{1}{4}(\\nabla\\mbox{div}\\, u_k,\\nabla h_k)\\Bigr]\\nonumber\\\\&=\\Bigl[\\f18(\\nabla {\\cal H}_{k}, u_k)+({\\cal H}_{k},h_k)\n+({\\cal G}_{k}, u_k)+\\f18({\\cal G}_{k}, \\nabla h_k)\\Bigr]\\nonumber\\\\\n&\\quad-(\\Delta_k(v\\cdot\\nabla u), u_k)\n-(\\Delta_k(v\\cdot\\nabla h),h_k)\\nonumber\\\\\n&\\quad-\\f18\\Bigl[(\\nabla\\Delta_k(v\\cdot\\nabla h), u_k)\n+(\\Delta_k(v\\cdot\\nabla u), \\nabla h_k)\\Bigr]\\nonumber\\\\\n&\\triangleq I+II+III+IV.\n\\end{align}\nNote that $2^k\\le 1$, we get by the Cauchy-Schwartz inequality that\n\\begin{eqnarray*}\n\\f18(u_k,\\nabla h_k)\\le \\f3{10}\\|u_k\\|_{2}\\|h_k\\|_{2}\\le \\f14\\|u_k\\|^2_{2}+\n\\f14\\|h_k\\|^2_{2},\n\\end{eqnarray*}\nhence, we get by the definition of $E_{lk}$ that\n\\begin{eqnarray}\\label{3.26}\n\\f14(\\|u_k\\|^2_{2}+\\|h_k\\|^2_{2})\\le E_{lk}^2\\le 2(\\|u_k\\|^2_{2}+\\|h_k\\|^2_{2}).\n\\end{eqnarray}\nSimilarly, we can prove\n\\begin{eqnarray*}\n\\frac{1}{4}(\\nabla\\mbox{div}\\, u_k,\\nabla h_k)\\le \\f3{5}\\|\\nabla u_k\\|_{2}\\|\\nabla h_k\\|_{2}\\le \\f{9} {10}\\|\\nabla u_k\\|_2^2+\\f1 {10}\\|\\nabla h_k\\|_2^2,\n\\end{eqnarray*}\nwhich together with (\\ref{3.26}) implies that\n\\begin{eqnarray}\\label{3.27}\n&&\\frac{1}{8}\\|\\nabla h_k\\|^2_{2}\n+\\f12\\|\\nabla u_k\\|^2_{2}+\\frac{11}{8}\\|\\mbox{div} u_k\\|^2_{2}\n-\\frac{1}{4}(\\nabla\\mbox{div}\\, u_k,\\nabla h_k)\\nonumber\\\\\n&&\\qquad\\ge \\frac{1}{160}2^{2k}(\\|u_k\\|^2_{2}+\\|\nh_k\\|^2_{2})\\ge \\frac{1}{320}2^{2k}E_{lk}^2.\n\\end{eqnarray}\n\nBy summing up (\\ref{3.25})-(\\ref{3.27}), we obtain\n\\begin{eqnarray}\\label{3.28}\n\\frac{d}{dt}E_{lk}^2+c2^{2k}E_{lk}^2\\le C|I+II+III+IV|.\n\\end{eqnarray}\n\nIn order to obtain (\\ref{3.3}), we use Lemma 5.1 to estimate the right hand terms of (\\ref{3.28}).\nUsing the fact that $2^k\\le 1$, we get by the Cauchy-Schwartz inequality and (\\ref{3.26}) that\n\\begin{eqnarray}\\label{3.29}\n|I|\\le C(\\|{\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2})E_{lk}.\n\\end{eqnarray}\nUsing Lemma 5.1 and (\\ref{3.26}), we have\n\\begin{eqnarray}\\label{3.30}\n|II+III+IV|\\le C(\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal F}^0_k(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2})E_{lk}.\n\\end{eqnarray}\n\nBy summing up (\\ref{3.28})-(\\ref{3.30}), we obtain\n\\begin{eqnarray*}\n\\frac{d}{dt}E_{lk}+c2^{2k}E_{lk}\\le C\\Bigl(\\|{\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal F}^0_k(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2}\\Bigr),\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray*}\nE_{lk}\\le e^{-c2^{2k}t}E_{lk}(0)+Ce^{-c2^{2k}t}\\ast_t\\Bigl(\\|{\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal F}^0_k(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2}\\Bigr).\n\\end{eqnarray*}\nTaking the $L^r$ norm with respect to $t$, we get by using the Young's inequality that\n\\begin{align}\n\\|E_{lk}\\|_{L^r_T}&\\le C2^{-2k\/r}e^r_k(T)\\Bigl(E_{lk}(0)+\\|{\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&\\quad+\\|{\\cal F}_k^1(t)\\|_{L^1_T(L^2)}+\\|{\\cal F}^0_k(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{L^1_T(L^2)}\\Bigr),\\nonumber\n\\end{align}\nfrom which and Lemma 5.1, it follows that\n\\begin{eqnarray}\\label{3.31}\n&&\\sum_{k<1}\\omega_k(T)\\|E_{lk}\\|_{L^\\infty_T}\\le C\\sum_{k<1}\\omega_k(T)E_{lk}(0)+\\sum_{k<1}\\omega_k(T)(\\|{\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)})\\nonumber\\\\\n&&\\qquad\\quad +C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{\\widetilde{E}^{0,1}_T}\\|v\\|_{L^1_T(B^2)}\\Big),\n\\end{eqnarray}\nand\n\\begin{align}\\label{3.32}\n&\\sum_{k<1}(2^{2k}\\|E_{lk}\\|_{L^1_T}+2^{k}\\|E_{lk}\\|_{L^2_T})\\nonumber\\\\&\\quad\\le C\\sum_{k<1}\\omega_k(T)E_{lk}(0)\n+\\sum_{k<1}\\omega_k(T)(\\|{\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)})\\nonumber\\\\\n&\\qquad+C\\Bigl(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{\\widetilde{E}^{0,1}_T}\\|v\\|_{L^1_T(B^2)}\\Bigr).\n\\end{align}\n\nOn the other hand, in order to obtain (\\ref{3.4}),\nwe use Proposition \\ref{Prop2.4} to deal with the right hand terms of (\\ref{3.28}).\nApplying (\\ref{2.17}) with $s_1=s_2=0$ to $II$, (\\ref{2.17}) with $s_1=0, s_2=1$ to $III$, (\\ref{2.19})\nwith $t_1=t_2=0, s_1=0, s_2=1$ to $IV$, we obtain\n\\begin{align}\\label{3.33}\n|II+III+IV|\\le CE_{lk}\\alpha_kV'(t)(\\|u\\|_{B^0}+\\|h\\|_{\\widetilde{B}^{0,1}}),\n\\end{align}\nwith $\\displaystyle\\sum_{k\\in {\\mathbf Z}}\\alpha_k\\le 1$ and $V(t)=\\|v(t')\\|_{L^1_t(B^2)}$. From (\\ref{3.32}) and (\\ref{3.36}), it follows that\n\\begin{eqnarray*}\n\\frac{d}{dt}E_{lk}+c2^{2k}E_{lk}\\le C\\Bigl(\\|{\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\alpha_kV'(t)(\\|u\\|_{B^0}+\\|h\\|_{\\widetilde{B}^{0,1}})\\Bigr),\n\\end{eqnarray*}\nfrom which and a similar proof of (\\ref{3.21}) ensure that\n\\begin{eqnarray}\\label{3.34}\n&&\\sum_{k<1}\\Bigl(2^{2k}\\|E_{lk}\\|_{L_T^1}+\\|E_{lk}\\|_{L_T^\\infty}\\Bigr)\\le \\sum_{k<1}E_{hk}(0)\\nonumber\\\\\n&&\\qquad+C\\Bigl(\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0,1})}\n+\\|{\\cal G}\\|_{L^1_T(B^0)}+\\int_0^TV'(t)(\\|u(t)\\|_{B^0}+\\|h(t)\\|_{\\widetilde{B}^{0,1}})dt\\Bigr).\n\\end{eqnarray}\n\n\\vspace{.2cm}\n\\noindent{\\bf The completion of the {\\it a-priori} estimates}\\vspace{.2cm}\n\n\\noindent Firstly, adding up \\eqref{3.17}, \\eqref{3.21}, \\eqref{3.31}, and \\eqref{3.32} yields that\n\\begin{eqnarray}\\label{3.35}\n&&\\|u\\|_{L^1_T(B^2)}+\\|u\\|_{L^2_T(B^1)}+\\|h\\|_{\\widetilde{E}^{0,1}_T}\\nonumber\\\\\n&&\\quad\\le C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)E_k(0)\n+C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\|{\\cal G}_k(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\sum_{k\\ge 1}\\omega_k(T)\\|\\nabla{\\cal H}_k(t)\\|_{L^1_T(L^2)}\n+C\\sum_{k<1}\\omega_k(T)\\|{\\cal H}_k(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+C\\|h\\|_{\\widetilde{E}^{0,1}_T}\\|v\\|_{L^1_T(B^2)},\n\\end{eqnarray}\nwhere we used the fact that\n$$\n\\|h\\|_{E^{1}_T}\\le C\\|h\\|_{\\widetilde{E}^{0,1}_T}.\n$$\nOn the other hand, adding up \\eqref{3.23} and \\eqref{3.34} gives rise to\n\\begin{eqnarray}\\label{3.36}\n&&\\|u\\|_{L^\\infty_T(B^0)}+\\|h\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h\\|_{L^1_T(\\widetilde{B}^{2,1})}\n\\nonumber\\\\&&\\qquad \\le  E_0+C\\Big(\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0, 1})}+\\|{\\cal G}\\|_{L^1_T(B^0)}+\n\\int_0^TV'(t)(\\|u\\|_{B^{0}}+\\|h\\|_{\\widetilde{B}^{0, 1}})dt\\Big),\n\\end{eqnarray}\nwhich together with the Gronwall inequality implies that\n\\begin{eqnarray}\\label{3.37}\n&&\\|u\\|_{L^\\infty_T(B^0)}+\\|h\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h\\|_{L^1_T(\\widetilde{B}^{2,1})}\\nonumber\\\\\n&&\\qquad\\qquad\\le Ce^{C\\|v\\|_{L^1_T(B^2)}}\\Big(E_0+\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0, 1})}+\\|{\\cal G}\\|_{L^1_T(B^0)}\\Big).\n\\end{eqnarray}\nFinally, let us remark that\n\\begin{eqnarray*}\nE_0\\thickapprox(\\|h_0\\|_{\\widetilde{B}^{0,1}}+\\|u_0\\|_{B^0}).\n\\end{eqnarray*}\n\n\n\\subsection{The uniform estimate of the approximate sequence of solutions}\n\nIn this subsection, we will construct the approximate solutions of (\\ref{3.1})\nand present the uniform estimate of the approximate solutions.\nLet us first define the approximate sequence $(h^n, u^n)_{n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits}$ of (\\ref{3.1}) by the following system:\n\\begin{align}\\label{3.38} \\left\\{\n\\begin{aligned}\n&\\partial_t h^{n+1}+u^n\\cdot\\nabla h^{n+1}+\\mbox{div} u^{n+1}=\n{\\cal H}^n,\\\\\n&\\partial_tu^{n+1}-(\\nabla\\cdot D(u^{n+1})+\\nabla \\textrm{div}\\,u^{n+1})+u^{n}\\cdot\\nabla u^{n+1}+\\nabla h^{n+1}\n={\\cal G}^{n},\\qquad\\quad\\\\\n&(h^{n+1}, u^{n+1})|_{t=0}=\\sum_{|k|\\le n+N}\\Delta_k(h_0,u_0),\n\\end{aligned}\n\\right.\n\\end{align}\nwhere\n$${\\cal H}^n\\triangleq -h^n\\mbox{div} u^n,\\qquad {\\cal G}^{n}\\triangleq \\frac{\\nabla h^n}{1+h^n}\\widetilde{\\nabla }u^n,\n\\quad \\textrm{with}\\quad \\widetilde{\\nabla} u^n=D(u^n)+\\mbox{div} u^n, $$\nand $N$ is a fixed large integer such that\n$$\n1+h^n(0)\\ge \\f34,\\qquad \\textrm{for} \\quad n\\ge 1.\n$$\nSet $(h^0, u^0)=(0,0)$ and solve the linear system, we can define  $(h^n, u^n)_{n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits_0}$ by the induction.\nNext, we are going to prove by the induction that there exist positive constants $\\eta$, $K$, and  $T$ such that\nthe following bounds hold for all $n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits_0$:\n\\begin{align}\n&1+h^n\\ge\\f12,\\label{3.39}\\\\\n&\\|u^{n}\\|_{L^1_T(B^2)\\cap L^2_T(B^1)}+\\|h^n\\|_{\\widetilde{E}^{0,1}_T}\n\\le \\eta,\\label{3.40}\\\\\n&\\|u^n\\|_{L^\\infty_T(B^0)}+\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})\\cap L^1_T(\\widetilde{B}^{2,1})}\n\\le KE_0.\\label{3.41}\n\\end{align}\nAssume that (\\ref{3.39})-(\\ref{3.41}) hold for $(h^n, u^n)$,\nwe need to prove that (\\ref{3.39})-(\\ref{3.41}) also hold for $(h^{n+1}, u^{n+1})$.\nApplying the {\\it a-priori} estimates (\\ref{3.35}) and (\\ref{3.37}) to $(h^{n+1}, u^{n+1})$, we obtain\n\\begin{eqnarray}\\label{3.42}\n&&\\|u^{n+1}\\|_{L^1_T(B^2)}+\\|u^{n+1}\\|_{L^2_T(B^1)}+\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}\\nonumber\\\\\n&&\\quad\\le C{\\cal Q}_0(T)+C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\|{\\cal G}_k^{n}(t)\\|_{L^1_T(L^2)}\n+C\\sum_{k\\ge 1}\\omega_k(T)\\|\\nabla {\\cal H}_k^{n}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\sum_{k<1}\\omega_k(T)\\|{\\cal H}_k^{n}(t)\\|_{L^1_T(L^2)}+C\\|u^{n+1}\\|_{L^2_T(B^1)}\\|u^n\\|_{L^2_T(B^1)}\\nonumber\\\\\n&&\\qquad+C\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}\\|u^n\\|_{L^1_T(B^2)},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{3.43}\n&&\\|u^{n+1}\\|_{L^\\infty_T(B^0)}+\\|h^{n+1}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h^{n+1}\\|_{L^1_T(\\widetilde{B}^{2,1})}\\nonumber\\\\\n&&\\qquad\\qquad\\le Ce^{C\\|u^{n}\\|_{L^1_T(B^2)}}\\Big(E_0+\\|{\\cal H}^{n}\\|_{L^1_T(\\widetilde{B}^{0,1})}+\\|{\\cal G}^{n}\\|_{L^1_T(B^0)}\\Big),\n\\end{eqnarray}\nwith\n$${\\cal Q}_0(T)\\triangleq\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)E_{k}(0).$$\nThanks to (\\ref{2.4}), we have\n\\begin{eqnarray*}\n\\|{\\cal H}^n\\|_{{B}^{0}}\\le C\\|{h}^n\\|_{{B}^{0}}\n\\|u^n\\|_{B^2},\\quad \\hbox{and}\\quad\n\\|{\\cal H}^n\\|_{{B}^{1}}\\le C\\|{h}^n\\|_{{B}^{1}}\n\\|u^n\\|_{B^2},\n\\end{eqnarray*} which together with the fact that $\\widetilde{B}^{0,1}={B}^{0}\\cap{B}^{1}$ yields\n\\begin{eqnarray}\\label{3.44}\n\\|{\\cal H}^n\\|_{L^1_T(\\widetilde{B}^{0,1})}\n\\le C\\|{h}^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\|u^n\\|_{L^1_T(B^2)}\\le CKE_0\\eta.\n\\end{eqnarray}\nWe rewrite ${\\cal G}^n$ as\n\\begin{eqnarray*}\\label{3.45}\n\\frac{\\nabla h^n}{1+h^n}\\widetilde{\\nabla} u^n=\n(1+h^n)\\nabla\\bigg(\\frac{h^n}{1+h^n}\\bigg)\\widetilde{\\nabla} u^n.\n\\end{eqnarray*}\nUsing (\\ref{2.4}) and  (\\ref{2.12}), we get\n\\begin{align}\\label{3.45}\n\\|{\\cal G}^{n}\\|_{L^1_T(B^0)}\n&\\le C\\Bigl\\|\\nabla\\bigg(\\frac{h^n}{1+h^n}\\bigg)\\Bigr\\|_{L^\\infty_T(B^0)}\\|(1+h^n)\\widetilde{\\nabla} u^n\\|_{L^1_T(B^1)}\\nonumber\\\\\n&\\le C(1+\\|h^n\\|_{L^\\infty_T(L^\\infty)})^2\\|h^n\\|_{L^\\infty_T(B^1)}(1+\\|h^n\\|_{L^\\infty_T(B^1)})\\|u^n\\|_{L^1_T(B^2)}\\nonumber\\\\\n&\\le C(1+\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})})^3\n\\|h^{n}\\|_{L^\\infty_T(B^1)}\\|u^n\\|_{L^1_T(B^2)}\\nonumber\\\\\n&\\le CKE_0(1+KE_0)^3\\eta.\n\\end{align}\nPlugging  (\\ref{3.44}) and (\\ref{3.45}) into (\\ref{3.43}) yields that\n\\begin{eqnarray}\\label{3.46}\n\\|u^{n+1}\\|_{L^\\infty_T(B^0)}+\\|h^{n+1}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h^{n+1}\\|_{L^1_T(\\widetilde{B}^{2,1})}\n\\le Ce^{C\\eta}\\Big(E_0+KE_0(1+KE_0)^3\\eta\\Big).\n\\end{eqnarray}\nWe take $T, \\eta>0$ small enough and $K=4C$ such that\n\\begin{align}\ne^{C\\eta}\\le 2,\\quad  K(1+KE_0)^3\\eta\\le 1,\\tag{$\\Re_1$}\n\\end{align}\nfrom which and (\\ref{3.46}), it follows that\n\\begin{eqnarray*}\n\\|u^{n+1}\\|_{L^\\infty_T(B^0)}+\\|h^{n+1}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h^{n+1}\\|_{L^1_T(\\widetilde{B}^{2,1})}\\le KE_0.\n\\end{eqnarray*}\nThis proves (\\ref{3.41}) for $(u^{n+1},h^{n+1})$.\n\nNext, we prove (\\ref{3.40}) for $(u^{n+1},h^{n+1})$.\nApplying Lemma \\ref{Lem5.2} with $s_1=0$ and $s_2=1$, (\\ref{2.4}) with $s_1=s_2=1$,\nand Lemma \\ref{Lem5.4} with $s=1$,  we obtain\n\\begin{align}\\label{3.47}\n\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\|{\\cal G}_k^{n}(t)\\|_{L^1_T(L^2)}\n&\\le C\\Bigl\\|\\nabla\\bigg(\\frac{h^n}{1+h^n}\\bigg)\\Bigr\\|_{E^0_T}\\|(1+h^n)\\widetilde{\\nabla} u^n\\|_{L^1_T(B^1)}\\nonumber\\\\\n&\\le C(1+\\|h^n\\|_{L^\\infty_T(L^\\infty)})^3\\|h^n\\|_{E^1_T}(1+\\|h^n\\|_{L^\\infty_T(B^1)})\\|u^n\\|_{L^1_T(B^2)}\\nonumber\\\\\n&\\le C(1+\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})})^4\n\\|h^{n}\\|_{\\widetilde{E}^{0,1}_T}\\|u^n\\|_{L^1_T(B^2)}\\nonumber\\\\\n&\\le C(1+KE_0)^4\\eta^2.\n\\end{align}\nOn the other hand, we apply Lemma 5.2 with $s_1=0, s_2=1$ to get\n\\begin{eqnarray*}\n&&\\sum_{k\\ge 1}\\omega_k(T)\\|\\nabla {\\cal H}_k^{n}(t)\\|_{L^1_T(L^2)}+\\sum_{k<1}\\omega_k(T)\\|{\\cal H}_k^{n}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\quad\\le C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)(\\|\\nabla h^n_k\\|_{L^\\infty_T(L^2)}+\n\\|h^n_k\\|_{L^\\infty_T(L^2)})\\|\\mbox{div} u^n\\|_{L^1_T(B^1)}\\nonumber\\\\\n&&\\qquad+C\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{2k}\\|u^n_k\\|_{L^1_T(L^2)}\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\nonumber\\\\\n&&\\quad\\triangleq I+II.\n\\end{eqnarray*}\nObviously, we have\n\\begin{eqnarray}\\label{3.48}\nI\\le C\\|h^n\\|_{\\widetilde{E}^{0,1}_T}\\|u^n\\|_{L^1_T(B^2)}\\le C\\eta^2.\n\\end{eqnarray}\nIn order to estimate $II$, we first fix $k_0\\ge 1$ such that\n\\begin{eqnarray}\\label{3.49}\n\\sum_{k\\ge k_0}\\|u_k(0)\\|_{2}\\le \\f {\\eta} {16CKE_0}.\n\\end{eqnarray}\nThen we write\n\\begin{align}\nII&=\\sum_{k\\ge k_0}\\omega_k(T)2^{2k}\\|u^n_k\\|_{L^1_T(L^2)}\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\n+\\sum_{k\\le k_0}\\omega_k(T)2^{2k}\\|u^n_k\\|_{L^1_T(L^2)}\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\nonumber\\\\\n&\\triangleq II_1+II_2.\\nonumber\n\\end{align}\nUsing (\\ref{3.18}), (\\ref{3.47}), and (\\ref{3.49}), we obtain\n\\begin{align}\\label{3.50}\nII_1&\\le CKE_0\\Bigl[\\sum_{k\\ge k_0}\\omega_k(T)\\|u_k(0)\\|_2+\n\\sum_{k\\ge k_0}\\omega_k(T)\\Bigl(\\|\\nabla h_k^n\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}^{n-1}\\|_{L^1_T(L^2)}\\Bigr)\\nonumber\\\\\n&\\quad+\\|u^n\\|_{L^2_T(B^1)}\\|u^{n-1}\\|_{L^2_T(B^1)}\\Bigr]\\nonumber\\\\\n&\\le CKE_0\\Bigl[\\f {\\eta} {16CKE_0}+\n\\sum_{k\\ge k_0}\\omega_k(T)\\|\\nabla h_k^n\\|_{L^1_T(L^2)}+(1+KE_0)^4\\eta^2\\Bigr].\n\\end{align}\nOn the other hand, thanks to (\\ref{3.19}) and Lemma \\ref{Lem5.1}, we have\n\\begin{align}\n\\sum_{k\\ge k_0}\\omega_k(T)\\|\\nabla h_k^n\\|_{L_T^1(L^2)}&\\le C(1-e^{-cT})\\sum_{k\\ge k_0}\\omega_k(T)E_{hk}(0)\\nonumber\\\\\n&\\quad+C(1-e^{-cT})\\sum_{k\\ge k_0}\\omega_k(T)\\Bigl(\\|\\nabla {\\cal H}_k^{n-1}(t)\\|_{L_T^1(L^2)}\n+\\|{\\cal G}_{k}^{n-1}(t)\\|_{L_T^1(L^2)}\\Bigr)\\nonumber\\\\\n&\\quad+C\\|u^n\\|_{L^2_T(B^1)}\\|u^{n-1}\\|_{L^2_T(B^1)}+C\\|h^n\\|_{E^1_T}\\|u^{n-1}\\|_{L^2_T(B^1)}\\nonumber\\\\\n&\\le C(1-e^{-cT})E_0+C(1-e^{-cT})KE_0\\eta+C(1+KE_0)^4\\eta^2,\\nonumber\n\\end{align}\nwhere we used (\\ref{3.44}) and (\\ref{3.47}) in the second inequality.\nPlugging the above inequality into (\\ref{3.50}) yields that\n\\begin{eqnarray}\\label{3.51}\nII_1\\le CKE_0\\Bigl[\\f {\\eta} {16CKE_0}+\n(1-e^{-cT})(E_0+KE_0\\eta)+(1+KE_0)^4\\eta^2\\Bigr].\n\\end{eqnarray}\nNote for $k\\le k_0$, we can choose $T>0$ small enough so that\n\\begin{align}\n\\omega_k(T)\\le \\f 1{16CKE_0\\eta},\\tag{$\\Re_2$}\n\\end{align}\nso we get\n\\begin{eqnarray}\\label{3.52}\n|II_2|\\le \\f \\eta {16}.\n\\end{eqnarray}\n\nPlugging (\\ref{3.47}), (\\ref{3.48}), (\\ref{3.51}), (\\ref{3.52}) into (\\ref{3.42}), we get\n\\begin{eqnarray}\\label{3.53}\n&&\\|u^{n+1}\\|_{L^1_T(B^2)}+\\|u^{n+1}\\|_{L^2_T(B^1)}+\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}\\nonumber\\\\\n&&\\quad\\le C{\\cal Q}_0(T)+\\f \\eta {8}+ C(1+KE_0)^5\\eta^2+\nCKE_0(1-e^{-cT})(E_0+KE_0\\eta)\\nonumber\\\\\n&&\\qquad+C\\eta(\\|u^{n+1}\\|_{L^2_T(B^1)}+\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}).\n\\end{eqnarray}\nNote that ${\\cal Q}_0(0)=0$, we can take $T, \\eta$ small enough such that\n\\begin{align}\n&C\\eta\\le \\f12,\\quad C{\\cal Q}_0(T)\\le \\f \\eta 8,\\quad C(1+KE_0)^5\\eta<\\f18,\n\\quad \\textrm{and}\\nonumber\\\\\n&CKE_0(1-e^{-cT})(E_0+KE_0\\eta)\\le \\f \\eta 8, \\tag{$\\Re_3$}\n\\end{align}\nwhich together with (\\ref{3.53}) gives\n\\begin{eqnarray*}\n\\|u^{n+1}\\|_{L^1_T(B^2)}+\\|u^{n+1}\\|_{L^2_T(B^1)}+\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}\\le\\eta.\n\\end{eqnarray*}\n\nFinally, let us prove (\\ref{3.39}) for $h^{n+1}$.\nWe  rewrite the first equation of \\eqref{3.38} as\n\\begin{eqnarray*}\n\\partial_t(1+h^{n+1})+u^{n}\\cdot\\nabla(1+h^{n+1})+\\mbox{div} u^{n+1}-{\\cal H}^{n}=0.\n\\end{eqnarray*}\nThen $1+h^{n+1}$ can be represented as\n\\begin{align}\\label{3.54}\n(1+h^{n+1})(t,x)=&(1+h^{n+1}_0)((\\psi^n)^{-1}_t(x))+\\int_0^t\\mbox{div}\\,u^{n+1}(\\tau,\\psi^n_\\tau((\\psi^n)^{-1}_t(x)))d\\tau\\nonumber\\\\\n&+\\int_0^t{\\cal H}^{n}(\\tau,\\psi^n_\\tau((\\psi^n)^{-1}_t(x)))d\\tau,\n\\end{align}\nwhere the flow map $\\psi^{n}_t$ is defined by\n\\begin{eqnarray*}\\left\\{\n\\begin{aligned}\n&\\partial_t\\psi^{n}_t(x)=u^{n}(t,\\psi^{n}_t(x))\\\\\n&\\psi^{n}_t|_{t=0}=x.\n\\end{aligned}\\right.\n\\end{eqnarray*}\nThanks to the inclusion map $B^1\\hookrightarrow L^\\infty$ and (\\ref{2.4}), we get\n\\begin{eqnarray*}\n&&\\int_0^t\\|\\mbox{div}\\,u^{n+1}(\\tau,\\psi^n_\\tau((\\psi^n)^{-1}_t(x)))\\|_{\\infty} d\\tau\\le \\|u^{n+1}\\|_{L^1_t(B^2)}\\le \\eta,\\\\\n&&\\int_0^t\\|{\\cal H}^{n}(\\tau,\\psi_\\tau(\\psi^{-1}_t(x)))\\|_\\infty d\\tau \\le\\|h^{n}\\mbox{div} u^{n}\\|_{L^1_t(B^1)}\\nonumber\\\\\n&&\\qquad\\qquad\\qquad\\qquad\\le C\\|h^{n}\\|_{L^\\infty_t(\\widetilde{B}^{0,1})}\\|u^{n}\\|_{L^1_t(B^2)}\\le CKE_0\\eta,\n\\end{eqnarray*}\nfrom which and (\\ref{3.54}), it follows that\n\\begin{eqnarray}\\label{3.55}\n1+h^{n+1}\\ge \\f34-(1+CKE_0)\\eta.\n\\end{eqnarray}\nWe take $\\eta$ small enough such that\n\\begin{align}\n(1+CKE_0)\\eta\\le \\f14, \\tag{$\\Re_4$}\n\\end{align}\nwhich together with (\\ref{3.55}) ensures that\n\\begin{eqnarray*}\n1+h^{n+1}\\ge \\f12.\n\\end{eqnarray*}\nSo far, we have show that $T$, $\\eta$ can be chosen small enough such that\nthe assumption $(\\Re_1)-(\\Re_4)$ hold under which\nthe approximate solutions $(u^n, h^n)_{n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits_0}$ is uniformly bounded in\n$${\\cal E}_T\\triangleq \\Bigl(L^\\infty_T(B^0)\\cap L^1_T(B^2)\\Bigr)\\times\n\\Bigl(L^\\infty_T(\\widetilde{B}^{0,1})\\cap L^1_T(\\widetilde{B}^{2,1})\\Bigr).$$\nIt should be pointed out that if $\\|u_0\\|_{B^0}+\n\\|h_0\\|_{\\widetilde{B}^{0,1}}$ is small enough, we can take $T=+\\infty$ such that\nthe assumption $(\\Re_1)-(\\Re_4)$ hold.\n\n\\subsection {The existence of the solution}\nNow let us  turn to prove the existence of the solution, and\nthe standard compact arguments will be used.\nIn the section 3.2, we have showed that the approximate solutions $(h^n,u^n)_{n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits}$\nsatisfy \\eqref{3.39}-\\eqref{3.41}, and without loss of generality, we can assume the following:\n\\begin{align}\n1&+h^n\\ge\\f12,\\label{3.56}\\\\\n\\|u^n\\|_{L^\\infty_T(B^0)\\cap L^1_T(B^2)}&+\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})\\cap L^1_T(\\widetilde{B}^{2,1})}\n\\le KE_0.\\label{3.57}\n\\end{align}\nUsing the interpolation and the fact that $B^0\\cap B^1=\\widetilde{B}^{0,1}$, we have\n\\begin{eqnarray*}\n&&\\|h^n\\|_{L^2_T(B^1)}\\lesssim\\|h^n\\|^{\\frac{1}{2}}_{L^\\infty_T(\\widetilde{B}^{0,1})}\n\\|h^n\\|^{\\frac{1}{2}}_{L^1_T(\\widetilde{B}^{2,1})},\\quad\n\\|u^n\\|_{L^2_T(B^1)}\\lesssim\n\\|u^n\\|^{\\frac{1}{2}}_{L^\\infty_T(B^0)}\\|u^n\\|^{\\frac{1}{2}}_{L^1_T(B^2)},\\\\\n&&\\|h^n\\|_{L^4_T(B^{\\frac{1}{2}})}\\lesssim\\|h^n\\|^{\\frac{1}{2}}_{L^\\infty_T(\\widetilde{B}^{0,1})}\n\\|h^n\\|^{\\frac{1}{2}}_{L^2_T(B^{1})},\\quad\\|u^n\\|_{L^\\frac{4}{3}_T(B^\\frac{3}{2})}\\lesssim\n\\|u^n\\|^{\\frac{1}{4}}_{L^\\infty_T(B^0)}\\|u^n\\|^{\\frac{3}{4}}_{L^1_T(B^2)},\n\\end{eqnarray*} from which and (\\ref{3.56}), it follows that\n\\begin{eqnarray}\\label{3.58}\n\\|h^n\\|_{L^2_T(B^1)}+\\|u^n\\|_{L^2_T(B^1)}+\\|h^n\\|_{L^4_T(B^{\\frac{1}{2}})}+\\|u^n\\|_{L^\\frac{4}{3}_T(B^\\frac{3}{2})}\n\\lesssim KE_0.\n\\end{eqnarray}\n\nNow, we  show that $(h^n, u^n)$ is uniformly bounded in $C^{\\frac{1}{2}}_{loc}(B^0)\\times\nC^{\\frac{1}{4}}_{loc}(B^{-\\frac{1}{2}})$. Using (\\ref{2.4}), (\\ref{3.57})\nand (\\ref{3.58}), it is easy to verify that\n\\begin{eqnarray*}\n&&\\|u^{n}\\cdot\\nabla h^{n+1}\\|_{L^2_T(B^{0})}\\lesssim\n\\|u^{n}\\|_{L^2_T(B^{1})}\\|h^{n+1}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\lesssim (KE_0)^2,\\\\\n&&\\|h^{n}\\mbox{div} u^{n}\\|_{L^2_T(B^{0})}\\lesssim\n\\|u^{n}\\|_{L^2_T(B^{1})}\\|h^{n}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\lesssim\n(KE_0)^2, \\end{eqnarray*}\nfrom which and the first equation of (\\ref{3.38}), it follows that $\\partial_t\nh^{n}$ is uniformly bounded in $L^2_T(B^0)$ which implies $h^{n}$ is\nuniformly bounded in $C^{\\frac{1}{2}}_{loc}(B^0)$.\nOn the other hand, thanks to (\\ref{2.4}), (\\ref{3.56}) and (\\ref{2.12}), we have\n\\begin{eqnarray*}\n&&\\|u^{n}\\cdot\\nabla u^{n+1}\\|_{L^\\frac{4}{3}_T(B^{-\\frac{1}{2}})}\\lesssim\n\\|u^{n}\\|_{L^\\infty_T(B^0)}\\|u^{n+1}\\|_{L^\\frac{4}{3}_T(B^\\frac{3}{2})}\\lesssim (KE_0)^2,\\\\\n&&\\bigg\\|\\frac{\\nabla h^{n}}{1+h^{n}}\\widetilde{\\nabla} u^{n}\\bigg\\|_{L^\\frac{4}{3}_T(B^{-\\frac{1}{2}})}\n\\lesssim C(1+\\|h^{n}\\|_{L^\\infty_T(B^1)})^3\\|u^{n}\\|_{L^\\frac{4}{3}_T(B^\\frac{3}{2})}\\lesssim C(1+KE_0)^3KE_0,\n\\end{eqnarray*}\nfrom which and the second equation of (\\ref{3.38}), it follows that\n$\\partial_t u^{n}$ is uniformly bounded in $L^\\frac{4}{3}_T(B^{-\\frac{1}{2}})$\nwhich implies $u^{n}$ is uniformly bounded\nin $C^{\\frac{1}{4}}_{loc}(B^{-\\frac{1}{2}})$.\n\nNext, we claim that the inclusions  $B^0\\cap B^1\\hookrightarrow L^2$ and\n$B^{-\\frac{1}{2}}\\cap B^0\\hookrightarrow \\dot{H}^{-\\frac{1}{2}}$ are locally\ncompact. Indeed, these can be proved by noting that for $s'<s$,\n$\\dot{H}^{s'}\\cap \\dot{H}^{s}\\hookrightarrow \\dot{H}^{s'}$ is locally compact and\nfor $s\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$,\n$B^s\\hookrightarrow \\dot{H}^s$.\nThen, by the Arzela-Ascoli theorem and Cantor's diagonal process, there\nexist a subsequence $(u^{n_k}, h^{n_k})$ and a function $(u, h)$ such that\n\\begin{eqnarray}\\label{3.59}\n(u^{n_k}, h^{n_k})\\rightarrow (u, h)\\qquad\n\\mbox{in}\\qquad C_{loc}(\\dot{H}^{-\\frac{1}{2}}_{loc})\\times C_{loc}(L^2_{loc}),\n\\end{eqnarray}\nas $n_k\\rightarrow\\infty$.\nOn the other hand, $(u^{n_k}, h^{n_k})$ is uniformly bounded\nin ${\\cal E}_T$, then there exists a\nsubsequence (which  still denoted by $(u^{n_k},h^{n_k})$)\nsuch that\n$$(u^{n_k}, h^{n_k})\\rightharpoonup (u, h)\\quad\n\\mbox{in}\\quad {\\cal E}_{T},$$\nwhere $``\\rightharpoonup\"$ denotes weak* convergence.\n\nFinally, let us  prove that $(u, h)$ solves (\\ref{1.1})\nin the sense of distribution. We only need to\nprove the nonlinear terms  such as  $u^n\\cdot\\nabla h^n$,\n$\\frac{\\nabla h^n}{1+h^n}\\widetilde{\\nabla} u^n$, etc tend\nto the corresponding nonlinear terms  in the sense of distribution.\nThis can be done by using the uniform\nestimates of $(u^n, h^n)$, $(u, h)$ in ${\\cal E}_T$ and the convergence result (\\ref{3.59}).\nHere, we only show the case of the term\n$Y(h^n)\\widetilde{\\nabla} u^n$\n(where $Y(z)\\triangleq \\nabla z\/(1+z) $), the other terms can be treated in the same way.\nFor any test function $\\theta\\in C_0^\\infty([0, T^*)\\times\\mathop{\\bf R\\kern 0pt}\\nolimits^2)$, we write\n\\begin{align}\n\\big<&Y(h^n)\\widetilde{\\nabla} u^n\n-Y(h)\\widetilde{\\nabla} u,\\,\\theta\\big>\\nonumber\\\\\n&=\n\\Big<(1+h^n)\\nabla\\Big(\\frac{h^n}{1+h^n}-\\frac{h}{1+h}\\Big)\\widetilde{\\nabla} u^n,\\,\\theta\\Big>\\nonumber\\\\&\\quad+\n\\Big<(h^n-h)\\nabla\\Big(\\frac{h}{1+h}\\Big)\\widetilde{\\nabla} u^n,\\,\\theta\\Big>+\n\\Big<(1+h)\\nabla\\Big(\\frac{h}{1+h}\\Big)\\widetilde{\\nabla} (u^n-u),\\,\\theta\\Big>\\nonumber\\\\&\n\\triangleq{I}_1+{I}_2+{I}_3.\\nonumber\n\\end{align}\nThanks to (\\ref{2.4}) and (\\ref{3.56}), we have\n\\begin{align}\n{I}_1&\\le\\bigg\\|\\frac{\\psi(h^n-h)}{(1+h^n)(1+h)}\\bigg\\|_2\n\\|\\nabla((1+h^n)\\widetilde{\\nabla} u^n\\theta)\\|_2\\lesssim\\|\\theta(h^n-h)\\|_2\\|(1+h^n)\\widetilde{\\nabla} u^n\\|_{B^1}\\nonumber\\\\\n&\\lesssim \\|\\theta(h^n-h)\\|_2(1+\\|h^n\\|_{\\widetilde{B}^{0,1}})\\|u^n\\|_{B^2},\\nonumber\n\\end{align}\nwhere $\\psi \\in C_0^\\infty([0, T^*)\\times\\mathop{\\bf R\\kern 0pt}\\nolimits^2)$,\\,and $\\psi=1$ on \\,$\\textrm{supp}\\,\\theta$.\nFor ${I}_2$, we have\n\\begin{align}\n{I}_2&\\le\\|\\theta(h^n-h)\\|_2\n\\bigg\\|\\nabla\\bigg(\\frac{h}{1+h}\\bigg)\\widetilde{\\nabla} u^n\\bigg\\|_2\n\\lesssim\\|\\theta(h^n-h)\\|_2\\|\\nabla h\\|_2\\|\\nabla u^n\\|_{L^\\infty}\\nonumber\\\\\n&\\lesssim \\|\\theta(h^n-h)\\|_2\\|h\\|_{\\widetilde{B}^{0,1}}\\|u^n\\|_{B^2}.\\nonumber\n\\end{align}\nUsing (\\ref{3.56}) and the interpolation, we get\n\\begin{align}\n{I}_3&\\le\\bigg\\|(1+h)\\nabla\\bigg(\\frac{h}{1+h}\\bigg)\\bigg\\|_2\n\\|\\widetilde{\\nabla} (u^n-u)\\theta\\|_2\\lesssim (1+\\|h\\|_\\infty)\\|\\nabla h\\|_2\\|(u^n-u)\\psi\\|_{\\dot{H}^1}\\nonumber\\\\\n&\\lesssim (1+\\|h\\|_{\\widetilde{B}^{0,1}})\\|h\\|_{B^1}\\|u^n-u\\|^{\\frac{3}{5}}_{\\dot{H}^2}\n\\|(u^n-u)\\theta\\|^{\\frac{2}{5}}_{\\dot{H}^{-\\frac{1}{2}}}.\\nonumber\n\\end{align}\nThus, by (\\ref{3.59}), we get as $n\\rightarrow 0$\n$$\n\\big<Y(h^n)\\widetilde{\\nabla} u^n\n-Y(h)\\widetilde{\\nabla} u,\\,\\theta\\big>\\longrightarrow 0.\n$$\nFollowing the argument in \\cite{D1}, we can also prove that $(u,h)$ is continuous in\ntime with values in $B^{0}\\times \\widetilde{B}^{0,1}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\n\\section{Uniqueness}\n\nIn this section, we will prove the uniqueness of the solution. Firstly, let us recall some known results.\n\\begin{Lemma}\\label{Lemma4.1}(Osgood's lemma) Let $\\rho$ be a measurable positive\nfunction and $\\gamma$ a positive locally integrable function, each defined on\nthe domain $[t_0, t_1]$. Let $\\mu: [0, \\infty)\\rightarrow [0,\\infty)$\nbe a continuous nondecreasing function, with $\\mu(0)=0$. Let $a\\ge0$,\nand assume that for all $t$ in $[t_0, t_1]$,\n$$\\rho(t)\\le a+\\int_{t_0}^{t}\\gamma(\\tau)\\mu(\\rho(\\tau))d\\tau.$$\nIf $a>0$, then\n$$-{\\cal M}(\\rho(t))+{\\cal M}(a)\\le \\int_{t_0}^{t}\\gamma(\\tau)d\\tau,\n\\quad\\mbox{where}\\quad{\\cal M}(x)=\\int_x^1\\frac{d\\tau}{\\mu(\\tau)}.$$\nIf $a=0$ and ${\\cal M}=\\infty$, then $\\rho\\equiv0.$\n\\end{Lemma}\nThis Lemma can be understood as a generalization of classical Gronwall Lemma and\ncan be found in \\cite{Ch1}.\n\\begin{Proposition}\\label{Prop4.1}\nLet $s\\in (-\\frac{d}{p}, 1+\\frac{d}{p})$, and $1\\le p,r\\le+\\infty$.\nLet $v$ be a vector field such that $\\nabla v\\in L^1_T(\\dot{B}^{\\frac{d}{p}}_{p,r}\\cap L^\\infty)$.\nAssume that $f_0\\in \\dot{B}^{s}_{p,r},$ $g\\in L^1_T(\\dot{B}^{s}_{p,r})$ and\n$f\\in L^\\infty_T(\\dot{B}^{s}_{p,r})\\cap C([0,T]; {\\cal S}')$ is the solution of\n\\begin{align}\n\\bigg\\{\\begin{aligned}\n&\\partial_t f+v\\cdot \\nabla f =g,\\\\\n&f(0,x)=f_0.\n\\end{aligned}\n\\bigg.\\nonumber\\end{align}\nThen there exists a constant $C(s,p,d)$ such that\nfor $t\\in[0,T]$\n\\begin{align}\\label{4.1}\n\\|f\\|_{\\widetilde{L}^\\infty_t(\\dot{B}^{s}_{p,r})}\\le Ce^{CV(t)}\\bigg(\n\\|f_0\\|_{\\dot{B}^{s}_{p,r}}+\\int_0^t e^{-CV(\\tau)}\\|g(\\tau)\\|_{\\dot{B}^{s}_{p,r}}d\\tau\\bigg),\n\\end{align}\nwhere $V(t)\\triangleq\\int_0^t\\|\\nabla v(\\tau)\\|_{\\dot{B}^{\\frac{d}{p}}_{p,r}\\cap L^\\infty}d\\tau.$\nIf $r<+\\infty$, then $f$ belongs to $C([0,T]; \\dot{B}^s_{p,r})$.\n\\end{Proposition}\nThe proof can be found in \\cite{D4}.\n\\begin{Proposition}\\label{Prop4.3}\nLet $T>0$, $s\\in \\mathop{\\bf R\\kern 0pt}\\nolimits$, and $1\\le q,r\\le+\\infty$. Assume that\n$u_0\\in \\dot{B}^{s}_{2,q},$ $g\\in \\widetilde{L}^1_T(\\dot{B}^{s}_{2,q})$ and\n$u$ is the solution of\n\\begin{align}\n\\,\\bigg\\{\\begin{aligned}\n&\\partial_t u-\\nu \\widetilde{\\Delta} u=g,\\nonumber\\\\\n&u(0,x)=u_0,\n\\end{aligned}\n\\bigg.\\end{align}\nwhere $\\widetilde{\\Delta} u=\\nabla\\cdot D(u)+\\nabla \\mbox{div} \\,u$.\nThen there exists a constant $C(s,d,\\nu)$ such that\n\\begin{align}\\label{4.2}\n(r\\nu)^{\\frac1r}\\|u\\|_{\\widetilde{L}^r_T(\\dot{B}^{s+\\frac2r}_{2,q})}\\le\n\\Big(&\\sum_{k\\in{\\mathbf Z}}\\big({1-e^{-r\\nu 2^{2k}T}}\\big)^{\\frac q r}2^{qks}\\|\\Delta_ku_0\\|_2^q\\Big)^{\\frac1q}\n\\nonumber\\\\&+C\\Big(\\sum_{k\\in{\\mathbf Z}}\\big({1-e^{-r\\nu 2^{2k}T}}\\big)^{\\frac qr}2^{qks}\\|\\Delta_kg\\|_{L^1_T(L^2)}^q\\Big)^{\\frac1q}.\n\\end{align}\nIf $q<+\\infty$, then $u$ belongs to $C([0,T]; \\dot{B}^{s}_{2,q})$.\n\\end{Proposition}\nThe proof is similar to the case when the diffusion term $\\widetilde{\\Delta} u$ is replaced by $\\Delta u$.\nWe can refer to \\cite{Ch2} see the details.\n\n\nNow we introduce the logarithmic interpolation inequality (see \\cite{D3})\n\\begin{Proposition}\\label{Prop4.4} For any $1\\le p,\\rho\\le+\\infty$, $s\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$ and $0<\\epsilon\\le 1$,\nwe have\n\\begin{eqnarray}\\label{4.3}\n\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^s_{p,1})}\\le\nC\\frac{\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^s_{p,\\infty})}}{\\epsilon}\n\\log\\bigg(e+\\frac{\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^{s-\\epsilon}_{p,\\infty})}\n+\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^{s+\\epsilon}_{p,\\infty})}}\n{\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^s_{p,\\infty})}}\\bigg).\n\\end{eqnarray}\n\n\\end{Proposition}\n\nNow, let us prove the uniqueness of the solution of (\\ref{3.1}).\nLet $(u_1, h_1)$, $(u_2, h_2)$ $\\in\\big(L^\\infty_T(B^0)\\cap\nL^1_T(B^2))\\times L^\\infty_T(\\widetilde{B}^{0,1})$\nbe two solutions of (\\ref{3.1}) with the same initial data.\nThe difference $\\vartheta\\triangleq h_2-h_1$, $w\\triangleq u_2-u_1$ satisfies the following system:\n\\begin{align}\\label{4.4}\n\\left\\{\n\\begin{aligned}\n&\\partial_t\\vartheta+u_2\\cdot\\nabla\\vartheta=\n-\\mbox{div} w-w\\nabla h_1-\\vartheta\\mbox{div} u_2-h_1\\mbox{div} w,\\\\\n&\\partial_tw-\\nu\\widetilde{\\Delta} w=-\\nabla\\vartheta-u_2\\cdot\\nabla w-w\\cdot\\nabla u_1\n+\\nu(1+h_1)\\nabla\\Big(\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} w\\\\ &\\qquad\\qquad\\qquad+\\nu(1+h_1)\n\\nabla\\Big(\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} u_2\n+\\nu\\vartheta\\nabla\\Big(\\frac{h_2}{1+h_2}\\Big)\\widetilde{\\nabla}u_2,\\\\\n&\\vartheta(0,x)=0,\\quad w(0,x)=0.\n\\end{aligned}\\right.\n\\end{align}\nWithout loss of generality, we assume that there holds for sufficiently small $T$\n\\begin{align}\\label{4.5}\n&1+h_1\\ge \\f12, \\\\\\label{4.6}\n&\\|h_1\\|_{\\widetilde{E}^{0,1}_T}\\le \\varepsilon,\n\\end{align}\nwhere $\\varepsilon>0$ is small enough. Applying the Proposition \\ref{Prop4.1} to the first\nequation of (\\ref{4.4}) yields\n\\begin{align}\\label{4.7}\n\\|\\vartheta(t)\\|_{\\dot{B}^0_{2,\\infty}}\\lesssim\n\\int_0^t\\!e^{C(V_2(t)-V_2(\\tau))}\\|w\\cdot\\nabla\nh_1+\\vartheta\\mbox{div} u_2\n+h_1\\mbox{div} w+\\mbox{div} w\\|_{\\dot{B}^0_{2,\\infty}}d\\tau,\n\\end{align} with $V_2(t)\\triangleq\\int_0^t\\|\\nabla\nu_2\\|_{\\dot{B}^1_{2,\\infty}\\cap L^\\infty}d\\tau$. It follows from\n(\\ref{2.5}) with $s=0$  that\n\\begin{align} \\|w\\cdot\\nabla\nh_1\\|_{\\dot{B}^0_{2,\\infty}}&\\lesssim\n\\|\\nabla h_1\\|_{\\dot{B}^0_{2,\\infty}}\\|w\\|_{B^1}\n\\lesssim\\|w\\|_{{B}^1}\\|h_1\\|_{{B}^1},\\nonumber\\\\\n\\|\\vartheta\\mbox{div} u_2\\|_{\\dot{B}^0_{2,\\infty}}&\\lesssim\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}\n\\|u_2\\|_{{B}^2},\\nonumber\\\\\n\\|h_1\\mbox{div} w\\|_{\\dot{B}^0_{2,\\infty}}&\\lesssim\n\\|\\mbox{div} w\\|_{\\dot{B}^0_{2,\\infty}}\n\\|h_1\\|_{{B}^1}\\lesssim\\|w\\|_{{B}^1}\n\\|h_1\\|_{{B}^1},\\nonumber\n\\end{align}\nwhere we have used $B^1\\hookrightarrow\\dot{B}^1_{2,\\infty}$.\nPlugging the above estimates into (\\ref{4.7}), we get\n\\begin{align}\\label{4.8}\n\\|\\vartheta(t)\\|_{\\dot{B}^0_{2,\\infty}}\\lesssim\n\\int_0^t\\!e^{C(V_2(t)-V_2(\\tau))}\\Bigl[\\|w\\|_{B^1}\n(1+\\|h_1\\|_{B^1})+\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}\n\\|u_2\\|_{B^2}\\Bigr]d\\tau.\n\\end{align}\nRecall that $u^i\\in L^1_T(B^2)$, we can take a $T\\in (0,\\infty)$  small enough so\nthat\n$$C\\|u_2\\|_{L^1_T(B^2)}\\le \\f1 4,$$\nwhich together with (\\ref{4.8}) implies that for $t\\le T$\n\\begin{align}\\label{4.9}\n\\|\\vartheta\\|_{L^\\infty_t(\\dot{B}^0_{2,\\infty})}\\lesssim\n\\|w\\|_{L^1_t(B^1)}(1+\\|h_1\\|_{L^\\infty_t(B^1)}).\n\\end{align}\nApplying (\\ref{4.3}) to the term\n$\\|w\\|_{L^1_t(B^1)}$ yields\n\\begin{align}\\label{4.10}\n\\|\\vartheta\\|_{L^\\infty_t(\\dot{B}^0_{2,\\infty})}\\lesssim&\n\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}\n\\log\\bigg(e+\\frac{\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^0_{2,\\infty})}\n+\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^2_{2,\\infty})}}\n{\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}}\\bigg)(1+\\|h_1\\|_{L^\\infty_t(B^1)}).\n\\end{align}\nThanks to $B^s\\hookrightarrow\\dot{B}^s_{2,\\infty}$\nand $\\widetilde{B}^{0,1}\\hookrightarrow B^1$, we have\n\\begin{align}\\label{4.11}\n\\|\\vartheta\\|_{L^\\infty_t(\\dot{B}^0_{2,\\infty})}\\lesssim\n\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}\n\\log\\bigg(e+\\frac{W(t)}\n{\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}}\\bigg)\n\\end{align}\nwith $$W(t)\\triangleq\\|u^i\\|_{\\widetilde{L}^1_t(B^0)}\n+\\|u^i\\|_{\\widetilde{L}^1_t(B^2)},$$ and for finite $t$, $W(t)< +\\infty$.\n\nNext, we deal with the second equation of (\\ref{4.4}). We get by\napplying (\\ref{2.7}) with $s=1$, $s=0$ respectively that\n\\begin{align}\\label{4.12}\n\\|u_2\\cdot\\nabla w\\|_{\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})}\\lesssim\n\\|u_2\\|_{L^2_t(B^1)}\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})},\\\\\\label{5.12}\n\\|w\\cdot\\nabla u_1\\|_{\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})}\\lesssim\n\\|u_1\\|_{L^2_t(B^1)}\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}.\n\\end{align}\nWe can deduce $h_i\\in C(0,T; \\mathop{\\bf R\\kern 0pt}\\nolimits^2)$ $(i=1,2)$ from the fact $B^1\\hookrightarrow C$.\nMoreover, due to (\\ref{4.5}), we can assume $h_1(t,x)+1\\ge \\delta$\nfor all $t\\le T$, $x\\in \\mathop{\\bf R\\kern 0pt}\\nolimits^2$. Since $h_1$, $h_2$ have the same initial data,\nfrom the continuity of $h_2$, there exists a $\\widetilde{T}\\le T$ such that\n$$h_2(x,t)+1\\ge \\delta,\\quad\\hbox{for all}\\quad t\\in[0, \\widetilde{T}],\\quad x\\in\\mathop{\\bf R\\kern 0pt}\\nolimits^2.$$\nIt follows from (\\ref{2.6}) with $s=1$, (\\ref{2.13}) and\n$B^1\\hookrightarrow\\dot{B}^1_{2,\\infty}\\cap L^\\infty$ that\n\\begin{align}\n&\\Big\\|(1+h_1)\\nabla\\Big(\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} u_2\\Big\\|\n_{\\dot{B}^{-1}_{2,\\infty}}\\nonumber\\\\&\\qquad\\lesssim\n\\Big\\|(1+h_1)\\nabla\\Big(\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big)\\Big\\|\n_{\\dot{B}^{-1}_{2,\\infty}}\\|\\widetilde{\\nabla} u_2\\|\n_{B^1}\\nonumber\\\\&\\qquad\\lesssim (1+\\|h_1\\|_{B^1})\n\\Big\\|\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big\\|\n_{\\dot{B}^0_{2,\\infty}}\\|u_2\\|_{B^2}\\nonumber\\\\&\\qquad\\lesssim (1+\\|h_1\\|_{B^1})\n(\\|h_1\\|_{B^1}+\\|h_2\\|_{B^1})\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}\n\\|u_2\\|_{B^2},\\nonumber\n\\end{align}\nwhich together with\n $L^1_t(\\dot{B}^{-1}_{2,\\infty})\\subset\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})$\nyields\\begin{align}\\label{4.14}\n\\Big\\|&(1+h_1)\\nabla\\Big(\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} u_2\\Big\\|\n_{\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})}\\nonumber\\\\&\n\\lesssim\\int_0^t(1+\\|h_1\\|_{B^1})(\\|h_1\\|_{B^1}+\\|h_2\\|_{B^1})\n\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}\\|u_2\\|_{B^2}d\\tau.\n\\end{align}\nThanks to (\\ref{2.6}),\n(\\ref{2.12}),   and\n$L^1_t(\\dot{B}^{-1}_{2,\\infty})\\subset\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})$, we get\n\\begin{align}\\label{4.15}\n&\\Big\\|\\vartheta\\nabla \\Big(\\frac{h_2}{1+h_2}\\Big)\\widetilde{\\nabla} u_2\\Big\\|\n_{\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})}\n\\lesssim\\int_0^t\\|\\vartheta\\|_{\\dot{B}^{0}_{2,\\infty}}\\|h_2\\|_{B^1}\\|u_2\\|_{B^2}d\\tau.\\end{align}\nThanks to Lemma 5.3 with $s_1=s_2=0$,  Lemma 5.4 with $s=1$, and (\\ref{2.5})  with $s=0$, we have\n\\begin{align} \\label{4.17}\n\\sup_{k\\in{\\mathbf Z}}&\\, \\omega_k(t)2^{-k}\\Big\\|\\Delta_k\\Big((1+h_1)\\nabla\\Big(\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} w\\Big)\\Big\\|_{L^1_t(L^2)}\n\\nonumber\\\\&\\lesssim\n\\Big\\|\\nabla\\Big(\\frac{h_1}{1+h_1}\\Big)\\Big\\|_{E^{0}_t}\n\\|(1+h_1)\\widetilde{\\nabla} w\\|_{\\widetilde{L}^1_t(\\dot{B}^0_{2,\\infty})}\\nonumber\\\\&\\lesssim\n\\Big\\|\\frac{h_1}{1+h_1}\\Big\\|_{E^{1}_t}(1+\\|h_1\\|_{L^\\infty_t(B^1)})\n\\|\\nabla w\\|_{\\widetilde{L}^1_t(\\dot{B}^0_{2,\\infty})}\n\\nonumber\\\\&\\lesssim\\|h_1\\|_{\\widetilde{E}^{0,1}_t}\n(1+\\|h_1\\|_{L^\\infty_t(\\widetilde{B}^{0,1})})^4\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}.\n\\end{align}\nIn terms of  Proposition 4.3, (\\ref{4.12})-(\\ref{4.17}) and\n$\\widetilde{B}^{0,1}\\hookrightarrow B^1$, we finally obtain\n\\begin{align}\\label{4.18}\n\\|w\\|&_{\\widetilde{L}^1_t(\\dot{B}^{1}_{2,\\infty})}+\n\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}\\nonumber\\\\ \\lesssim &\n\\|u_2\\|_{L^2_t(B^1)}\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}\n+\\|u_1\\|_{L^2_t(B^1)}\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}\\nonumber\\\\&+\\|h_1\\|_{\\widetilde{E}^{0,1}_t}\n\\big(1+\\|h_1\\|_{L^\\infty_t(\\widetilde{B}^{0,1})}\\big)^4\n\\|w\\|_{\\widetilde{L}^1_t(B^1_{2,\\infty})}\\nonumber\\\\\n&+\\int_0^t(1+\\|h_1\\|_{\\widetilde{B}^{0,1}})(1+\\|h_1\\|_{\\widetilde{B}^{0,1}}\n+\\|h_2\\|_{\\widetilde{B}^{0,1}})(1+\\|u_2\\|_{B^2})\n\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}d\\tau.\n\\end{align}\nLet us define $$Z(t)\\triangleq \\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^{1}_{2,\\infty})}\n+\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}.$$\nDue to (\\ref{4.6}), if $T$ is chosen small enough,\nthen the first four terms of the right side of (\\ref{4.18}) can be absorbed by the left\nside $Z(t)$. Noting that $r\\log(e+\\frac{W(T)}{r})$ is increasing, from\n(\\ref{4.11}) and (\\ref{4.18}), it follows that\n\\begin{align}\\label{4.19}\nZ(t)&\\lesssim\\int_0^t(1+W'(\\tau))\nZ(\\tau)\\log\\Big(e+\\frac{W(\\tau)}{Z(\\tau)}\\Big)d\\tau\\nonumber\\\\&\n\\lesssim \\int_0^t(1+W'(\\tau))\nZ(\\tau)\\log\\Big(e+\\frac{W(T)}{Z(\\tau)}\\Big)d\\tau.\n\\end{align}\nIt is easy to verify that\n$$1+W'(\\tau)\\in L^1_{loc}(\\mathop{\\bf R\\kern 0pt}\\nolimits^+)\\quad\\hbox{and}\n\\quad\\int_0^1\\frac{dr}{r\\log(e+\\frac{W(T)}{r})}=+\\infty.$$\nHence by Osgood Lemma, we have $Z\\equiv0$ on $[0,\\widetilde{T}]$, i.e. $w\\equiv0$, then from\n(\\ref{4.9}),\n$\\vartheta=h_2-h_1\\equiv0$. Then a standard\ncontinuous argument gives the uniqueness.\n\n\n\\setcounter{equation}{0}\n\\section{Appendix}\n\nIn this appendix, we prove some  multilinear estimates in the weighted Besov space.\n\n\\begin{Lemma}\\label{Lem5.1}\nLet $A$ be a homogeneous smooth function of degree $m$. Assume that  $-\\frac d2<\\rho\\le\\frac d2$. Then there hold\n\\begin{eqnarray}\n&&\\Big|(A(D)\\Delta_k(v\\cdot\\nabla h),A(D)\\Delta_kh)\\Big|\\le C\\|{\\cal\nF}_k^m(t)\\|_2\\|A(D)\\Delta_kh\\|_2,\\label{5.1a}\\\\\n&&\\Big|(A(D)\\Delta_k(v\\cdot\\nabla u),A(D)\\Delta_ku)\\Big|\\le C\\|{\\cal\n\\widetilde{F}}_k^m(t)\\|_2\\|A(D)\\Delta_ku\\|_2,\\label{5.2a}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n&&\\Big|(A(D)\\Delta_k(v\\cdot\\nabla h),\\Delta_k u)+(\\Delta_k(v\\cdot\\nabla\nu),A(D)\\Delta_kh)\\Big|\\nonumber\\\\\n&&\\qquad\\qquad\\le C\\big(\\|{\\cal {F}}_k^m(t)\\|_2+\\|{\\cal \\widetilde{F}}_k^0(t)\\|_2\\big)\\big(\\|\\Delta_k\nu\\|_2+\\|A(D) \\Delta_kh\\|_2\\big),\\label{5.3a}\n\\end{eqnarray}\nwhere ${\\cal F}_k^m(t)$ and ${\\cal \\widetilde{F}}_k^m(t)$ satisfy\n\\begin{eqnarray}\\label{5.1}\n&&\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(\\rho-m)}\\|{\\cal\nF}_k^m(t)\\|_{L^1_T(L^2)}\\le C\\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\frac{d}2+1})},\\\\\n&&\\sum_{k\\in{\\mathbf Z}}2^{k(\\rho-m)}\\|{\\cal \\widetilde{F}}_k^m(t)\\|_{L^1_T(L^2)}\n\\le C\\|u\\|_{L^2_T(B^{\\rho+1})}\\|v\\|_{L^2_T(B^{\\frac{d}2})}.\\label{5.2}\n\\end{eqnarray}\n\\end{Lemma}\n\n\\noindent{\\it Proof.}\\quad Let us first prove (\\ref{5.1a}).\nUsing the Bony's paraproduct decomposition, we write\n\\begin{align}\\label{5.3}\n\\big(A(D)\\Delta_k(v\\cdot\\nabla h), A(D)\\Delta_kh\\big)&=\\big(A(D)\\Delta_k(T'_{\\pa_j h}v^j),  A(D)\\Delta_kh\\big)+J_k,\n\\end{align}\nwhere\n\\begin{eqnarray*}\n&&T'_fg=T_fg+R(f,g),\\quad \\textrm{and} \\\\\n&&J_k=\\sum_{|k'-k|\\le3}([A(D)\\Delta_k, S_{k'-1}v^j]\\Delta_{k'}\\pa_jh,  A(D)\\Delta_kh)\\\\\n&&\\qquad+\\sum_{|k'-k|\\le3}((S_{k'-1}-S_{k-1})v^jA(D)\\Delta_{k}\\Delta_{k'}\\pa_jh,  A(D)\\Delta_kh)\\\\\n&&\\qquad+(S_{k-1}v^jA(D)\\Delta_{k}\\pa_jh,  A(D)\\Delta_kh)\n\\end{eqnarray*}\nWe get by integration by parts that\n$$(S_{k-1}v^jA(D)\\Delta_{k}\\pa_jh,  A(D)\\Delta_kh)=-\\frac12\\big(S_{k-1}\\mbox{div}\\, vA(D)\\Delta_{k}h,  A(D)\\Delta_kh\\big).$$\nLet us set\n\\begin{eqnarray*}\n&&{\\cal F}_{k,0}^m(t)=A(D)\\Delta_k(T'_{\\pa_j h}v^j),\\\\\n&&{\\cal F}_{k,1}^m(t)=\\sum_{|k'-k|\\le3}[A(D)\\Delta_k, S_{k'-1}v^j]\\Delta_{k'}\\pa_jh,\\\\\n&&{\\cal F}_{k,2}^m(t)=\\sum_{|k'-k|\\le3}(S_{k'-1}-S_{k-1})v^jA(D)\\Delta_{k}\\Delta_{k'}\\pa_jh,\\\\\n&&{\\cal F}_{k,4}^m(t)=-\\frac12 S_{k-1}\\mbox{div}\\, vA(D)\\Delta_{k}h.\n\\end{eqnarray*}\nBy the Cauchy-Schwartz inequality, we get\n\\begin{eqnarray*}\n\\big(A(D)\\Delta_k(v\\cdot\\nabla h), A(D)\\Delta_kh\\big)\\le \\|{\\cal\nF}_k^m(t)\\|_2\\|A(D)\\Delta_kh\\|_2,\n\\end{eqnarray*}\nwith ${\\cal F}_k^m(t)=\\displaystyle\\sum_{i=0}^3{\\cal F}_{k,i}^m(t).$ So,\nit remains to prove that ${\\cal F}_k^m(t)$ satisfies (\\ref{5.1}). For the simplicity, we set\n$$\\widetilde{\\Delta}_{k}=\\displaystyle\\sum_{|k'-k|\\le 1}\\Delta_{k'},\\quad\n\\widetilde{\\widetilde{\\Delta}}_{k}=\\displaystyle\\sum_{|k'-k|\\le 3}\\Delta_{k'}.$$\nThanks to the definition of ${\\cal F}_{k,0}^m(t)$ and Lemma \\ref{Lem2.1}, we have\n\\begin{align}\n\\|{\\cal F}_{k,0}^m(t)\\|_{L^1_T(L^2)}&\\le \\sum_{|k'-k|\\le3}2^{km}\\|S_{k'-1}\\pa_jh\\|_{L^\\infty_T(L^\\infty)}\\|\\Delta_{k'}v^j\\|_{L^1_T(L^2)}\\nonumber\\\\\n&\\quad+\\sum_{k'\\ge k-2}2^{(m+\\frac d2)k}\\|\\Delta_k(\\Delta_{k'}\\pa_jh\\widetilde{\\Delta}_{k'}v^j)\\|_{L^1_T(L^1)}\\nonumber\\\\\n&\\triangleq I+II.\\nonumber\n\\end{align}\nThanks to Lemma \\ref{Lem2.1}, we have\n\\begin{align}\n2^{k(\\rho-m)}I &\\lesssim 2^{k\\rho}\n\\sum_{k'\\le k+1}2^{k'(1+\\f d2)}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|\\widetilde{\\widetilde{\\Delta}}_{k}v\\|_{L^1_T(L^2)}\\nonumber\\\\\n&\\lesssim\\sum_{k'\\le k+1}2^{(k'-k)(1+\\f d2-\\rho)}2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}2^{k(1+\\f d2)}\\|\\widetilde{\\widetilde{\\Delta}}_{k}v\\|_{L^1_T(L^2)},\\nonumber\n\\end{align}\nfrom which and the definition of $\\omega_k(T)$, it follows that\n\\begin{eqnarray}\\label{5.4}\n&&\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)2^{k(\\rho-m)}I\\nonumber\\\\\n&&\\lesssim \\sum_{k'\\in {\\mathbf Z}}2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\sum_{k\\ge k'-1}\\omega_k(T)2^{(k'-k)(1+\\f d2-\\rho)}2^{k(1+\\f d2)}\\|\\widetilde{\\widetilde{\\Delta}}_{k}v\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\lesssim \\sum_{k'\\in {\\mathbf Z}}\\omega_{k'}(T)2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\sum_{k\\ge k'-1}2^{(k'-k)(\\f d2-\\rho)}2^{k(1+\\f d2)}\\|\\widetilde{\\widetilde{\\Delta}}_{k}v\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\lesssim \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}, \\end{eqnarray} where we\nused the assumption $\\rho\\le \\f d 2$ in the last inequality. Set\n$e_k(T)=e^1_{k}(T)+e^2_{k}(T)$. Using Lemma \\ref{Lem2.1}, we also have\n\\begin{align}\n\\omega_k(T)2^{k(\\rho-m)}II&\\lesssim \\omega_k(T)2^{k(\\rho+\\f d2)}\n\\sum_{k'\\ge k-2}2^{k'}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|\\widetilde{\\Delta}_{k'}v\\|_{L^1_T(L^2)}\\nonumber\\\\\n&\\lesssim 2^{k(\\rho+\\f d2)} \\sum_{k'\\ge\nk-2}2^{k'}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|\\widetilde{\\Delta}_{k'}v\\|_{L^1_T(L^2)}\n\\sum_{k'\\ge \\widetilde{k}\\ge k}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T)\\nonumber\\\\\n&\\quad+2^{k(\\rho+\\f d2)} \\sum_{k'\\ge\nk-2}2^{k'}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|\\widetilde{\\Delta}_{k'}v\\|_{L^1_T(L^2)}\n\\sum_{\\widetilde{k}\\ge k, \\widetilde{k}\\ge k'}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T)\\nonumber\\\\\n&\\triangleq II_1+II_2.\\nonumber\n\\end{align}\nNote that for $\\widetilde{k}\\le k'$\n$$\ne_{\\widetilde{k}}(T)\\le e_{k'}(T)\\le \\omega_{k'}(T),\n$$\nfrom which and $\\rho>-\\f d 2$, we deduce that\n\\begin{align}\\label{5.5}\n\\sum_{k\\in Z}II_1&\\lesssim \\sum_{k'\\in Z}\\omega_{k'}(T)2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}2^{k'(\\f d2+1)}\n\\|\\widetilde{\\Delta}_{k'}v\\|_{L^1_T(L^2)}\n\\sum_{k\\le k'+2}2^{(k-k')(\\rho+\\f d2)}\\nonumber\\\\\n&\\lesssim \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}.\n\\end{align}\nSimilarly, we can obtain\n\\begin{align}\\label{5.6}\n\\sum_{k\\in {\\mathbf Z}}II_2&\\lesssim \\sum_{k'\\in {\\mathbf Z}}2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\sum_{k\\le k'+2}2^{(k-k')(\\f d2+\\rho)}\n\\sum_{\\widetilde{k}\\ge k'}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T)\\|v\\|_{L^1_T(B^{\\f d2+1})}\\nonumber\\\\\n&\\lesssim \\sum_{k'\\in {\\mathbf Z}}\\omega_{k'}(T)2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\sum_{k\\le k'+2}2^{(k-k')(\\f d2+\\rho+1)}\n\\|v\\|_{L^1_T(B^{\\f d2+1})}\\nonumber\\\\\n&\\lesssim  \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}.\n\\end{align}\n\nBy summing up (\\ref{5.4})-(\\ref{5.6}), we obtain\n\\begin{eqnarray}\\label{5.7}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(\\rho-m)}\\|{\\cal F}_{k,0}^m(t)\\|_{L^1_T(L^2)}\\lesssim \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}.\n\\end{eqnarray}\n\nNote that $A(D)\\Delta_k=2^{km}\\widetilde{\\varphi}(2^{-k}D)$ with $\\widetilde{\\varphi}(\\xi)=A(\\xi)\\varphi(\\xi)$.\nSet $\\tilde\\theta={\\cal F}^{-1}\\tilde\\varphi$, we get by using the Taylor's formula that\n\\begin{align}\\label{5.8}\n{\\cal F}_{k,1}^m(t)=\\sum_{|k'-k|\\le3}2^{k(m-1)}\\int_{\\mathop{\\bf R\\kern 0pt}\\nolimits^d}\\int_0^1\\tilde\\theta(y)\n(y\\cdot S_{k'-1}\\nabla v^j(x-2^{-k}\\tau y))\\Delta_{k'}\\pa_jh(x-2^{-k}y) d\\tau dy,\\nonumber\n\\end{align}\nfrom which and Lemma \\ref{Lem2.1}, it follows that\n\\begin{align}\n\\|{\\cal F}_{k,1}^m(t)\\|_{L^1_T(L^2)}&\\lesssim2^{k(m-1)}\\sum_{|k'-k|\\le3}\n\\|S_{k'-1}\\nabla v^j\\|_{L^1_T(L^\\infty)}\\|\\Delta_{k'}\\pa_jh\\|_{L^\\infty_T(L^2)}\\nonumber\\\\&\n\\lesssim2^{km}\\sum_{|k'-k|\\le3}\n\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|v\\|_{L^1_T(B^{\\frac d 2 +1})},\\nonumber\n\\end{align}\nthus, we get\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(\\rho-m)}\\|{\\cal F}_{k,1}^m(t)\\|_{L^1_T(L^2)}\n\\lesssim  \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}.\n\\end{align}\nThanks to  the fact $|k'-k|\\le3$ and Lemma \\ref{Lem2.1}, we have\n\\begin{align}\n\\|(S_{k'-1}-S_{k-1})v^jA(D)\\Delta_{k}\\Delta_{k'}\\pa_jh\\|_{L^1_T(L^2)}\n\\lesssim 2^{km}\\|\\Delta_{k}h\\|_{L^\\infty_T(L^2)}\\|v\\|_{L^1_T(B^{\\f d2+1})},\\nonumber\n\\end{align}\nfrom which, it follows that\n\\begin{align}\\label{5.9}\n\\sum_{k\\in{\\mathbf Z}}&\\omega_k(T)2^{k(\\rho-m)}\\big(\\|{\\cal F}_{k,2}^m(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal F}_{k,3}^m(t)\\|_{L^1_T(L^2)}\\big)\\lesssim\\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})},\n\\end{align}\nwhich together with (\\ref{5.7}) and (\\ref{5.8}) yields (\\ref{5.1}).\n\nUsing the decomposition (\\ref{5.3}) with $h$ instead of $u$ and Lemma \\ref{Lem2.1}, (\\ref{5.2a}) can be easily proved. We omit it here.\nIn order to prove (\\ref{5.3a}), we use the decomposition\n\\begin{align}\n\\big(A(D)\\Delta_k(v\\cdot\\nabla h),\\Delta_k u\\big)+\\big(\\Delta_k(v\\cdot\\nabla u),A(D)\\Delta_kh\\big)=I_k+J_k,\\nonumber\n\\end{align}\nwith\n\\begin{align}\nI_k=&\\big(A(D)\\Delta_k(T'_{\\pa_j h}v^j),\\Delta_k u\\big)+\\big(\\Delta_k(T'_{\\pa_j u}v^j),A(D)\\Delta_kh\\big)\\nonumber\\\\\n\\triangleq&\\big({\\cal F}_{k,0}^m(t),\\Delta_k u\\big)+\\big({\\cal \\widetilde{F}}_{k,0}^0(t),A(D)\\Delta_kh\\big)\\nonumber\\\\\nJ_k=&\\sum_{|k'-k|\\le3}\\Big([A(D)\\Delta_k, S_{k'-1}v^j]\\Delta_{k'}\\pa_jh,  \\Delta_ku\\Big)+\n\\Big((S_{k'-1}-S_{k-1})v^jA(D)\\Delta_{k}\\Delta_{k'}\\pa_jh,  \\Delta_ku\\Big)\\nonumber\\\\\n&+\\sum_{|k'-k|\\le3}\\Big([\\Delta_k, S_{k'-1}v^j]\\Delta_{k'}\\pa_ju,  A(D)\\Delta_kh\\Big)+\n\\Big((S_{k'-1}-S_{k-1})v^j\\Delta_{k}\\Delta_{k'}\\pa_ju,  A(D)\\Delta_kh\\Big)\\nonumber\\\\\n&-\\Big(S_{k-1}\\mbox{div} vA(D)\\Delta_{k}h,  \\Delta_ku\\Big)\\nonumber\\\\\n\\triangleq&\\big({\\cal F}_{k,1}^m(t),\\Delta_k u\\big)+\\big({\\cal F}_{k,2}^m(t),\\Delta_k u\\big)+\n\\big({\\cal \\widetilde{F}}_{k,1}^0(t),A(D)\\Delta_kh\\big)\n+\\big({\\cal\\widetilde{ F}}_{k,2}^0,A(D)\\Delta_kh\\big)+\\big({\\cal F}_{k,3}^m(t), \\Delta_ku\\big),\n\\nonumber\\end{align}\nfrom which, a similar proof of (\\ref{5.1}) gives (\\ref{5.3a}). This completes the proof of Lemma \\ref{Lem5.1}. \\hfill $ \\blacksquare $ \\vskip 3mm\n\n\n\\begin{Lemma}\\label{Lem5.2}\nLet  $s_1\\le\\frac d2-1$, $s_2\\le\\frac d2$, and $s_1+s_2>0$. Then there holds\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(s_1+s_2-\\frac d2)}\\|\\Delta_k(fg)\\|_{L^1_T(L^2)}\n\\le C\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks_1}\\|\\Delta_kf\\|_{L^{r_1}_T(L^2)}\\|g\\|_{L^{r_2}_T(B^{s_2})},\\label{5.10}\n\\end{align}\nwhere $1\\le r_1,r_2\\le \\infty$ and $\\f 1 {r_1}+\\f 1 {r_2}=1$.\n\\end{Lemma}\n\n\\noindent{\\it Proof.}\\quad Using the Bony's paraproduct decomposition, we write\n\\begin{align}\n\\Delta_k(fg)&=\\sum_{|k'-k|\\le3}\\Delta_k(S_{k'-1}f\\Delta_{k'}g)+\n\\sum_{|k'-k|\\le3}\\Delta_k(S_{k'-1}g\\Delta_{k'}f)\\nonumber\\\\\n&\\quad+\\sum_{k'\\ge k-2}\\Delta_k(\\Delta_{k'}f\\widetilde{\\Delta}_{k'}g)\\triangleq I+II+III.\\nonumber\n\\end{align}\nA similar proof of (\\ref{5.4}) ensures that for $s_1\\le\\frac d2-1$\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(s_1+s_2-\\frac d2)}\\|I\\|_{L^1_T(L^2)}\\lesssim\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks_1}\\|\\Delta_kf\\|_{L^{r_1}_T(L^2)}\\|g\\|_{L^{r_2}_T(B^{s_2})},\\nonumber\n\\end{align}\nwhile $II$ can be directly deduced for $s_2\\le\\frac d2$. On the other hand,\na similar proof of (\\ref{5.5}) and (\\ref{5.6}) gives for $s_1+s_2>0$\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(s_1+s_2-\\frac d2)}\\|III\\|_{L^1_T(L^2)}\\lesssim \\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks_1}\\|\\Delta_kf\\|_{L^{r_1}_T(L^2)}\\|g\\|_{L^{r_2}_T(B^{s_2})}.\\nonumber\n\\end{align}\nThis completes the proof of Lemma \\ref{Lem5.2}. \\hfill $ \\blacksquare $ \\vskip 3mm\n\nSimilarly, we can also prove the following lemma.\n\n\\begin{Lemma}\\label{Lem5.3}\nLet $s_1\\le\\frac d2-1$, $s_2<\\frac d2$ and $s_1+s_2\\ge0$. Then there holds\n\\begin{align}\n\\sup_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(s_1+s_2-\\frac d2)}\\|\\Delta_k(fg)\\|_{L^1_T(L^2)}\n\\le C\\|f\\|_{E^{s_1}_T}\\|g\\|_{\\widetilde{L}^1_T(\\dot B^{s_2}_{2,\\infty})}.\\label{5.11}\n\\end{align}\n\\end{Lemma}\n\n\\begin{Lemma}\\label{Lem5.4}\nLet $s>0$. Assume that $F\\in W^{[s]+3,\\infty}_{loc}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ with  $F(0)=0$. Then\nthere holds\n\\begin{eqnarray}\n\\|F(f)\\|_{E^s_T}\\le C(1+\\|f\\|_{L^\\infty_T(L^\\infty)})^{[s]+2}\\|f\\|_{E^s_T}.\\label{5.12}\n\\end{eqnarray}\n\\end{Lemma}\n\n\\noindent{\\it Proof.}\\quad We decompose $F(f)$ as\n\\begin{align}\nF(f)=\\sum_{k'\\in{\\mathbf Z}}F(S_{k'+1}f)-F(S_{k'}f)&=\\sum_{k'\\in{\\mathbf Z}}\\Delta_{k'}f\\int_0^1F'(S_{k'}f+\\tau\\Delta_{k'}f)d\\tau\\nonumber\\\\\n&\\triangleq\\sum_{k'\\in{\\mathbf Z}}\\Delta_{k'}f\\, m_{k'},\\nonumber\n\\end{align}\nwhere $m_{k'}=\\int_0^1F'(S_{k'}f+\\tau\\Delta_{k'}f)d\\tau$. Furthermore, we write\n\\begin{align}\n\\Delta_kF(f)=\\sum_{k'<k}\\Delta_k(\\Delta_{k'}f\\,m_{k'})+\\sum_{k'\\ge k}\\Delta_k(\\Delta_{k'}f\\, m_{k'})\\triangleq I+II.\\nonumber\n\\end{align}\nBy Lemma \\ref{Lem2.1}, we have\n\\begin{align}\\label{5.13}\n\\|I\\|_{L^\\infty_T(L^2)}&\\le\\sum_{k'<k}\\|\\Delta_k(\\Delta_{k'}f\\,m_{k'})\\|_{L^\\infty_T(L^2)}\\nonumber\\\\\n&\\le \\sum_{k'<k}2^{-k|\\al|}\\sup_{|\\gamma|=|\\al|}\\|D^\\gamma\\Delta_k(\\Delta_{k'}f\\,m_{k'})\\|_{L^\\infty_T(L^2)},\n\\end{align}\nwith $\\al$ to be determined later. Note that for $|\\gamma|\\ge0$, we have\n$$\\|D^\\gamma m_{k'}\\|_\\infty\\lesssim2^{k'|\\gamma|}(1+\\|f\\|_\\infty)^{|\\gamma|}\\|F'\\|_{W^{|\\gamma|,\\infty}},$$\nfrom which and (\\ref{5.13}), it follows that\n\\begin{align}\n2^{ks}\\|I\\|_{L^\\infty_T(L^2)}\\lesssim2^{k(s-|\\al|)}\\sum_{k'<k}2^{k'|\\al|}\n\\|\\Delta_{k'}f\\|_{L^\\infty_T(L^2)}(1+\\|f\\|_{L^\\infty_T(L^\\infty)})^{|\\al|}\\|F'\\|_{W^{|\\al|,\\infty}},\\nonumber\n\\end{align}\nthus, if we take $|\\al|=[s]+2$, we get\n\\begin{align}\n&\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks}\\|I\\|_{L^\\infty_T(L^2)}\\nonumber\\\\\n&\\quad\\lesssim\\sum_{k'\\in{\\mathbf Z}}2^{k's}\\omega_{k'}(T)\\|\\Delta_{k'}f\\|_{L^\\infty_T(L^2)}\n\\sum_{k>k'}2^{(k-k')(s-|\\al|+1)}(1+\\|f\\|_{L^\\infty_T(L^\\infty)})^{|\\al|}\\|F'\\|_{W^{|\\al|,\\infty}}\n\\nonumber\\\\&\\quad \\lesssim(1+\\|f\\|_{L^\\infty_T(L^\\infty)})^{[s]+2}\\|F'\\|_{W^{[s]+2,\\infty}}\\|f\\|_{E^s_T}.\\label{5.14}\n\\end{align}\n\nNext, let us turn to the proof of $II$. We get by using Lemma 2.1 that\n\\begin{align}\n\\|II\\|_{L^\\infty_T(L^2)}&\\lesssim\\sum_{k\\ge k'}\\|\\Delta_{k'}f\\|_{L^\\infty_T(L^2)}.\\nonumber\n\\end{align}\nThen we write\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks}\\|II\\|_{L^\\infty_T(L^2)}\n&\\lesssim \\sum_{k\\in {\\mathbf Z}}2^{ks}\\sum_{k'\\ge k}\\|\\Delta_{k'}f\\|_{L^\\infty_T(L^2)}\n\\sum_{k'\\ge \\widetilde{k}\\ge k}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T)\\nonumber\\\\\n&\\quad+\\sum_{k\\in {\\mathbf Z}}2^{ks}\\sum_{k'\\ge k}\\|\\Delta_{k'}f\\|_{L^\\infty_T(L^2)}\n\\sum_{\\widetilde{k}\\ge k', \\widetilde{k}\\ge k}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T),\\nonumber\n\\end{align}\nfrom which, a similar proof of (\\ref{5.5}) and (\\ref{5.6}) ensures that\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks}\\|II\\|_{L^\\infty_T(L^2)}\\lesssim\\|f\\|_{E^s_T}.\\label{5.15}\n\\end{align}\n\nBy summing up (\\ref{5.14}) and (\\ref{5.15}), we deduce the inequality (\\ref{5.12}).\nThis completes the proof of Lemma \\ref{Lem5.4}. \\hfill $ \\blacksquare $ \\vskip 3mm\n\n\n\\bigskip\n\n\\noindent {\\bf Acknowledgments.}  The authors thank the referees\n for their invaluable comments and suggestions which helped improve the paper greatly.\n  Q. Chen and C.\nMiao  were partially  supported by the NSF of China (No.10571016).\nZ. Zhang is supported by the NSF of China(No.10601002).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}}\n\n\\IEEEPARstart{T}{here} has been a fundamental paradigm shift in the landscape of how we build software over the last 2 decades.\nThe compute stack was originally envisaged with the assumption that a single program would run on a single machine.\nIn this traditional model, system software abstractions subsumed the management of processor, memory, disk and physically connected peripherals within the context of the machine.\nHowever, this landscape rapidly changed with the evolution toward software being served on the backbone provided by the internet.\nNow, an `application' is realized through the cooperation of multiple distinct sub-applications (services) running collaboratively.\nFor example a single application my contain self one or more self-contained database, memcache, logging, application logic, AI model applications interfacing each other over APIs as shown in Figure~\\ref{fig:intro} (left).\nWe call these applications \\emph{diffuse applications}.\n\nThis work contends that the fundamental programming paradigms in computing has not evolved at pace.\nThe abstractions envisioned during the era of the single machine computational model is still present at the programming interface and throughout the runtime stack leading to significant and costly complexity.\n\n\nTo address this complexity, two keystone abstractions have recently emerged to facilitate the development of these diffuse applications.\nThe first of these abstractions is the introduction and rapid dissemination of containerization service platforms.\nWith what started as a key insight articulated in ``The Datacenter as a Computer,'' Google would innovate their Borg system and ultimately released it open source as Kubernetes.\nWith Kubernetes, the underlying hardware resources would be abstracted away with the introduction of \\emph{pods} (virtual machines), and other resources that can be virtually networked together and otherwise configured irrespective of the physical hardware.\nToday, Kubernetes is the most prevalent containerized service abstraction layer in cloud computing.\nThe second of keystone abstraction would be coined ``Severless Computing'' and gained prominence with the introduction of Amazons Lamda functions.\nThis FaaS abstraction would facilitate the development of diffuse applications at the level of functions and abstract away the underlying containerized service ecosystem.\nA programer can simply make function calls in their favorite language without every needing to be aware of where the function will run nor the system level resources that would be allocated or managed.\n\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/jaseci_stack.pdf}\n    \\caption{Comparison between status quo development of production grade \\emph{diffuse} applications (left), and the Jaseci technology stack that hides and automates an expanded set of subsystems through raising the level of abstraction (right).}\n    \\label{fig:intro}\n\\end{figure}\n\n\n\nThough these two abstractions have been highly impactful, these innovations in our stack architecture represent a bottom up evolution of abstractions.\nAs a result, programmers are still left with single-machine abstractions at the programming interface and must grapple with a significant amount of complexity.\nFor example, traditional languages and their runtime stacks are predominately designed with the goal of hiding and managing intra-machine resources while what is needed for diffuse applications is the hiding and management of inter-machine resources.\nAnalogous to the virtualization and management of allocated memory on the heap provided by garbage collectors in modern languages (intra-machine), the virtualization and management of resources such as microservice creation, scheduling and orchestration alongside policies for organizing distributed databases, mem caches, logging and other highly complex subsystems (inter-machine) is not only needed, but as we show in this work, possible and practical.\nWithout this raising of the level of abstraction, it has become prohibitively difficult for a single engineer to invent, build, deploy, launch, and scale modern cutting edge applications.\n\nTo the best of our knowledge, we are not aware of a thorough, wholistic, and top-down design of a serverless programming paradigm and computational stack from the language level down through the system runtime stack to hide this expanded set of resources.\n\nIn this work, we present a wholistic design approach with the goal of abstracting away and automating a new class of underlying systems, allowing a programmer to articulate solutions and diffuse applications at the problem level.\nWe present the design of a \\emph{diffuse runtime execution engine} we call \\textbf{Jaseci}, and a \\emph{data-spacial programming language} we call \\textbf{Jac}.\nThe design of Jaseci and Jac has initially been inspired to by sophisticated emerging AI applications at scale and is driven by two key insightss.\n\\begin{itemize}\n   \n   \n\n    \\item \\emph{Higher level abstractions are needed at the language level to allow single creators to work at the problem level to build end-to-end diffuse AI products.}\n    \\item \\emph{A new set of abstractions across the language runtime and system stack is needed to automate and hide the class of inter-machine resources from the programmer.}\n\n         \n\\end{itemize}\n\n\\noindent\nTo this end we present techniques across two categories,\n\n\\begin{enumerate}\n   \n   \n\n    \\item \\emph{Jac Language} - A language that introduces a new set of abstractions, namely \\textbf{data-spacial scoping} and \\textbf{agent oriented programming}. These abstractions natively facilitates the emerging need to reason about and solve problems with graph representations as well as the need for algorithmic modularity and encapsulation to hide a new class of inter-machine resources.\n    \\item \\emph{Jaseci Diffuse Runtime Engine} - A runtime that raises the abstraction layer to the problem solving level where the runtime engine subsumes responsibility for not only for the optimization of program code, but the orchestration, configuration, and optimization of the full cloud compute stack and inter-machine resources (such tasks as container formation, scaling and optimization).\n\n         \n\\end{enumerate}\n\nJaseci and Jac is fully functional, open-source~\\cite{jaseci-website,jaseci-github,jaseci-pypi}, and used in production for four real-world products today.\nThese commercial products were built entirely on the Jaseci staci and includes Myca~\\cite{myca-website}, HomeLendingPal~\\cite{hlp-website},  ZeroShotBot~\\cite{zsb-website} and TrueSelph~\\cite{ts-website}.\nAcross these and other projects, the Jac language has been used by dozens of programmers in the creation of production software and Jaseci deployments support tens of thousands of production queries per day currently.\nIn practice, our initial infrastructure has been leveraged in practice to achieve 10x reduction in development time and near 100\\% elimination of typical backend code needed for a complicated AI based application.\n\nThe specific contributions of this paper include:\n\\begin{itemize}\n    \\item We formulate the problem of development complexity and present a top down programing paradigm and runtime stack for diffuse applications.\n    \\item We describe the design and implementation Jaseci's \\textbf{diffuse runtime execution engine}.\n    \\item We introduce Jac, a language that implements a \\textbf{data-spacial} programming paradigm (the first of its kind).\n    \\item We describe the utility of Jaseci and Jac through real world case studies of building out a real production scale-out product.\n\\end{itemize}\n\nWe find that the wholistic design philosophy and resulting paradigm of Jaseci and Jac is a promising one.\nMultiple development teams have adopted the data spacial programming model of Jac and the diffuse runtime execution engine in Jaseci to build sophisticated AI products with significantly reduced complexity and teaming.\n\n\n\\section{The Case for a New Computational Model}\\label{sec:motivation}\n\nThough recent advancements in serverless computing has been instrumental in improving the ability of teams to more rapidly develop software, significant challenges remain in the development of cutting edge applications and products in our current compute landscape.\nAn demonstrative problem domain with this challenge are those characterized by applications that include sophisticated AI pipelines on their critical path.\n\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/team_sizes.pdf}\n    \\caption{Comparison of typical development team required to realize production grade AI application today (left), and the ability of a single software developer to realize such an application with Jaseci (right). }\n    \\label{fig:dev}\n\\end{figure}\n\n\n\\subsection{Problem Scenario}\nFigure~\\ref{fig:dev}A shows the typical set of  often siloed roles needed to create software in this environment.\nThe first critical role needed is an \\emph{architect \/ tech-lead} responsible for architecting the software solution across disparate components, programming languages, frameworks, and SDKs.\nIf a microservice ecosystem is needed (which is a must for modern\nAI applications), the architect will also decide what will and won't be its own service (container) and define the interfaces between these disparate  services.\nFor the AI model work, the role of a \\emph{data scientist \/ ML engineer} is needed.\nThis role typically works primarily in Jupyter notebooks selecting, creating, training and tuning ML models to support application features.\nProduction software engineering is typically outside of the scope of this expertise in practice.\nThe role of a \\emph{backend engineer} is needed for implementing the main services of the application and taking the code out of Jupyter notebooks to build the models into the backend (server-side) of the application.\nThe \\emph{backend engineer} is also responsible for supporting new features and creating their API interfaces for \\emph{frontend} engineers.\nOne of the key roles any software team needs to deploy an AI product is a \\emph{DevOps engineer}.\nThis role is solely responsible for deploying and configuring containers to run on a cloud and ensure these containers are operational and scaled to the load requirements of the software.\nThis responsibility covers configuring software instance pods, database pods, caching layers, logging services, and parameterizing replicas and auto-scaling heuristics.\n\n\n\nIn this traditional model of software engineering, many challenges and complexity emerge.\nAn example is the (quite typical) scenario of the first main server-side implementation of the application being a monoservice while DB, caching, and logging are microservices.\nAs the \\emph{ML engineer} introduces models of increasing size, the \\emph{dev-ops} person alerts the team that the cloud instances, though designated as \\emph{large}, only have 8gb of ram.\nMeanwhile new AI models being integrated exceed this limit.\nThis event leads to a re-architecture of the main monoservice to be split out AI models into microservices and interfaces being designed or adopted leading to significant backend work \/ delays.\nIn this work, we aim to create a solution that would move all of this decisioning and work under the purview of the automated runtime system.\n\n\n\n\n\nUltimately, the mission of Jaseci is to accelerate and democratize the development and deployments of end-to-end scalable AI applications as presented in Figure~\\ref{fig:dev}B.\nTo this end, we present a novel set of higher level abstractions for programming sophisticated software in a micro-service\/serverless AI and a full stack architecture and programming model that abstracts away and automates much of the complexity of building diffuse applications on a distributed compute substrate of potentially thousands of compute nodes.\n\n\\section{Data Spacial Programming Model with Jac}\\label{sec:jac}\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/data_spacial.pdf}\n    \\caption{A visualization of the behavior of scopes and problem solving abstractions provided by the near ubiquitous function \/ method based languages (left) and the data spacial programming model (right).}\n    \\label{fig:benefit}\n\\end{figure}\n\nTraditionally in computer science, the task of raising the level abstraction in a computational model has primarily been for the goal of increasing programmer productivity.\nThis productivity comes from allowing engineers to function at the problem level while hiding the complexity of the underlying system.\nThe Jac language introduces a set of new abstractions guided by these principles based on two key insights.\nFirst, Jac recognizes the emerging need for programmers to reason about and solve problems with graph representations of data.\nSecond, Jac further supports the need for algorithmic modularity and encapsulation to change and prototype production software in place of prior running codebases.\nBased on these insights, we introduce two new sets of abstractions.\nAs shown in Figure~\\ref{fig:benefit}b, Jac's \\textbf{data-spacial scoping} natively facilitates graph based problem solving by replacing the traditional \\emph{temporal} notion of scope with a function's activation record with scoping that is flattened and spatially laid out in graph structure.\nThis type of scoping allows for richer semantics for the organization of the data relevant to the problem being solved.\nFigure~\\ref{fig:benefit}b also depicts Jac's \\textbf{agent oriented programming} as little robots.\nEach robot carries scope with it as it walks and performs compute relevant to where it sits on the graph.\nThese `agent' abstractions capture the need for algorithmic modality and encapsulation when introducing solutions to already sophisticated codebases.\nJac can be used solely to build out complete solutions or as glue code with components built in other languages.\nBy leveraging these new language abstractions, HomeLendingPal~\\cite{hlp-website} was able to create a production grade conversational AI experience with $\\sim$300 lines of code in contrast to the tens of thousands it would take to build in a traditional programming language.\n\n\\section{Diffuse Runtime Execution with Jaseci}\\label{sec:arch}\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/jaseci_arch.pdf}\n    \\caption{The architecture of the Jaseci diffuse runtime execution. The runtime stack includes and combines information from interpreter level profiling, cloud monitoring and profiling, microservice orchestrator and optimizer. Container linked libraries are also depicted. }\n    \\label{fig:benefit2}\n\\end{figure}\n\nJaseci's cloud-scale runtime engine presents a higher level abstraction of the software stack.\nThe \\emph{diffuse} runtime engine subsumes responsibility not only for the optimization of program code, but also the orchestration, configuration, and optimization of constituent micro services and the full cloud compute stack.\nDuties such as container formation, microservice scaling, scheduling and optimization are automated by the runtime.\nFor example, as shown in Figure~\\ref{fig:benefit2}c Jaseci introduces the concept of \\textbf{container linked libraries} to complement traditional notions of statically and dynamically linked libraries.\nFrom the programmers perspective, they need not know whether a call to a library is fused with the running programming instance or a remote call to a microservice somewhere in a cluster.\nThe decisioning of what \\emph{should} be a microservice and what should be statically in the programs object scope is made automatically and seamlessly by the Jaseci \\textbf{microservice orchestration engine}.\nUnderlying in-cluster microservices are encapsulated and hidden with this abstraction.\nWith the runtime having full visibility and control over the diffuse application, high complexity runtime decisions and heuristics such as autoscaling is brought under the purview of the runtime software stack, relieving the need of manual configuration.\nWith this Jaseci runtime, a single frontend engineer was able to implement the full ZeroShotBot~\\cite{zsb-website} application (which uses a number of transformer neural networks) without writing a single line of traditional `backend' code.\nThis implementation currently support tens of thousands of queries a day across about $\\sim$12 business customers with tens of thousands of individual end users in a single production environment.\n\n\\section{Case Studies and Discussion}\\label{sec:casestudy}\n\nJaseci is available on Github~\\cite{jaseci-github} under MIT open source license and is composed of an ecosystem of tools spanning 3 packages.\nThese include \\textbf{Jaseci Core}, its core execution engine,  \\textbf{Jaseci Serv}, its diffuse runtime cloud-scale execution engine, and \\textbf{Jaseci Kit}, a collection of cutting edge AI engines provided by the Jaseci community.\nIn addition to these main codebases, an experimental toolkit we call \\textbf{Jaseci Studio} is in development to provide visual programming and debugging tooling for developers building with Jaseci.\n\nThere are a number of notable examples of Jaseci's use in production.\nThese users include  four selected start-up companies that have adopted Jac and Jaseci as their development engine and have already launched their products built using Jaseci.\n\n\\indent \\textbf{ myca.ai}~\\cite{myca-website} - a B2C personal productivity platform that uses AI to understand personal behavior trends and help users allocate their time, prioritize their tasks and achieve personal growth goals. Using Jaseci, myca.ai's back-end development only took 1 month and myca.ai was launched within 3 months' development to the public. Myca.ai is one of the fast growing personal growth tool and has received positive feedback from their users.\n\n\\indent \\textbf{ZeroShotBot}~\\cite{zsb-website} - a B2B company that develops a cutting edge conversational AI platform using Jaseci. The product development took 2 months and was done by frontend engineers. Zeroshotbot has gained significant market traction and has been in business discussions with major logos such as Volaris, Pizzahut to provide readily deployable FAQ chatbots.\n\n\n\n\\indent \\textbf{Truselph}~\\cite{ts-website} - A minority founded startup. Truselph creates an avatar of the person and builds conversational intelligence that allows the general public to interact with the avatar and ask questions, while the avatar will be able to provide personalized answers with emotions and facial expressions. Truselph is in partnership with Lenovo to co-develop Truselph powered Kiosks for retail stores and is in business discussions with chains such as Sephora.\n\n\n\\indent \\textbf{ Home Lending Pal}~\\cite{hlp-website} - an AI Powered Mortgage Advisor. Home Lending Pal is a minority founded start-up that helps people, especially under-served minority population to navigate through the mortgage and home purchase process. Home Lending Pal adopted Jaseci to provide two main product features: 1 - personalized mortgage advice and 2 - Kev, an AI-powered chatbot that will answer users questions about the process and give them a plan to improve their finances.\n\n\\balance\n\\section{Conclusion}\nJaseci is a novel computational model invented, designed and implemented to address this challenge.\nJaseci includes a novel programming model we call \\emph{data-spacial programming} and a runtime engine we call the \\emph{diffuse execution environment} to enable rapid development of large scale and nimble AI applications.\nOur initial infrastructure has been used in practice to achieve 10x reduction in development time and near 100\\% elimination of typical backend code needed for a complicated AI based application.\nJaseci~\\cite{jaseci-website} was open sourced in 2021~\\cite{jaseci-github}~\\cite{jaseci-pypi}.\nToday Jaseci is in production with 4 distinct commercial products built on the engine, including Myca~\\cite{myca-website}, HomeLendingPal~\\cite{hlp-website},  ZeroShotBot~\\cite{zsb-website} and TrueSelph~\\cite{ts-website}.\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n    \\newpage\n\\fi\n\n\n\n\\section{Introduction}\\label{sec:introduction}}\n\\else\n\\section{Introduction}\n\\label{sec:introduction}\n\\fi\n\n\n\n\n\\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''\nfor IEEE Computer Society journal papers produced under \\LaTeX\\ using\nIEEEtran.cls version 1.8b and later.\nI wish you the best of success.\n\n\\hfill mds\n \n\\hfill August 26, 2015\n\n\\subsection{Subsection Heading Here}\nSubsection text here.\n\n\n\\subsubsection{Subsubsection Heading Here}\nSubsubsection text here.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nThe conclusion goes here.\n\n\n\n\n\n\n\n\n\\appendices\n\\section{Proof of the First Zonklar Equation}\nAppendix one text goes here.\n\n\\section{}\nAppendix two text goes here.\n\n\n\\ifCLASSOPTIONcompsoc\n \n  \\section*{Acknowledgments}\n\\else\n \n  \\section*{Acknowledgment}\n\\fi\n\n\nThe authors would like to thank...\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\n\\section{Introduction}\nThis demo file is intended to serve as a ``starter file''\nfor IEEE Computer Society conference papers produced under \\LaTeX\\ using\nIEEEtran.cls version 1.8b and later.\nI wish you the best of success.\n\n\\hfill mds\n \n\\hfill August 26, 2015\n\n\\subsection{Subsection Heading Here}\nSubsection text here.\n\n\n\\subsubsection{Subsubsection Heading Here}\nSubsubsection text here.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nThe conclusion goes here.\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcompsoc\n \n  \\section*{Acknowledgments}\n\\else\n \n  \\section*{Acknowledgment}\n\\fi\n\n\nThe authors would like to thank...\n\n\n\n\n\n\n\n\n\\section{Introduction}\\label{sec:introduction}}\n\n\n\n\n\\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''\nfor IEEE Computer Society journal papers produced under \\LaTeX\\ using\nIEEEtran.cls version 1.8b and later.\nI wish you the best of success.\n\n\\hfill mds\n \n\\hfill August 26, 2015\n\n\\subsection{Subsection Heading Here}\nSubsection text here.\n\n\n\\subsubsection{Subsubsection Heading Here}\nSubsubsection text here.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nThe conclusion goes here.\n\n\n\n\n\n\n\n\n\\appendices\n\\section{Proof of the First Zonklar Equation}\nAppendix one text goes here.\n\n\\section{}\nAppendix two text goes here.\n\n\n\\ifCLASSOPTIONcompsoc\n \n  \\section*{Acknowledgments}\n\\else\n \n  \\section*{Acknowledgment}\n\\fi\n\n\nThe authors would like to thank...\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}