diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzziofe" "b/data_all_eng_slimpj/shuffled/split2/finalzziofe" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzziofe" @@ -0,0 +1,5 @@ +{"text":"\\section{Statement of main theorem}\n\n\nIn this note, we study the initial and Neumann boundary value problem of a quasilinear diffusion equation with a linear reaction term:\n\\begin{equation}\\label{ibvp}\n\\begin{cases} u_t =\\dv (\\sigma(Du))-c(t)u &\\mbox{in $\\Omega\\times (0,T]$},\\\\\n{\\partial u}\/{\\partial \\n}=0 & \\mbox{on $\\partial \\Omega\\times (0,T]$},\\\\\nu(x,0)=u_0(x) &\\mbox{for $x\\in \\Omega$}.\n\\end{cases}\n\\end{equation}\nHere, $\\Omega\\subset \\R^n$ ($n\\ge 1$) is a bounded convex domain with $C^{2}$ boundary, $T>0$ is any fixed number, $u=u(x,t)$ is the unknown function with $u_t$ and $Du=(u_{x_1},\\cdots,u_{x_n})$ denoting its rate of change and spatial gradient respectively, $\\n$ is the outer unit normal on $\\partial\\Omega$, $u_0\\in C^{2}(\\bar\\Omega)$ is a given initial function satisfying the compatibility condition:\n\\begin{equation}\\label{ibvp-1}\n{\\partial u_0}\/{\\partial \\n}=0\\;\\; \\mbox{on $\\partial \\Omega,$}\n\\end{equation}\n$c=c(t)\\in W^{1,q_0}(0,T)$ is nonnegative for some $n+20$ are \\emph{ellipticity} constants. We easily have\n\\[\n \\sigma^i_{p_j}(p) = f(|p|^2)\\delta_{ij} + 2f'(|p|^2) p_ip_j \\quad (i,j=1,2,\\cdots,n;\\; p\\in \\R^n)\n\\]\nand hence the {\\em uniform ellipticity} condition:\n\\begin{equation*\n {\\lambda} |q|^2 \\le \\sum_{i,j=1}^n\\sigma^i_{p_j}(p) q_iq_j\\le \\Lambda |q|^2\\quad \\forall\\; p, \\, q\\in\\R^n;\n\\end{equation*}\nthat is, (\\ref{ibvp}) is a quasilinear uniformly parabolic problem with conormal boundary condition.\nThus the existence, uniqueness and regularity of a classical solution $u$ to (\\ref{ibvp}) follow from the standard theory such as in \\cite[Theorem 13.24]{Ln} under suitable H\\\"older regularity assumptions on $u_0$ and $\\partial\\Omega$.\n\nThe main result of this note is the following theorem.\n\n\n\n\\begin{thm}[Gradient Maximum Principle]\\label{thm:main}\nIf $u\\in C^{2,1}(\\bar\\Omega_T)$ is a classical solution to problem (\\ref{ibvp}), where $\\Omega_T=\\Omega\\times(0,T]$, then it satisfies the \\emph{gradient maximum principle:}\n\\begin{equation}\\label{gmp-1}\n\\|Du\\|_{L^\\infty(\\Omega_T)}=\\|Du_0\\|_{L^\\infty(\\Omega)}.\n\\end{equation}\n\\end{thm}\n\n\nGradient estimates for parabolic equations are usually given as {\\em a priori} estimates depending on the initial datum, domain and ellipticity constants. Our result, Theorem \\ref{thm:main}, gives an estimate independent of the {\\em convex} domain and ellipticity constants. In case of the heat equation ($f\\equiv 1$ and $c\\equiv 0$), (\\ref{gmp-1}) was proved in \\cite{Ka} for $C^{3,1}$ solutions and convex $C^3$ domains. Theorem \\ref{thm:main} extends such a result to a large class of uniformly parabolic equations for $C^{2,1}$ solutions and convex $C^2$ domains. It is also important to note that the convexity assumption on the domain $\\Omega$ in our result cannot be dropped in general; see a counterexample in \\cite[Theorem 4.1]{AR}. Also, we refer the reader to \\cite{PW, PS} for more extensive studies on the maximum principles in elliptic and parabolic differential equations.\n\nOur motivation of (\\ref{gmp-1}) is in the application of its pure diffusion case ($c\\equiv 0$) to the study of the Neumann problem of some forward-backward diffusion equations \\cite{KY, KY1, KY2}.\nAlthough the proof of Theorem \\ref{thm:main} would become much easier if $u$ belonged to $C^{3,1}(\\bar\\Omega_T)$, the existence of such a solution $u$ often requires the initial datum $u_0$ lie in $C^{3+\\alpha}(\\bar\\Omega)$ for some $0<\\alpha<1$ and satisfy, in addition to (\\ref{ibvp-1}), the second compatibility condition:\n\\begin{equation}\\label{ibvp-3}\n\\partial (\\dv(\\sigma(Du_0)))\/\\partial \\n=0 \\;\\; \\mbox{ on $\\partial\\Omega.$}\n\\end{equation}\nThese requirements give rise to a subtle but critical issue on the application of the convex integration method for constructing infinitely many Lipschitz solutions to certain forward-backward parabolic Neumann problems. For example, dealing with Perona-Malik type equations in \\cite{KY}, condition (\\ref{ibvp-3}) was posted for nonconstant radial initial data $u_0\\in C^{3+\\alpha}(\\bar\\Omega)$ when $\\Omega$ is a ball. Also, an earlier version of the main existence theorem in \\cite{KY1} for the Perona-Malik equation assumed that initial data $u_0\\in C^{3+\\alpha}(\\bar\\Omega)$ with compatibility conditions (\\ref{ibvp-1}) and (\\ref{ibvp-3}) satisfy some technical restrictions, which cannot handle the cases with $\\|Du_0\\|_{L^\\infty(\\partial\\Omega)}\\ge 1$ or with $0<\\|Du_0\\|_{L^\\infty(\\partial\\Omega)}< 1$ and $\\|Du_0\\|_{L^\\infty(\\Omega)}\\ge \\|Du_0\\|_{L^\\infty(\\partial\\Omega)}^{-1}.$ Our main result of this note removes these requirements and restrictions on nonconstant initial data $u_0$: the only requirement is that initial data $u_0\\in C^{2+\\alpha}(\\bar\\Omega)$ fulfill (\\ref{ibvp-1}).\n\nAnother purpose of studying (\\ref{gmp-1}) (when $c\\equiv 0$) is to confirm the validity of \\cite[Theorem 6.1]{KK} for \\emph{convex} domains. It has been a general belief that the initial-Neumann boundary value problem of a forward-backward parabolic equation in \\cite{KK} admits a unique global \\emph{classical} solution if the initial datum $u_0\\in C^{2+\\alpha}(\\bar\\Omega)$ satisfies (\\ref{ibvp-1}) and $\\|Du_0\\|_{L^\\infty(\\Omega)}0$ is the threshold at which the forward parabolicity of governing equation turns into the backward one. Regarding this, many authors often reported that such a problem is well-posed for \\emph{subcritical} (or \\emph{subsonic}) initial data. However, the proof of \\cite[Theorem 6.1]{KK} on such a result should be based on the gradient maximum principle (\\ref{gmp-1}) for a modified uniformly parabolic problem of type (\\ref{ibvp}), and so the convexity of the domain $\\Omega$ should not be overlooked in the proof as pointed out above.\n\n\nWe finish this section with some comments on notations. We mainly follow the notations in the book \\cite{LSU} for function spaces, with one exception that the letter $C$ is used instead of $H$ regarding suitable (parabolic) H\\\"older spaces.\nFor integers $k,l\\ge 0$ with $2l\\le k$, we denote by $C^{k,l}(\\bar\\Omega_T)$ [resp. $C^{k,l}(\\Omega_T)$] the space of functions $u\\in C^0(\\bar\\Omega_T)$ [$u\\in C^0(\\Omega_T)$] such that $D^a_x D^j_t u\\in C^0(\\bar\\Omega_T)$ [$D^a_x D^j_t u\\in C^0(\\Omega_T)$] for all multiindices $|a|\\le k$ and integers $0\\le j\\le l$ with $|a|+2j\\le k$.\nWe also adopt the summation convention that repeated indices in a term represent the sum.\n\n\n\n\n\\section{Proof of main theorem}\n\nWe follow the notations and assumptions of Theorem \\ref{thm:main} and introduce two useful lemmas.\nThe convexity assumption on the domain $\\Omega$ enters into the result (\\ref{gmp-1}) through the following lemma from \\cite[Lemma 2.1]{AR} or from \\cite[Theorem 2]{Ka}; we do not reproduce the proof here.\n\n\\begin{lem}\\label{lem-neumann}\nLet $\\Omega\\subset\\R^n$ be a bounded convex domain with $\\partial\\Omega$ of class $C^2$. If $w\\in C^2(\\bar\\Omega)$ satisfies\n$\\partial w\/\\partial \\n=0$ on $\\partial\\Omega,$\nthen $\\partial(|Dw|^2)\/\\partial\\n\\le 0$ on $\\partial\\Omega.$\n\\end{lem}\n\n\n\nThe next lemma gives an improved interior regularity of the solution $u\\in C^{2,1}(\\bar\\Omega_T)$ to problem (\\ref{ibvp}) that enables us to apply classical Hopf's Lemma for parabolic equations in a suitable setup. Its proof is postponed until the end of this section.\n\n\\begin{lem}\\label{lem-ireg}\nOne has\n\\begin{equation}\\label{lem-ireg-1}\nu\\in C^{3+\\beta_0,\\frac{3+\\beta_0}{2}}(\\Omega_T)\n\\end{equation}\nfor some $0<\\beta_0<1$.\n\\end{lem}\n\nWe now prove Theorem \\ref{thm:main} based on the two lemmas above.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\n\nLet $v=|Du|^2$ on $\\bar\\Omega_T.$ By Lemma \\ref{lem-ireg}, $v \\in C^{1,0}(\\bar\\Omega_T)\\cap C^{2,1}(\\Omega_T).$ We compute, within $\\Omega_T$,\n\\[\n\\Delta v =2 Du \\cdot D(\\Delta u) + 2 |D^2u|^2,\n\\]\n\\[\nu_t= \\dv (f(v)Du)-cu = f'(v) Dv \\cdot Du + f(v) \\Delta u- cu,\n\\]\n\\[\n\\begin{split}\nDu_t =& f''(v) (Dv\\cdot Du) Dv + f'(v) (D^2u) Dv \\\\\n&+ f'(v) (D^2v) Du + f'(v) (\\Delta u) Dv + f(v) D(\\Delta u) - c Du.\\end{split}\n\\]\nFrom these equations, using $v_t= 2Du\\cdot Du_t$, we obtain\n\\begin{equation}\\label{para-1}\nv_t- \\mathcal L (v) - V\\cdot Dv =-2f(v) |D^2u|^2 - 2c |Du|^2\\le 0 \\;\\;\\mbox{in}\\;\\;\\Omega_T,\n\\end{equation}\nwhere $\\mathcal L(v)$ and $V$ are defined by\n\\[\n\\mathcal L(v)= f(|Du|^2)\\Delta v + 2 f'(|Du|^2) Du\\cdot (D^2 v)Du,\n\\]\n\\[\nV= 2f''(v) (Dv\\cdot Du)Du + 2 f'(v) (D^2u) Du +\n 2 f'(v) (\\Delta u) Du.\n\\]\nSet $\\mathcal L (v)= a_{ij} v_{x_ix_j}$ with coefficients $a_{ij}=a_{ij}(x,t)$, given by\n\\[\na_{ij}=\\sigma^i_{p_j}(Du)=f(|Du|^2)\\delta_{ij} + 2 f'(|Du|^2)u_{x_i}u_{x_j} \\quad (i,j=1,\\cdots,n).\n\\]\nThen, on $\\bar\\Omega_T$, all eigenvalues of the matrix $(a_{ij})$ lie in $[\\lambda,\\Lambda]$.\n\nWe now show\n\\[\n \\max_{(x,t)\\in\\bar\\Omega_T}v(x,t) = \\max_{x\\in \\bar\\Omega} v(x,0),\n\\]\nwhich completes the proof. We argue by contradiction; suppose\n\\begin{equation}\\label{claim-1}\nM:=\\max_{(x,t)\\in\\bar\\Omega_T}v(x,t) > \\max_{x\\in \\bar\\Omega} v(x,0).\n\\end{equation}\n Let $(x_0,t_0)\\in\\bar\\Omega_T$ with $v(x_0,t_0)=M;$ then $t_0>0.$ If $x_0\\in \\Omega$, then the strong maximum principle \\cite{Ev} applied to (\\ref{para-1}) would imply that $v$ is constant on $\\bar\\Omega\\times[0,t_0],$ which yields $v(x,0)\\equiv M$ on $\\bar\\Omega$, a contradiction to (\\ref{claim-1}). Consequently, $x_0\\in \\partial\\Omega$ and thus $v(x_0,t_0)=M>v(x,t)$ for all $(x,t)\\in \\Omega_T.$ We can then apply Hopf's Lemma for parabolic equations \\cite{PW} to (\\ref{para-1}) to deduce $\n \\partial v(x_0,t_0) \/\\partial \\n >0,$\nwhich contradicts the conclusion of Lemma \\ref{lem-neumann}.\n \\end{proof}\n\nWe finally give the proof of Lemma \\ref{lem-ireg}, although it may be well known to the experts in regularity theory.\n\n\\begin{proof}[Proof of Lemma \\ref{lem-ireg}]\nWe rely on \\cite[Theorem III.12.1]{LSU} for the bootstrap of interior regularity for the solution $u\\in C^{2,1}(\\bar\\Omega_T)$ to problem (\\ref{ibvp}).\nWe divide the proof into several steps.\n\n1. In $\\Omega_T$,\n\\begin{equation}\\label{ireg-1}\nu_t=\\dv (f(|Du|^2)Du)-cu =a_{ij}u_{x_i x_j}-cu,\n\\end{equation}\nwhere $a_{ij}=\\sigma^i_{p_j}(Du)=f(|Du|^2)\\delta_{ij}+2f'(|Du|^2)u_{x_i}u_{x_j} \\in C^{1,0}(\\bar\\Omega_T)$ and $c\\in W^{1,q_0}(0,T)\\subset C^{\\alpha_0}([0,T])$ with $n+20$, where $\\Gamma_T=\\bar\\Omega_T\\setminus\\Omega_T$ is the parabolic boundary of $\\Omega_T$.\n\n3. Fix any $\\epsilon>0$ sufficiently small, and let\n\\[\n\\Omega^\\epsilon=\\{x\\in\\Omega\\,:\\,\\dist(x,\\partial\\Omega)>\\epsilon\\},\\;\\;\\Omega_T^\\epsilon =\\Omega^\\epsilon\\times(\\epsilon,T].\n\\]\nAlso, fix any two indices $k,l\\in\\{1,\\cdots,n\\}$, and set $w=u_{x_k x_l}\\in C^{0}(\\bar\\Omega_T^\\epsilon)$. Then by Step 2, we have $w\\in V^{1,0}_2(\\Omega_T^\\epsilon)$.\nTaking \\emph{formal} derivative of (\\ref{ireg-3}) in terms of $x_l$, we have\n\\begin{equation}\\label{ireg-6}\nw_t-\\frac{\\partial}{\\partial x_j}(a_{ij} w_{x_i})+b_i w_{x_i}+cw =h,\n\\end{equation}\nwhere\n\\[\nh= (a_{ij})_{x_k x_l}u_{x_i x_j}+(a_{ij})_{x_k}u_{x_i x_j x_l}+(a_{ij})_{x_l}u_{x_i x_j x_k}.\n\\]\nSince $f\\in C^3([0,\\infty))$, Step 2 implies\n\\begin{equation}\\label{ireg-7}\nh\\in L^q(\\Omega_T^\\epsilon)\\;\\;\\forall 1\\le q<\\infty.\n\\end{equation}\nObserve that coefficients of equation (\\ref{ireg-6}) are the same as those of equation (\\ref{ireg-3}). Thus as in Step 2, with (\\ref{ireg-7}), we see that the admissible criteria (1.3)--(1.6) in Chapter III of \\cite{LSU} are satisfied by coefficients and free term of equation (\\ref{ireg-6}). Also, $w\\in V^{1,0}_2(\\Omega_T^\\epsilon)$ is a weak solution to (\\ref{ireg-6}).\n\nAs in Step 2, we also rewrite equation (\\ref{ireg-6}) in non-divergence form:\n\\begin{equation}\\label{ireg-8}\nw_t-a_{ij}w_{x_ix_j}+cw=h.\n\\end{equation}\nLikewise, coefficients of (\\ref{ireg-8}) are equal to those of (\\ref{ireg-5}), and free term $h$ satisfies (\\ref{ireg-7}).\n\nAgain with (\\ref{ireg-2}), it follows from \\cite[Theorem III.12.1]{LSU} that weak derivatives $w_t,\\,w_{x_i x_j}$ $(i,j=1,\\cdots,n)$ exist and belong to $L^q(Q)$ for all $1\\le q<\\infty$ and domains $Q\\subset\\Omega_T^\\epsilon$ with $\\dist(Q,\\Gamma_T^\\epsilon)>0$, where $\\Gamma_T^\\epsilon=\\bar\\Omega_T^\\epsilon\\setminus\\Omega_T^\\epsilon$ is the parabolic boundary of $\\Omega_T^\\epsilon$.\n\n4. Set $\\tilde w=u_t\\in C^0(\\bar\\Omega_T^\\epsilon)$. By Step 2, we have $\\tilde w\\in V^{1,0}_2(\\Omega_T^\\epsilon)$.\nDifferentiating (\\ref{ireg-1}) \\emph{formally} with respect to $t$,\n\\begin{equation*\n\\tilde w_t - \\frac{\\partial}{\\partial x_j}(a_{ij}\\tilde w_{x_i}) + b_i\\tilde w_{x_i}+c\\tilde w=\\tilde h,\n\\end{equation*}\nwhere\n\\[\n\\tilde h=(a_{ij})_t u_{x_i x_j}-c' u.\n\\]\nFrom Step 2 and $c'\\in L^{q_0}(0,T)$, we have\n\\begin{equation*\n\\tilde h \\in L^{q_0}(\\Omega_T^\\epsilon).\n\\end{equation*}\n\nAs above, we obtain from \\cite[Theorem III.12.1]{LSU} that $\\tilde w_t=u_{tt}$ exists and belongs to $L^{q_0}(Q)$ for all domains $Q\\subset\\Omega_T^\\epsilon$ with $\\dist(Q,\\Gamma_T^\\epsilon)>0$.\n\n\n5. By Steps 2--4, we conclude that\n\\[\nu\\in W^{4,2}_{q_0}(\\Omega_T^{2\\epsilon})\\;\\;\\forall\\epsilon>0.\n\\]\nBy the parabolic Sobolev embedding theorem \\cite[Lemma II.3.3]{LSU}, we obtain\n\\[\nu\\in C^{3+\\beta_0,\\frac{3+\\beta_0}{2}}(\\Omega_T),\n\\]\nwhere $0<\\beta_0<1-\\frac{n+2}{q_0}$; hence (\\ref{lem-ireg-1}) holds.\n\\end{proof}\n\\vspace{2ex}\n\n\n{\\bf Acknowledgments.} The author would like to thank Professor Baisheng Yan and the referee for many helpful comments and suggestions to improve the presentation of the paper.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{#1}\\label{#2}}\n\n\\def\\operatorname{Perm}{\\operatorname{Perm}}\n\\def\\operatorname{Cov}{\\operatorname{Cov}}\n\\def\\operatorname{CSep}{\\operatorname{CSep}}\n\\def\\operatorname{Sep}{\\operatorname{Sep}}\n\\def\\operatorname{Ob}{\\operatorname{Ob}}\n\\def\\operatorname{ind}{\\operatorname{ind}}\n\\def\\operatorname{sep}{\\operatorname{sep}}\n\\def\\operatorname{Br}{\\operatorname{Br}}\n\\def\\operatorname{dBr}{\\operatorname{dBr}}\n\\def\\rightarrow{\\rightarrow}\n\n\\let\\oldmarginpar\\marginpar\n\\def\\marginpar#1{\\oldmarginpar{\\tiny #1}}\n\\def\\operatorname{DPic}{\\operatorname{DPic}}\n\\def\\operatorname{Pic}{\\operatorname{Pic}}\n\\def\\operatorname{pr}{\\operatorname{pr}}\n\\def\\mathbin{\\overset{L}{\\otimes}}{\\mathbin{\\overset{L}{\\otimes}}}\n\\def\\operatorname{cosk}{\\operatorname{cosk}}\n\\def\\operatorname{rk}{\\operatorname{rk}}\n\\def\\operatorname{Supp}{\\operatorname{Supp}}\n\\def\\operatorname{Spec}{\\operatorname{Spec}}\n\n\\def\\multiset#1#2{\\ensuremath{\\left(\\kern-.3em\\left(\\genfrac{}{}{0pt}{}{#1}{#2}\\right)\\kern-.3em\\right)}}\n\n\\begin{document}\n\n\\title[Recent developments on noncommutative motives]{Recent developments on noncommutative motives}\n\\author{Gon{\\c c}alo~Tabuada}\n\n\\address{Gon{\\c c}alo Tabuada, Department of Mathematics, MIT, Cambridge, MA 02139, USA}\n\\email{tabuada@math.mit.edu}\n\\urladdr{http:\/\/math.mit.edu\/~tabuada}\n\\thanks{The author is very grateful to the organizers Nitu Kitchloo, Mona Merling, Jack Morava, Emily Riehl, and W. Stephen Wilson, for the kind invitation to present some of this work at the second Mid-Atlantic Topology Conference. The author was supported by~a~NSF~CAREER~Award.}\n\n\n\\date{\\today}\n\n\\abstract{This survey covers some of the recent developments on noncommutative motives and their applications. Among other topics, we compute the additive invariants of relative cellular spaces and orbifolds; prove Kontsevich's semi-simplicity conjecture; prove a far-reaching noncommutative generalization of the Weil conjectures; prove Grothendieck's standard conjectures of type $C^+$ and $D$, Voevodsky's nilpotence conjecture, and Tate's conjecture, in several new cases; embed the (cohomological) Brauer group into secondary $K$-theory; construct a noncommutative motivic Gysin triangle; compute the localizing $\\mathbb{A}^1$-homotopy invariants of corner skew Laurent polynomial algebras and of noncommutative projective schemes; relate Kontsevich's category of noncommutative mixed motives to Morel-Voevodsky's stable $\\mathbb{A}^1$-homotopy category, to Voevodsky's triangulated category of mixed motives, and to Levine's triangulated category of mixed motives; prove the Schur-finiteness conjecture for quadric fibrations over low-dimensional bases; and finally extend Grothendieck's theory of periods to the setting of dg categories.}\n}\n\n\\maketitle\n\n\\smallskip\n\n\\vskip-\\baselineskip\n\n\\vspace{-0.3cm}\n\n\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\,{\\em {\\small To Lily, for being by my side.}}\n\\section*{Introduction}\nAfter the release of the monograph [Noncommutative Motives. With a preface by Yuri I. Manin. University Lecture Series {\\bf 63}, American Mathematical Society, 2015], several important results on the theory of noncommutative motives have been established. The purpose of this survey, written for a broad mathematical audience, is to give a rigorous overview of some of these recent results. We will follow closely the notations, as well as the writing style, of the monograph \\cite{book}. Therefore, we suggest the reader to have it at his\/her desk while reading this survey. The monograph \\cite{book} is divided into the following chapters:\n\n\\vspace{0.2cm}\n\n\\begin{itemize}\n\\item[] Chapter 1. Differential graded categories.\n\\item[] Chapter 2. Additive invariants.\n\\item[] Chapter 3. Background on pure motives.\n\\item[] Chapter 4. Noncommutative pure motives.\n\\item[] Chapter 5. Noncommutative (standard) conjectures.\n\\item[] Chapter 6. Noncommutative motivic Galois groups.\n\\item[] Chapter 7. Jacobians of noncommutative Chow motives.\n\\item[] Chapter 8. Localizing invariants.\n\\item[] Chapter 9. Noncommutative mixed motives.\n\\item[] Chapter 10. Noncommutative motivic Hopf dg algebras\n\\item[] Appendix A. Grothendieck derivators.\n\\end{itemize}\n\n\\vspace{0.2cm}\n\nIn this survey we cover some of the recent developments concerning the Chapters 2, 4, 5, 6, 8, and 9. These developments are described in Sections \\ref{sec:additive}, \\ref{sec:pure}, \\ref{sec:conjectures}, \\ref{sec:Galois}, \\ref{sec:localizing}, and \\ref{sec:NCmixed}, respectively. The final Section \\ref{sec:periods}, entitled ``Noncommutative realizations and periods'', discusses a recent research subject which was not addressed in \\cite{book}.\n\\subsection*{Preliminaries}\nThroughout the survey, $k$ will denote a base field. We will assume the reader is familiar with the language of differential graded (=dg) categories; for a survey on dg categories, we invite the reader to consult Keller's ICM address \\cite{ICM-Keller}. In particular, we will freely use the notions of {\\em Morita equivalence} of dg categories (see \\cite[\\S1.6]{book}) and {\\em smooth\/proper dg category} in the sense of Kontsevich (see \\cite[\\S1.7]{book}). We will write $\\mathsf{dgcat}(k)$ for the category of (small) dg categories and $\\mathsf{dgcat}_\\mathrm{sp}(k)$ for the full subcategory of smooth proper dg categories. Given a $k$-scheme $X$ (or more generally an algebraic stack ${\\mathcal X}$), we will denote by $\\mathrm{perf}_\\mathrm{dg}(X)$ the canonical dg enhancement of the category of perfect complexes $\\mathrm{perf}(X)$; see \\cite[Example~1.27]{book}.\n\\section{Additive invariants}\\label{sec:additive}\nRecall from \\cite[\\S2.3]{book} the construction of the {\\em universal additive invariant} of dg categories $U\\colon \\mathsf{dgcat}(k) \\to \\Hmo_0(k)$. In \\cite[\\S2.4]{book} we described the behavior of $U$ with respect to semi-orthogonal decompositions, full exceptional collections, purely inseparable field extensions, central simple algebras, sheaves of Azumaya algebras, twisted flag varieties, nilpotent ideals, finite-dimensional algebras of finite global dimension, etc. In \\S\\ref{sub:relative}-\\ref{sub:orbifolds} we describe the behavior of $U$ with respect to relative cellular spaces and orbifolds. As explained in \\cite[Thm.~2.9]{book}, all the results in \\S\\ref{sub:relative}-\\ref{sub:orbifolds} are {\\em motivic} in the sense that they hold similarly for every additive invariant such as algebraic $K$-theory, mod-$n$ algebraic $K$-theory, Karoubi-Villamayor $K$-theory, nonconnective algebraic $K$-theory, homotopy $K$-theory, {\\'e}tale $K$-theory, Hochschild homology, cyclic homology, negative cyclic homology, periodic cyclic homology, topological Hochschild homology, topological cyclic homology, topological periodic cyclic homology, etc. Consult \\S\\ref{sec:NCrealizations} for further examples of additive invariants.\n\n\\begin{notation} Given a $k$-scheme $X$ (or more generally an algebraic stack ${\\mathcal X}$), we will write $U(X)$ instead of $U(\\mathrm{perf}_\\mathrm{dg}(X))$.\n\\end{notation}\n\\subsection{Relative cellular spaces}\\label{sub:relative}\nA flat morphism of $k$-schemes $p\\colon X \\to Y$ is called an {\\em affine fibration} of relative dimension $d$ if for every point $y\\in Y$ there exists a Zariski open neighborhood $y \\in V$ such that $X_V:=p^{-1}(V) \\simeq Y \\times \\mathbb{A}^d$ with $p_V\\colon X_V \\to Y$ isomorphic to the projection onto the first factor. Following Karpenko \\cite[Def.~6.1]{Karpenko}, a smooth projective $k$-scheme $X$ is called a {\\em relative cellular space} if it admits a filtration by closed subschemes\n\\begin{equation*\n\\varnothing = X_{-1} \\hookrightarrow X_0 \\hookrightarrow \\cdots \\hookrightarrow X_i \\hookrightarrow \\cdots \\hookrightarrow X_{n-1} \\hookrightarrow X_n=X\n\\end{equation*}\nand affine fibrations $p_i\\colon X_i \\backslash X_{i-1} \\to Y_i, 0 \\leq i \\leq n$, of relative dimension $d_i$ with $Y_i$ a smooth projective $k$-scheme. The smooth $k$-schemes $X_i \\backslash X_{i-1}$ are called the {\\em cells} and the smooth projective $k$-schemes $Y_i$ the {\\em bases} of the cells.\n\\begin{example}[$\\mathbb{G}_m$-schemes]\\label{example:bb}\nThe celebrated Bialynicki-Birula decomposition \\cite{BB} provides a relative cellular space structure on smooth projective $k$-schemes equipped with a $\\mathbb{G}_m$-action. In this case, the bases of the cells are given by the connected components of the fixed point locus.\nThis class of relative cellular spaces includes the isotropic flag varieties considered by Karpenko in \\cite{Karpenko} as well as the isotropic homogeneous spaces considered by Chernousov-Gille-Merkurjev~in~\\cite{CGM}.\n\\end{example}\n\\begin{theorem}[{\\cite[Thm.~2.7]{Gysin}}]\\label{thm:cellular}\nGiven a relative cellular space $X$, we have an isomorphism $U(X) \\simeq \\bigoplus_{i=0}^n U(Y_i)$. \n\\end{theorem}\nTheorem \\ref{thm:cellular} shows that the additive invariants of relative cellular spaces $X$ are completely determined by the basis $Y_i$ of the cells $X_i \\backslash X_{i-1}$. Among other ingredients, its proof makes use of Theorem \\ref{thm:Gysin}; consult \\cite[\\S9]{Gysin} for details.\n\\begin{example}[Kn\\\"orrer periodicity] \nLet $q=fg+q'$, where $f$, $g$, and $q'$, are\n forms of degrees $a>0$, $b>0$, and $a+b$, in disjoint sets\n of variables $(x_i)_{i=1,\\ldots,m}$, $(y_j)_{j=1,\\ldots,n}$, and\n $(z_l)_{l=1,\\ldots,p}$, respectively. Such a decomposition holds, for example, in the case of isotropic quadratic forms $q$. Let us write $Q$ and $Q'$ for the projective\n hypersurfaces defined by $q$ and $q'$, respectively. Assume that $Q$ is smooth. Under this assumption, we have a $\\mathbb{G}_m$-action on $Q$ given by $\\lambda\\cdot\n (\\underline{x},\\underline{y},\\underline{z}):=(\\lambda^b\\underline{x},\\lambda^{-a}\\underline{y},\\underline{z})$\n with fixed point locus $\\mathbb{P}^{m-1}\\amalg \\mathbb{P}^{n-1}\\amalg Q'$; this implies that $Q'$ is also smooth. By combining Theorem \\ref{thm:cellular} and Example \\ref{example:bb} with the fact that $U(\\mathbb{P}^n)\\simeq U(k)^{\\oplus (n+1)}$ (see \\cite[\\S2.4.2]{book}), we obtain an induced isomorphism $U(Q)\\simeq U(k)^{\\oplus (m+n)}\\oplus U(Q')$. Morally speaking, this shows that (modulo $k$) the additive invariants of $Q$ and $Q'$ are the same.\n\\end{example}\n\\subsection{Orbifolds}\\label{sub:orbifolds}\nLet $G$ be a finite group of order $n$ (we assume that $1\/n \\in k$), $\\varphi$ the set of all cyclic subgroups of $G$, ${\\varphi\\!}{\/}{\\!\\sim}$ a set of representatives of the conjugacy classes in $\\varphi$, $X$ a smooth $k$-scheme equipped with a $G$-action, and $[X\/G]$ the associated orbifold. As explained in \\cite[\\S3]{Orbifold}, the assignment $[V] \\mapsto V\\otimes_k -$, where $V$ stands for a $G$-representation, gives rise to an action of the representation ring $R(G)$ on $U([X\/G])$. Given $\\sigma \\in \\varphi$, let $e_\\sigma$ be the unique idempotent of the $\\mathbb{Z}[1\/n]$-linearized representation ring $R(\\sigma)_{1\/n}$ whose image under all the restrictions $R(\\sigma)_{1\/n}~\\to~R(\\sigma')_{1\/n}$, with $\\sigma' \\subsetneq \\sigma$, is zero. \nThe normalizer $N(\\sigma)$ of $\\sigma$ acts naturally on $[X^\\sigma\/\\sigma]$ and hence on $U([X^\\sigma\/\\sigma])$. By functoriality, this action restricts to the direct summand $e_\\sigma U([X^\\sigma\/\\sigma])_{1\/n}$.\n\\begin{theorem}[{\\cite[Thm.~1.1 and Cor.~1.6]{Orbifold}}]\\label{thm:orbifold}\nThe following computations hold:\n\\begin{itemize}\n\\item[(i)] We have an induced isomorphism\n\\begin{equation}\\label{eq:formula1}\nU([X\/G])_{1\/n}\\simeq \\bigoplus_{\\sigma \\in \\varphi\\!\/\\!\\sim} (e_\\sigma U([X^\\sigma\/\\sigma])_{1\/n})^{N(\\sigma)}\n\\end{equation}\nin the $\\mathbb{Z}[1\/n]$-linearized (and idempotent completed) category $\\Hmo_0(k)_{1\/n}$.\n\\item[(ii)] If $k$ contains the $n^{\\mathrm{th}}$ roots of unity, then \\eqref{eq:formula1} reduces to an isomorphism\n\\begin{equation}\\label{eq:formula2}\nU([X\/G])_{1\/n} \\simeq \\bigoplus_{\\sigma \\in \\varphi\\!\/\\!\\sim}(U(X^\\sigma)_{1\/n} \\otimes_{\\mathbb{Z}[1\/n]} e_\\sigma R(\\sigma)_{1\/n})^{N(\\sigma)}\\,,\n\\end{equation}\nwhere $-\\otimes_{\\mathbb{Z}[1\/n]}-$ stands for the canonical action of the category of finitely generated projective $\\mathbb{Z}[1\/n]$-modules on $\\Hmo_0(k)_{1\/n}$.\n\\item[(iii)] If $k$ contains the $n^{\\mathrm{th}}$ roots of unity and $F$ is a field which contains the $n^{\\mathrm{th}}$ roots of unity and $1\/n \\in F$, then we have induced isomorphisms\n\\begin{equation}\\label{eq:formula3}\nU([X\/G])_F\\simeq \\bigoplus_{g \\in G\\!\/\\!\\sim} U(X^g)_F^{C(g)} \\simeq (\\bigoplus_{g \\in G} U(X^g)_F)^G\n\\end{equation}\nin the category $\\Hmo_0(k)_F$, where $C(g)$ stands for the centralizer of $g$.\n\\end{itemize}\nMoreover, \\eqref{eq:formula2}-\\eqref{eq:formula3} are isomorphisms of (commutative) monoids.\n\\end{theorem}\nRoughly speaking, Theorem \\ref{thm:orbifold} shows that the additive invariants of orbifolds can be computed using solely ``ordinary'' schemes.\n\\begin{example}[McKay correspondence]\nIn many cases, the dg category $\\mathrm{perf}_\\mathrm{dg}([X\/G])$ is known to be Morita equivalent \nto $\\mathrm{perf}_\\mathrm{dg}(Y)$ for a crepant \n resolution $Y$ of the (singular) geometric quotient $X\/\\!\\!\/G$; see \\cite{BezKal2,BKR,KV,\n Kawamata}. This is generally referred to as the ``McKay correspondence''.\nWhenever it holds, we can replace $[X\/G]$ by $Y$ in the formulas \\eqref{eq:formula1}-\\eqref{eq:formula3}. Here is an illustrative example (with $k$ algebraically closed): the cyclic group $G=C_2$ acts on any abelian surface $S$ by the\n involution $a \\mapsto -a$ and the Kummer surface $\\mathrm{Km}(S)$ is defined as\nthe blow-up of $S\/\\!\\!\/C_2$ at its $16$ singular points.\nIn this case, the dg category $\\mathrm{perf}_\\mathrm{dg}([S\/C_2])$ is Morita equivalent to \n $\\mathrm{perf}_\\mathrm{dg}(\\mathrm{Km}(S))$. Consequently, Theorem \\ref{thm:orbifold}(ii) leads to an isomorphism:\n\\begin{equation}\\label{eq:McKay}\nU(\\operatorname{Km}(S))_{1\/2}\\simeq U(S)_{1\/2}^{C_2} \\oplus U(k)_{1\/2}^{\\oplus 16}\\,.\n\\end{equation}\nNote that since the Kummer surface is Calabi-Yau, the category $\\mathrm{perf}(\\operatorname{Km}(S))$ does {\\em not} admit non-trivial semi-orthogonal decompositions. This shows that the isomorphism \\eqref{eq:McKay} is {\\em not} induced from a semi-orthogonal decomposition.\n\\end{example}\n\\begin{corollary}[Algebraic $K$-theory]\nIf $k$ contains the $n^{\\mathrm{th}}$ roots of unity, then we have the following isomorphism of $\\mathbb{Z}$-graded commutative $\\mathbb{Z}[1\/n]$-algebras:\n\\begin{equation}\\label{eq:formula4}\n K_\\ast([X\/G])_{1\/n} \\simeq \\bigoplus_{\\sigma \\in \\varphi\\!\/\\!\\sim}(K_\\ast(X^\\sigma)_{1\/n} \\otimes_{\\mathbb{Z}[1\/n]} e_\\sigma R(\\sigma)_{1\/n})^{N(\\sigma)}\\,.\n\\end{equation}\n\\end{corollary}\nThe formula \\eqref{eq:formula4} was originally established by Vistoli in \\cite[Thm.~1]{Vistoli}. Among other ingredients, Vistoli's proof makes essential use of d\\'evissage. The proof of Theorem \\ref{thm:orbifold}, and hence of \\eqref{eq:formula4}, is not only different but moreover avoids the use of d\\'evissage; consult \\cite[\\S6]{Orbifold} for details.\n\\begin{corollary}[Cyclic homology]\nIf $k$ contains the $n^{\\mathrm{th}}$ roots of unity, then we have the following isomorphisms of $\\mathbb{Z}$-graded commutative $k$-algebras: \n\\begin{equation}\\label{eq:formula5}\nHC_\\ast([X\/G]) \\simeq \\bigoplus_{g\\in G\\!\/\\!\\sim} HC_\\ast(X^g)^{C(g)} \\simeq (\\bigoplus_{g\\in G} HC_\\ast(X^g))^G\\,.\n\\end{equation}\n\\end{corollary}\nThe formula \\eqref{eq:formula5} was originally established by Baranovsky in \\cite[Thm.~1.1]{Baranovsky}. Baranovsky's proof is very specific to cyclic homology. In constrast, the proof of Theorem \\ref{thm:orbifold}, and hence of \\eqref{eq:formula5}, avoids all the specificities of cyclic homology and is moreover quite conceptual; consult \\cite[\\S6]{Orbifold} for details.\n\\begin{corollary}[Topological periodic cyclic homology]\nLet $k$ be a perfect field of characteristic $p>0$, $W(k)$ the associated ring of $p$-typical Witt vectors, and $K:=W(k)[1\/p]$ the fraction field of $W(k)$. If $k$ contains the $n^{\\mathrm{th}}$ roots of unity, then we have the following isomorphisms of $\\mathbb{Z}\/2$-graded commutative $K$-algebras: \n\\begin{equation}\\label{eq:formula6}\nTP_\\ast([X\/G])_{1\/p} \\simeq \\bigoplus_{g\\in G\\!\/\\!\\sim} TP_\\ast(X^g)_{1\/p}^{C(g)} \\simeq (\\bigoplus_{g\\in G} TP_\\ast(X^g)_{1\/p})^G\\,.\n\\end{equation}\n\\end{corollary}\nTo the best of the author's knowledge, the formula \\eqref{eq:formula6} is new in the literature; consult \\cite[\\S1]{Orbifold} for further corollaries of Theorem \\ref{thm:orbifold}.\n\\subsubsection{Twisted analogues}\\label{sec:twisted}\nGiven a sheaf of Azumaya algebras ${\\mathcal F}$ over $[X\/G]$, \\textsl{i.e.}\\ a $G$-equivariant sheaf of Azumaya algebras over $X$, all the computations of Theorem \\ref{thm:orbifold} admit ${\\mathcal F}$-twisted analogues; consult \\cite[Thm.~1.27 and Cor.~1.29]{Orbifold} for details.\n\\section{Noncommutative pure motives}\\label{sec:pure}\nIn \\S\\ref{sub:recollections} we recall the definition of the different categories of noncommutative pure motives. Subsections \\S\\ref{sub:Brauer}-\\ref{sub:rigidity} are devoted to three structural properties of these categories (relation with the Brauer group, semi-simplicity and rigidity). In \\ref{sub:zeta} we prove to a far-reaching noncommutative generalization of the Weil conjectures; see Theorem \\ref{thm:zeta}. Finally, in \\S\\ref{sub:equivariant} we describe some of the equivariant analogues of the theory of noncommutative pure motives.\n\\subsection{Recollections}\\label{sub:recollections}\nRecall from \\cite[\\S4.1]{book} that the category of {\\em noncommutative Chow motives $\\NChow(k)$} is defined as the idempotent completion of the full subcategory of $\\Hmo_0(k)$ consisting of the objects $U({\\mathcal A})$, with ${\\mathcal A}$ a smooth proper dg category. By construction, this category is additive, rigid symmetric monoidal, and comes equipped with a symmetric monoidal functor $U\\colon \\mathsf{dgcat}_{\\mathrm{sp}}(k) \\to \\NChow(k)$. Moreover, given smooth proper dg categories ${\\mathcal A}$ and ${\\mathcal B}$, we have isomorphisms:\n\\begin{equation}\\label{eq:star}\n\\mathrm{Hom}_{\\NChow(k)}(U({\\mathcal A}),U({\\mathcal B})) \\simeq K_0({\\mathcal D}_c({\\mathcal A}^\\mathrm{op} \\otimes {\\mathcal B}))=: K_0({\\mathcal A}^\\mathrm{op} \\otimes {\\mathcal B})\\,.\n\\end{equation}\nGiven a rigid symmetric monoidal category $({\\mathcal C}, \\otimes, {\\bf 1})$, consider the $\\otimes$-ideal\n$$\\otimes_{\\mathrm{nil}}(a,b) := \\{ f \\in \\mathrm{Hom}_{\\mathcal C}(a,b) \\,|\\,f^{\\otimes n}=0\\,\\,\\mathrm{for}\\,\\,\\mathrm{some}\\,\\,n\\gg 0\\}\\,.$$\nRecall from \\cite[\\S4.4]{book} that the category of {\\em noncommutative $\\otimes$-nilpotent motives $\\mathrm{NVoev}(k)$} is defined as the idempotent completion of the quotient $\\NChow(k)\/\\otimes_{\\mathrm{nil}}$.\n\nAs explained in \\cite[\\S4.5]{book}, periodic cyclic homology gives rise to an additive symmetric monoidal functor $HP^\\pm\\colon \\NChow(k) \\to \\mathrm{Vect}_{\\mathbb{Z}\/2}(k)$, with values in the category of finite-dimensional $\\mathbb{Z}\/2$-graded $k$-vector spaces. Recall from {\\em loc. cit.} that the category of {\\em noncommutative homological motives $\\mathrm{NHom}(k)$} is defined as the idempotent completion of the quotient $\\NChow(k)\/\\mathrm{Ker}(HP^\\pm)$.\n\nGiven a rigid symmetric monoidal category $({\\mathcal C}, \\otimes, {\\bf 1})$, consider the $\\otimes$-ideal\n$$\n{\\mathcal N}(a,b) := \\{ f \\in \\mathrm{Hom}_{\\mathcal C}(a,b)\\,|\\,\\forall g \\in \\mathrm{Hom}_{\\mathcal C}(b,a)\\,\\,\\mathrm{we}\\,\\,\\mathrm{have}\\,\\,\\mathrm{tr}(g\\circ f)=0\\}\\,,\n$$\nwhere $\\mathrm{tr}(g\\circ f)$ stands for the categorical trace of the endomorphism $g\\circ f$. Recall from \\cite[\\S4.6]{book} that the category of {\\em noncommutative numerical motives $\\NNum(k)$} is defined as the idempotent completion of the quotient $\\NChow(k)\/{\\mathcal N}$.\n\\subsection{Relation with the Brauer group}\\label{sub:Brauer}\nLet $\\mathrm{Br}(k)$ be the Brauer group of the base field $k$. Given a central simple $k$-algebra $A$, we write $[A]$ for its Brauer class.\n\\begin{example}[Local fields]\nA local field $k$ is isomorphic to $\\mathbb{R}$, to $\\mathbb{C}$, to a finite field extension of $\\mathbb{Q}_p$, or to a finite field extension of $\\mathbb{F}_p(\\!(t)\\!)$. Thanks to local class field theory, we have $\\mathrm{Br}(\\mathbb{R})\\simeq \\mathbb{Z}\/2$, $\\mathrm{Br}(\\mathbb{C})=0$, and $\\mathrm{Br}(k)\\simeq \\mathbb{Q}\/\\mathbb{Z}$ in all the remaining cases. Moreover, every element of $\\mathrm{Br}(k)$ can be represented by a cyclic $k$-algebra.\n\\end{example}\nRecall from \\cite[\\S2.4.4]{book} that we have the following equivalence\n\\begin{equation}\\label{eq:equivalence-1}\n[A]=[B] \\Leftrightarrow U(A) \\simeq U(B)\n\\end{equation}\nfor any two central simple $k$-algebras $A$ and $B$. Intuitively speaking, \\eqref{eq:equivalence-1} shows that the Brauer class $[A]$ and the noncommutative Chow motive $U(A)$ contain exactly the same information. Let $K_0(\\NChow(k))$ be the Grothendieck ring of the additive symmetric monoidal category $\\NChow(k)$. Given a central simple $k$-algebra $A$, we write $[U(A)]$ for the Grothendieck class of $U(A)$. The (proof of the) next result is contained in \\cite[Thm.~6.12]{Secondary}\\cite[Thm.~1.3]{SecondaryII}:\n\\begin{theorem}\\label{thm:new}\nGiven central simple $k$-algebras $A$ and $B$, we have the equivalence:\n\\begin{equation}\\label{eq:equivalence-2}\nU(A)\\simeq U(B) \\Leftrightarrow [U(A)] \\simeq [U(B)]\\,.\n\\end{equation}\n\\end{theorem}\nRoughly speaking, Theorem \\ref{thm:new} shows that the noncommutative Chow motives of central simple $k$-algebras are insensitive to the Grothendieck group relations. By combining the equivalences \\eqref{eq:equivalence-1} and \\eqref{eq:equivalence-2}, we obtain the following result:\n\\begin{corollary}\\label{cor:new}\nThe following map is injective:\n\\begin{eqnarray*}\n\\mathrm{Br}(k) \\longrightarrow K_0(\\NChow(k)) && [A] \\mapsto [U(A)]\\,.\n\\end{eqnarray*}\n\\end{corollary}\nConsult \\S\\ref{sub:secondary} for some applications of Corollary \\ref{cor:new} to secondary $K$-theory.\n\\begin{remark}[Generalizations]\\label{rk:generalization1}\nTheorem \\ref{thm:new} and Corollary \\ref{cor:new} hold more generally with $k$ replaced by a base $k$-scheme $X$. Furthermore, instead of the Brauer group $\\mathrm{Br}(X)$, we can consider the second \\'etale cohomology group\\footnote{As proved by Gabber \\cite{Gabber} and de Jong \\cite{deJong}, in the case where $X$ admits an ample line bundle (\\textsl{e.g.}\\ $X$ affine), the Brauer group $\\mathrm{Br}(X)$ may be identified with the torsion subgroup~of~$H^2_{\\mathrm{et}}(X,\\mathbb{G}_m)$.} $H^2_{\\mathrm{et}}(X,\\mathbb{G}_m)$; consult \\cite{Secondary,SecondaryII} for details. In the case of an affine cone over a smooth irreducible plane complex curve of degree $\\geq 4$, the latter \\'etale cohomology group contains non-torsion classes. The same phenomenon occurs, for example, in the case of Mumford's (celebrated) singular surface \\cite[page~75]{Mumford}; see \\cite[Example~1.32]{SecondaryII}.\n\\end{remark}\n\\begin{remark}[Jacques Tits' motivic measure]\nThe Grothendieck ring of varieties $K_0\\mathrm{Var}(k)$, introduced in a letter from Grothendieck to Serre in the sixties, is defined as the quotient of the free abelian group on the set of isomorphism classes of $k$-schemes by the ``cut-and-paste'' relations. Although very important, the structure of this ring still remains poorly understood. Among other ingredients, Theorem \\ref{thm:new} was used in the construction of a new motivic measure $\\mu_T$ entitled {\\em Tits motivic measure}; consult \\cite{Tits} for details. This new motivic measure led to the proof of several new structural properties of $K_0\\mathrm{Var}(k)$. For example, making use of $\\mu_T$, it was proved in {\\em loc. cit.} that two quadric hypersurfaces (or more generally involution varieties), associated to quadratic forms of degree $6$, have the same Grothendieck class if and only if they are isomorphic. In the same vein, it was proved in {\\em loc. cit.} that two products of conics have the same Grothendieck class if and only if they are isomorphic; this refines a previous result of Koll\\'ar \\cite{Kollar}.\n\\end{remark}\n\\subsection{Semi-simplicity}\\label{sub:semi-simple}\nLet $F$ be a field of characteristic zero. The following result is obtained by combining \\cite[Thm.~4.27]{book} with \\cite[Thm.~1.1]{positive}:\n\\begin{theorem}\\label{thm:semi-simple}\nThe category $\\NNum(k)_F$ is abelian semi-simple.\n\\end{theorem}\nAssuming certain (polarization) conjectures, Kontsevich conjectured in his seminal talk \\cite{IAS} that the category $\\NNum(k)_F$ was abelian semi-simple. Theorem \\ref{thm:semi-simple} not only proves this conjecture but moreover shows that Kontsevich's insight holds unconditionally. Let $\\Num(k)_F$ be the (classical) category of numerical motives; see \\cite[\\S4]{Andre}. The next result is obtained by combining \\cite[Rk.~4.32]{book} with \\cite[Cor.~1.2]{positive}:\n\\begin{corollary}\\label{cor:semi-simple}\nThe category $\\Num(k)_F$ is abelian semi-simple.\n\\end{corollary}\nAssuming certain (standard) conjectures, Grothendieck conjectured in the sixties that the category $\\Num(k)_F$ was abelian semi-simple. This conjecture was proved unconditionally by Jannsen \\cite{Jannsen} in the nineties using \\'etale cohomology. Corollary \\ref{cor:semi-simple} provides us with an alternative proof of Grothendieck's conjecture\n\\subsubsection{Numerical Grothendieck group}\nThe Grothendieck group $K_0({\\mathcal A})$ of a proper dg category ${\\mathcal A}$ comes equipped with the following Euler bilinear pairing:\n\\begin{eqnarray*}\\label{eq:pairing}\n\\chi \\colon K_0({\\mathcal A}) \\times K_0({\\mathcal A}) \\longrightarrow \\mathbb{Z} && ([M],[N]) \\mapsto \\sum_n (-1)^n \\mathrm{dim}_k \\mathrm{Hom}_{{\\mathcal D}_c({\\mathcal A})}(M,N[n])\\,.\n\\end{eqnarray*}\nThis bilinear pairing is, in general, not symmetric neither skew-symmetric. Nevertheless, when ${\\mathcal A}$ is moreover smooth the associated left and right kernels of $\\chi$ agree; see \\cite[Prop.~4.24]{book}. Consequently, under these assumptions on ${\\mathcal A}$, we have a well-defined {\\em numerical Grothendieck group} $K_0({\\mathcal A})\/_{\\!\\!\\sim \\mathrm{num}}:=K_0({\\mathcal A})\/\\mathrm{Ker}(\\chi)$. Following \\cite[Thm.~4.26]{book}, given smooth proper dg categories ${\\mathcal A}$ and ${\\mathcal B}$, we have isomorphisms:\n\\begin{equation}\\label{eq:natural-iso}\n\\mathrm{Hom}_{\\NNum(k)}(U({\\mathcal A}),U({\\mathcal B})) \\simeq K_0({\\mathcal A}^\\mathrm{op} \\otimes {\\mathcal B})\/\\mathrm{Ker}(\\chi)\\,.\n\\end{equation}\nThe next result, whose proof makes use of Theorem \\ref{thm:semi-simple}, is obtained by combining \\cite[Thm.~1.2]{Separable} with \\cite[Thm.~6.2]{positive}:\n\\begin{theorem}\\label{thm:free}\n$K_0({\\mathcal A})\/_{\\!\\!\\sim \\mathrm{num}}$ is a finitely generated free abelian group.\n\\end{theorem}\nGiven a smooth proper $k$-scheme $X$, let us write ${\\mathcal Z}^\\ast(X)\/_{\\!\\!\\sim \\mathrm{num}}$ for the (graded) group of algebraic cycles on $X$ up to numerical equivalence. By combining Theorem \\ref{thm:free} with the Hirzebruch-Riemann-Roch theorem, we obtain the following result:\n\\begin{corollary}\n${\\mathcal Z}^\\ast(X)\/_{\\!\\!\\sim \\mathrm{num}}$ is a finitely generated free abelian (graded) group.\n\\end{corollary}\n\\subsection{Rigidity}\\label{sub:rigidity}\nRecall that a field extension $l\/k$ is called {\\em primary} if the algebraic closure of $k$ in $l$ is purely inseparable over $k$. When $k$ is algebraically closed, every field extension $l\/k$ is primary.\n\\begin{theorem}[{\\cite[Thm.~2.1(i)]{rigidity}}]\\label{thm:rigidity}\nGiven a primary field extension $l\/k$ and a field $F$ of characteristic zero, the base-change functor $-\\otimes_k l: \\NNum(k)_F \\to \\NNum(l)_F$ is fully-faithful. The same holds integrally when $k$ is algebraically closed. \n\\end{theorem}\nIntuitively speaking, Theorem \\ref{thm:rigidity} shows that the theory of noncommutative numerical motives is ``rigid'' under base-change along primary field extensions. Alternatively, thanks to the isomorphisms \\eqref{eq:natural-iso}, Theorem \\ref{thm:rigidity} shows that the numerical Grothendieck group is ``rigid'' under primary field extensions. The commutative counterpart, resp. mixed analogue, of Theorem \\ref{thm:rigidity} was established by Kahn in \\cite[Prop.~5.5]{Kahn2}, resp. is provided by Theorem \\ref{thm:rigidity2}.\n\\begin{remark}[Extra functoriality]\nLet $l\/k$ be a primary field extension. As proved in \\cite[Thm.~2.3]{rigidity}, Theorems \\ref{thm:semi-simple} and \\ref{thm:rigidity} imply that the base-change functor admits a left=right adjoint. Without the assumption that the field extension $l\/k$ is primary, such an adjoint functor does {\\em not} exists in general; consult \\cite[Rk.~2.4]{rigidity} for details.\n\\end{remark}\n\\subsection{Zeta functions of endomorphisms}\\label{sub:zeta}\nLet $N\\!\\!M \\in \\NChow(k)_\\mathbb{Q}$ be a noncommutative Chow motive and $f$ an endomorphism of $N\\!\\!M$. Following Kahn \\cite[Def.~3.1]{Zeta}, the {\\em zeta function of $f$} is defined as the following formal power series\n\\begin{equation}\\label{eq:zeta}\nZ(f;t):= \\mathrm{exp}\\left(\\sum_{n \\geq 1} \\mathrm{tr}(f^{\\circ n}) \\frac{t^n}{n}\\right) \\in \\mathbb{Q}\\llbracket t \\rrbracket\\,,\n\\end{equation}\nwhere $f^{\\circ n}$ stands for the composition of $f$ with itself $n$-times, $\\mathrm{tr}(f^{\\circ n}) \\in \\mathbb{Q}$ stands for the categorical trace of $f^{\\circ n}$, and $\\mathrm{exp}(t):=\\sum_{m \\geq 0} \\frac{t^m}{m!} \\in \\mathbb{Q}\\llbracket t\\rrbracket$. \n\\begin{remark}\nWhen $N\\!\\!M = U({\\mathcal A})_\\mathbb{Q}$ and $f=[\\mathrm{B}]_\\mathbb{Q}$, with $\\mathrm{B} \\in {\\mathcal D}_c({\\mathcal A}^\\mathrm{op} \\otimes {\\mathcal A})$ a dg ${\\mathcal A}\\text{-}{\\mathcal A}$-bimodule (see \\S\\ref{sub:recollections}), we have the following computation\n\\begin{equation}\\label{eq:integers}\n\\mathrm{tr}(f^{\\circ n}) =[HH({\\mathcal A}; \\underbrace{\\mathrm{B}\\otimes^{\\bf L}_{\\mathcal A} \\cdots \\otimes^{\\bf L}_{\\mathcal A} \\mathrm{B}}_{n\\text{-}\\text{times}})] \\in K_0(k) \\simeq \\mathbb{Z}\\,,\n\\end{equation}\nwhere $HH({\\mathcal A}; \\mathrm{B}\\otimes^{\\bf L}_{\\mathcal A} \\cdots \\otimes^{\\bf L}_{\\mathcal A} \\mathrm{B})$ stands for the Hochschild homology of ${\\mathcal A}$ with coefficients in $\\mathrm{B}\\otimes^{\\bf L}_{\\mathcal A} \\cdots \\otimes^{\\bf L}_{\\mathcal A} \\mathrm{B}$; see \\cite[Prop.~2.26]{book}. Intuitively speaking, the integer \\eqref{eq:integers} is the ``number of fixed points'' of the dg ${\\mathcal A}\\text{-}{\\mathcal A}$-bimodule $\\mathrm{B}\\otimes^{\\bf L}_{\\mathcal A} \\cdots \\otimes^{\\bf L}_{\\mathcal A} \\mathrm{B}$.\n\\end{remark}\n\\begin{example}[Zeta function]\\label{ex:schemes}\nLet $k=\\mathbb{F}_q$ be a finite field, $X$ a smooth proper $k$-scheme, and $\\mathrm{Fr}$ the geometric Frobenius. When ${\\mathcal A}= \\mathrm{perf}_\\mathrm{dg}(X)$ and $\\mathrm{B}$ is the dg bimodule associated to the pull-back dg functor $\\mathrm{Fr}^\\ast\\colon \\mathrm{perf}_\\mathrm{dg}(X) \\to \\mathrm{perf}_\\mathrm{dg}(X)$, \\eqref{eq:integers} reduces to $[HH(X;\\Gamma_{\\mathrm{Fr}^{\\circ n}})]=\\langle \\Delta \\cdot \\Gamma_{\\mathrm{Fr}^{\\circ n}}\\rangle= |X(\\mathbb{F}_{q^n})|$. Consequently, \\eqref{eq:zeta} reduces to the (classical) zeta function $Z_X(t):= \\mathrm{exp}(\\sum_{n \\geq 1} |X(\\mathbb{F}_{q^n})| \\frac{t^n}{n})$ of $X$.\n\\end{example}\n\\begin{remark}[Witt vectors]\nRecall from \\cite{Hazewinkel} the definition of the ring of (big) Witt vectors $\\mathrm{W}(\\mathbb{Q})=(1 + t \\mathbb{Q}\\llbracket t \\rrbracket, \\times, \\ast)$. Since the leading term of \\eqref{eq:zeta} is equal to $1$, the zeta function $Z(f;t)$ of $f$ belongs to $\\mathrm{W}(\\mathbb{Q})$. Moreover, given endomorphisms $f$ and $f'$ of noncommutative Chow motives $N\\!\\!M$ and $N\\!\\!M'$, we have $Z(f\\oplus f';t)=Z(f;t) \\times Z(f';t)$ and $Z(f\\otimes f'; t) = Z(f;t) \\ast Z(f';t)$ in $\\mathrm{W}(\\mathbb{Q})$.\n\\end{remark}\nLet $B = \\prod_i B_i$ be a finite-dimensional semi-simple $\\mathbb{Q}$-algebra, $Z_i$ the center of $B_i$, $\\delta_i$ for the degree $[Z_i:\\mathbb{Q}]$, and $d_i$ the index $[B_i: Z_i]^{1\/2}$. Given a unit $b \\in B^\\times$, its {\\em $i^{\\mathrm{th}}$ reduced norm} $\\mathrm{Nrd}_i(b) \\in \\mathbb{Q}$ is defined as the composition $(\\mathrm{N}_{Z_i\/\\mathbb{Q}}\\circ \\mathrm{Nrd}_{B_i\/Z_i})(b_i)$. \n\nLet $N\\!\\!M \\in \\NChow(k)_\\mathbb{Q}$ be a noncommutative Chow motive. Thanks to Theorem \\ref{thm:semi-simple}, $B:=\\mathrm{End}_{\\NNum(k)_\\mathbb{Q}}(N\\!\\!M)$ is a finite-dimensional semi-simple $\\mathbb{Q}$-algebra; let us write $e_i \\in B$ for the central idempotent corresponding to the summand $B_i$. Given an invertible endomorphism $f$ of $N\\!\\!M$, its {\\em determinant $\\mathrm{det}(f) \\in \\mathbb{Q}$} is defined as the following product $\\prod_i \\mathrm{Nrd}_i(f)^{\\mu_i}$, where $\\mu_i :=\\frac{\\mathrm{tr}(e_i)}{\\delta_i d_i}$.\n\\begin{theorem}[{\\cite[Thm.~5.8]{positive}}]\\label{thm:zeta}\n\\begin{itemize}\n\\item[(i)] The series $Z(f;t) \\in \\mathbb{Q} \\llbracket t \\rrbracket$ is {\\em rational}, \\textsl{i.e.}\\ $Z(f;t)=\\frac{p(t)}{q(t)}$ with $p(t), q(t) \\in \\mathbb{Q}[t]$. Moreover, $\\mathrm{deg}(q(t)) - \\mathrm{deg}(p(t))= \\mathrm{tr}(\\id_{N\\!\\!M})$. \n\\item[(ii)] When $f$ is invertible, we have the following functional equation:\n$$ Z(f^{-1};t^{-1}) = (-t)^{\\mathrm{tr}(\\id_{N\\!\\!M})} \\mathrm{det}(f) Z(f;t)\\,.$$\n\\end{itemize}\n\\end{theorem}\n\\begin{corollary}[Weil conjectures]\\label{cor:zeta}\nLet $k=\\mathbb{F}_q$ be a finite field, $X$ a smooth proper $k$-scheme $X$ of dimension $d$, and $\\mathrm{E}:=\\langle \\Delta \\cdot \\Delta \\rangle \\in \\mathbb{Z}$ the self-intersection number of the diagonal $\\Delta$ of $X \\times X$. \n\\begin{itemize}\n\\item[(i)] The zeta function $Z_X(t)$ of $X$ is rational. Moreover, $\\mathrm{deg}(q(t))-\\mathrm{deg}(p(t))= \\mathrm{E}$.\n\\item[(ii)] We have the following functional equation $Z_X(\\frac{1}{q^d t}) = \\pm t^{\\mathrm{E}} q^{\\frac{d}{2} \\mathrm{E}} Z_X(t)$.\n\\end{itemize}\n\\end{corollary}\nWeil conjectured\\footnote{Weil conjectured also that the zeta function $Z_X(t)$ of $X$ satisfied an analogue of the Riemann hypothesis. This conjecture was proved by Deligne \\cite{Deligne-IHES} using, among other tools, Lefschetz pencils.} in \\cite{Weil} that the zeta function $Z_X(t)$ of $X$ was rational and that it satisfied a functional equation. These conjectures were proved independently by Dwork \\cite{Dwork} and Grothendieck \\cite{Grothendieck-Bourbaki} using $p$-adic analysis and \\'etale cohomology, respectively. Corollary \\ref{cor:zeta} provides us with an alternative proof of the Weil conjectures; see \\cite[Cor.~5.12]{positive}. Moreover, Theorem \\ref{thm:zeta} proves a far-reaching noncommutative generalization of the Weil conjectures. \n\\subsection{Equivariant noncommutative motives}\\label{sub:equivariant}\nLet $G$ be a finite group of order $n$ (we assume that $1\/n \\in k$). Recall from \\cite[Def.~4.1]{Equivariant} the definition of a {\\em $G$-action} on a dg category ${\\mathcal A}$. Given a $G$-action $G \\circlearrowright {\\mathcal A}$, we have an associated dg category ${\\mathcal A}^G$ of $G$-equivariant objects. From a topological viewpoint, ${\\mathcal A}^{G}$ may be understood as the ``homotopy fixed points'' of the $G$-action on ${\\mathcal A}$. Here are two examples:\n\\begin{example}[$G$-schemes]\nGiven a $G$-scheme $X$, the dg category $\\mathrm{perf}_\\mathrm{dg}(X)$ inherits a $G$-action. In this case, the dg category $\\mathrm{perf}_\\mathrm{dg}(X)^G$ is Morita equivalent to the dg category of $G$-equivariant perfect complexes $\\mathrm{perf}^{G}_\\mathrm{dg}(X)=\\mathrm{perf}_\\mathrm{dg}([X\/G])$.\n\\end{example}\n\\begin{example}[Cohomology classes]\nGiven a cohomology class $[\\alpha] \\in H^2(G,k^\\times)$, the dg category $k$ inherits a $G$-action $G \\circlearrowright_\\alpha k$. In this case, the dg category of $G$-equivariant objects is Morita equivalent to the twisted group algebra $k_\\alpha[G]$. \n\\end{example}\nLet $\\mathsf{dgcat}^{G}(k)$ be the category of (small) dg categories equipped with a $G$-action, and $\\mathsf{dgcat}^{G}_{\\mathrm{sp}}(k)$ the full subcategory of smooth proper dg categories. As explained in \\cite[\\S5]{Equivariant}, the category $\\NChow(k)$ admits a $G$-equivariant counterpart $\\NChow^{G}(k)$. Recall from {\\em loc. cit.} that the latter category is additive, rigid symmetric monoidal, and comes equipped with a symmetric monoidal functor $U^{G}\\colon \\mathsf{dgcat}_{\\mathrm{sp}}^{G}(k) \\to \\NChow^{G}(k)$. Moreover, we have isomorphisms\n$$\\mathrm{Hom}_{\\NChow^{G}(k)}(U^{G}(G \\circlearrowright {\\mathcal A}), U^{G}(G \\circlearrowright {\\mathcal B})) \\simeq K_0^{G}({\\mathcal A}^\\mathrm{op} \\otimes {\\mathcal B})\\,,$$ \nwhere the right-hand side stands for the $G$-equivariant Grothendieck group. In particular, the ring of endomorphisms of the $\\otimes$-unit $U^{G}(G \\circlearrowright_1 k)$ agrees with the representation ring\\footnote{Recall that when $k=\\mathbb{C}$ and $G$ is abelian, we have an isomorphism $R(G)\\simeq \\mathbb{Z}[\\widehat{G}]$.} $R(G)$. Let us write $I$ for the augmentation ideal associated to the rank homomorphism $R(G) \\twoheadrightarrow \\mathbb{Z}$.\n\\subsubsection{Relation with equivariant Chow motives}\nMaking use of Edidin-Graham's work \\cite{EG} on equivariant intersection theory, Laterveer \\cite{Laterveer}, and Iyer and M\\\"uller-Stach \\cite{Iyer-Muller}, extended the theory of Chow motives to the $G$-equivariant setting. In particular, they constructed a category of $G$-equivariant Chow motives $\\mathrm{Chow}^{G}(k)$ and a (contravariant) symmetric monoidal functor $\\mathfrak{h}^{G}\\colon \\mathrm{SmProj}^{G}(k) \\to \\mathrm{Chow}^{G}(k)$, defined on smooth projective $G$-schemes.\n\\begin{theorem}[{\\cite[Thm.~8.4]{Equivariant}}]\\label{thm:bridge-equivariant}\nThere exists a $\\mathbb{Q}$-linear, fully-faithful, symmetric monoidal $\\Phi_\\mathbb{Q}^{G}$ making the following diagram commute\n\\begin{equation*\n\\xymatrix{\n\\mathrm{SmProj}^{G}(k) \\ar[rrr]^-{X \\mapsto G \\circlearrowright \\mathrm{perf}_\\mathrm{dg}(X)} \\ar[d]_-{\\mathfrak{h}^{G}(-)_\\mathbb{Q}} &&& \\mathsf{dgcat}_{\\mathrm{sp}}^{G}(k) \\ar[d]^-{U^{G}(-)_\\mathbb{Q}} \\\\\n\\mathrm{Chow}^{G}(k)_\\mathbb{Q} \\ar[d] &&& \\NChow^{G}(k)_\\mathbb{Q} \\ar[d]^-{(-)_{I_\\mathbb{Q}}} \\\\\n\\mathrm{Chow}^{G}(k)_\\mathbb{Q}\/_{\\!-\\otimes \\mathbb{Q}(1)} \\ar[rrr]_-{\\Phi_\\mathbb{Q}^{G}} &&& \\NChow^{G}(k)_{\\mathbb{Q}, I_\\mathbb{Q}}\\,,\n}\n\\end{equation*}\nwhere $\\mathrm{Chow}^{G}(k)_\\mathbb{Q}\/_{\\!-\\otimes \\mathbb{Q}(1)}$ stands for the orbit category of $\\mathrm{Chow}^{G}(k)_\\mathbb{Q}$ with respect to the $G$-equivariant Tate motive $\\mathbb{Q}(1)$ (see \\cite[\\S4.2]{book}), and $(-)_{I_\\mathbb{Q}}$ for the localization functor associated to the augmentation ideal $I_\\mathbb{Q}$.\n\\end{theorem}\nRoughly speaking, Theorem \\ref{thm:bridge-equivariant} shows that in order to compare the equivariant commutative world with the equivariant noncommutative world, we need to ``$\\otimes$-trivialize'' the $G$-equivariant Tate motive $\\mathbb{Q}(1)$ on one side and to localize at the augmentation ideal $I_\\mathbb{Q}$ on the other side. Only after these two reductions, the equivariant commutative world embeds fully-faithfully into the equivariant noncommutative world. As illustrated in \\S\\ref{sub:exceptional}, this shows that the $G$-equivariant Chow motive $\\mathfrak{h}^{G}(X)_\\mathbb{Q}$ and the $G$-equivariant noncommutative Chow motive $U^{G}(G \\circlearrowright \\mathrm{perf}_\\mathrm{dg}(X))$ contain (important) independent information~about~$X$.\n\\subsubsection{Full exceptional collections}\\label{sub:exceptional}\nLet $X$ be a smooth projective $G$-scheme. In order to study it, we can proceed into two distinct directions. On one direction, we can associate to $X$ its $G$-equivariant Chow motive $\\mathfrak{h}^{G}(X)_\\mathbb{Q}$. On another direction, we can associate to $X$ the $G$-action $G \\circlearrowright\\mathrm{perf}_\\mathrm{dg}(X)$. The following result, whose proof makes use of Theorem \\ref{thm:bridge-equivariant}, relates these two distinct directions of study:\n\\begin{theorem}[{\\cite[Thm.~1.2]{Equivariant}}]\\label{thm:exceptional}\nIf the category $\\mathrm{perf}(X)$ admits a full exceptional collection $({\\mathcal E}_1, \\ldots, {\\mathcal E}_n)$ of $G$-invariant objects ($\\neq$ $G$-equivariant objects), then there exists a choice of integers $r_1, \\ldots, r_n \\in \\{0, \\ldots, \\mathrm{dim}(X)\\}$ such that \n\\begin{equation}\\label{eq:decomp-motivic}\n\\mathfrak{h}^{G}(X)_\\mathbb{Q} \\simeq \\mathbb{L}^{\\otimes r_1} \\oplus \\cdots \\oplus \\mathbb{L}^{\\otimes r_n}\\,,\n\\end{equation}\nwhere $\\mathbb{L}\\in \\mathrm{Chow}^{G}(k)_\\mathbb{Q}$ stands for the $G$-equivariant Lefschetz motive.\n\\end{theorem}\nTheorem \\ref{thm:exceptional} can be applied, for example, to any $G$-action on projective spaces, quadrics, Grassmannians, etc; consult \\cite[Examples 9.9-9.11]{Equivariant} for details. Morally speaking, Theorem \\ref{thm:exceptional} shows that the existence of a full exceptional collection of $G$-invariant objects completely determines the $G$-equivariant Chow motive $\\mathfrak{h}^{G}(X)_\\mathbb{Q}$. In particular, $\\mathfrak{h}^{G}(X)_\\mathbb{Q}$ loses all the information about the $G$-action on $X$. In contrast, as explained in \\cite[Rmk.~9.4 and Prop.~9.8]{Equivariant}, the $G$-invariant objects ${\\mathcal E}_1, \\ldots, {\\mathcal E}_n$ yield (non-trivial) cohomology classes $[\\alpha_1], \\ldots, [\\alpha_n] \\in H^2(G, k^\\times)$ such that \n\\begin{equation}\\label{eq:motivic-decomp-NC}\nU^{G}(G \\circlearrowright\\mathrm{perf}_\\mathrm{dg}(X))\\simeq U^{G}(G \\circlearrowright_{\\alpha_1} k) \\oplus \\cdots \\oplus U^{G}(G \\circlearrowright_{\\alpha_n} k)\\,.\n\\end{equation}\nTaking into account \\eqref{eq:decomp-motivic}-\\eqref{eq:motivic-decomp-NC}, the $G$-equivariant Chow motive $\\mathfrak{h}^{G}(X)_\\mathbb{Q}$ and the $G$-equivariant noncommutative Chow motive $U^{G}(G \\circlearrowright \\mathrm{perf}_\\mathrm{dg}(X))$ should be considered as complementary. While the former keeps track of the Tate twists but not of the $G$-action, the latter keeps track of the $G$-action but not of the Tate twists.\n\\section{Noncommutative (standard) conjectures}\\label{sec:conjectures}\nIn \\S\\ref{sub:recollections2} we recall some important conjectures of Grothendieck, Voevodsky, and Tate. Subsection \\S\\ref{sub:counterparts} is devoted to their noncommutative counterparts. As a first application of the noncommutative viewpoint, we prove that the original conjectures of Grothendieck, Voevodsky, and Tate, are invariant under homological projective duality. This leads to a proof of these original conjectures in several new cases. As a second application, we extend the original conjectures from schemes to algebraic stacks and prove them in the case of ``low-dimensional'' orbifolds.\n\\subsection{Recollections}\\label{sub:recollections2}\nLet $k$ be a base field of characteristic zero. Given a smooth proper $k$-scheme $X$ and a Weil cohomology theory $H^\\ast$, let us write $\\pi_X^n$ for the $n^{\\mathrm{th}}$ K\\\"unneth projector of $H^\\ast(X)$, $Z^\\ast(X)_\\mathbb{Q}$ for the $\\mathbb{Q}$-vector space of algebraic cycles on $X$, and $Z^\\ast(X)_\\mathbb{Q}\/_{\\!\\sim \\mathrm{nil}}$, $Z^\\ast(X)_\\mathbb{Q}\/_{\\!\\sim \\mathrm{hom}}$, and $Z^\\ast(X)_\\mathbb{Q}\/_{\\!\\sim \\mathrm{num}}$, for the quotient of $Z^\\ast(X)_\\mathbb{Q}$ with respect to the smash-nilpotence, homological, and numerical equivalence relation, respectively. Recall from \\cite[\\S3.0.8-3.0.11]{book} that:\n\\begin{itemize}\n\\item[(i)] The Grothendieck's standard conjecture\\footnote{The standard conjecture of type $C^+$ is also known as the {\\em sign conjecture}. If the even K\\\"unneth projector $\\pi_X^+$ is algebraic, then the odd K\\\"unneth projector $\\pi^-_X:=\\sum_n \\pi_X^{2n+1}$ is also algebraic.} of type $C^+$, denoted by $C^+(X)$, asserts that the even K\\\"unneth projector $\\pi^+_X:=\\sum_n \\pi^{2n}_X$ is algebraic.\n\\item[(ii)] The Grothendieck's standard conjecture of type $D$, denoted by $D(X)$, asserts that $Z^\\ast(X)_\\mathbb{Q}\/_{\\!\\sim \\mathrm{hom}}=Z^\\ast(X)_\\mathbb{Q}\/_{\\!\\sim \\mathrm{num}}$.\n\\item[(iii)] The Voevodsky's nilpotence conjecture $V(X)$ (which implies Grothendieck's conjecture $D(X)$) asserts that $Z^\\ast(X)_\\mathbb{Q}\/_{\\!\\sim \\mathrm{nil}}=Z^\\ast(X)_\\mathbb{Q}\/_{\\!\\sim \\mathrm{num}}$.\n\\item[(iv)] The Schur-finiteness conjecture\\footnote{Consult \\S\\ref{sec:Schur} for the mixed analogue of the Schur-finiteness conjecture.}, denoted by $S(X)$, asserts that the Chow motive $\\mathfrak{h}(X)_\\mathbb{Q}$ is Schur-finite in the sense of Deligne \\cite[\\S1]{Deligne}.\n\\end{itemize}\n\\begin{remark}[Status]\n\\begin{itemize}\n\\item[(i)] Thanks to the work of Grothendieck and Kleiman (see \\cite{Grothendieck,Kleim1,Kleim}), the conjecture $C^+(X)$ holds when $\\mathrm{dim}(X) \\leq 2$, and also for abelian varieties. Moreover, this conjecture is stable under products.\n\\item[(ii)] Thanks to the work of Lieberman \\cite{Lieberman}, the conjecture $D(X)$ holds when $\\mathrm{dim}(X)\\leq 4$, and also for abelian varieties.\n\\item[(iii)] Thanks to the work Voevodsky \\cite{Voevodsky-IMRN} and Voisin \\cite{Voisin-nilpotence}, the conjecture $V(X)$ holds when $\\mathrm{dim}(X)\\leq 2$. Thanks to the work of Kahn-Sebastian \\cite{KS}, the conjecture $V(X)$ holds moreover when $X$ is an abelian $3$-fold.\n\\item[(iv)] Thanks to the work of Kimura \\cite{Kimura} and Shermenev \\cite{Shermenev}, the conjecture $S(X)$ holds when $\\mathrm{dim}(X)\\leq 1$, and also for abelian varieties.\n\\end{itemize}\n\\end{remark}\nLet $k=\\mathbb{F}_q$ be a finite base field of characteristic $p>0$. Given a smooth proper $k$-scheme $X$ and a prime number $l\\neq p$, recall from \\cite{Tate-motives, Tate} that the Tate conjecture, denoted by $T^l(X)$, asserts that the cycle class map is surjective:\n$$\n{\\mathcal Z}^\\ast(X)_{\\mathbb{Q}_l} \\longrightarrow H^{2\\ast}_{l\\text{-}\\text{adic}}(X_{\\overline{k}}, \\mathbb{Q}_l(\\ast))^{\\mathrm{Gal}(\\overline{k}\/k)}\\,.\n$$\n\\begin{remark}[Status]\nThanks to the work of Tate \\cite{Tate}, the conjecture $T^l(X)$ holds when $\\mathrm{dim}(X)\\leq 1$, and also for abelian varieties. Thanks to the work of several other people (consult Totaro's survey \\cite{Totaro}), the conjecture $T^l(X)$ holds moreover when $X$ is a $K3$-surface (and $p\\neq 2$).\n\\end{remark}\n\\subsection{Noncommutative counterparts}\\label{sub:counterparts}\nLet $k$ be a base field of characteristic zero. Recall from \\S\\ref{sec:pure} that periodic cyclic homology descends to the category of noncommutative Chow motives yielding a functor $HP^\\pm\\colon \\NChow(k)_\\mathbb{Q} \\to \\mathrm{Vect}_{\\mathbb{Z}\/2}(k)$. Given a smooth proper dg category ${\\mathcal A}$, consider the following $\\mathbb{Q}$-vector spaces\n$$\nK_0({\\mathcal A})_\\mathbb{Q}\/_{\\!\\sim ?}:= \\mathrm{Hom}_{?}(U(k)_\\mathbb{Q}, U({\\mathcal A})_\\mathbb{Q})\\,,\n$$\nwhere $?$ belongs to $\\{\\mathrm{nil}, \\mathrm{hom}, \\mathrm{num}\\}$ and $\\{\\mathrm{NVoev}(k)_\\mathbb{Q}, \\mathrm{NHom}(k)_\\mathbb{Q}, \\NNum(k)_\\mathbb{Q}\\}$, respectively. Under these notations, the important conjectures in \\S\\ref{sub:recollections2} admit the following noncommutative counterparts:\n\n\\vspace{0.1cm}\n\n{\\bf Conjecture $C^+_{\\mathrm{nc}}({\\mathcal A})$:} The even K\\\"unneth projector $\\pi^+_{\\mathcal A}$ of $HP^\\pm({\\mathcal A})$ is {\\em algebraic}, \\textsl{i.e.}\\ there exists an endomorphism $\\underline{\\pi}^+_{\\mathcal A}$ of $U({\\mathcal A})_\\mathbb{Q}$ such that $HP^\\pm(\\underline{\\pi}^+_{\\mathcal A})=\\pi^+_{\\mathcal A}$.\n\n\\vspace{0.1cm}\n\n{\\bf Conjecture $D_{\\mathrm{nc}}({\\mathcal A})$:} The equality $K_0({\\mathcal A})_\\mathbb{Q}\/_{\\!\\sim \\mathrm{hom}}=K_0({\\mathcal A})_\\mathbb{Q}\/_{\\!\\sim \\mathrm{num}}$ holds.\n\n\\vspace{0.1cm}\n\n{\\bf Conjecture $V_{\\mathrm{nc}}({\\mathcal A})$:} The equality $K_0({\\mathcal A})_\\mathbb{Q}\/_{\\!\\sim \\mathrm{nil}}=K_0({\\mathcal A})_\\mathbb{Q}\/_{\\!\\sim \\mathrm{num}}$ holds.\n\n\\vspace{0.1cm}\n\n{\\bf Conjecture $S_{\\mathrm{nc}}({\\mathcal A})$:} The noncommutative Chow motive $U({\\mathcal A})_\\mathbb{Q}$ is Schur-finite.\n\n\\vspace{0.1cm}\n\nLet $k=\\mathbb{F}_q$ be a finite base field of characteristic $p>0$. Given a smooth proper dg category ${\\mathcal A}$ and a prime number $l\\neq p$, consider the following abelian groups \n\\begin{eqnarray}\\label{eq:abeliangroups}\n\\mathrm{Hom}\\left(\\mathbb{Z}(l^\\infty), \\pi_{-1} L_{KU}K({\\mathcal A}\\otimes_{\\mathbb{F}_q} \\mathbb{F}_{q^n})\\right) && n \\geq 1\\,,\n\\end{eqnarray}\nwhere $\\mathbb{Z}(l^\\infty)$ stands for the Pr\\\"ufer $l$-group and $L_{KU}K({\\mathcal A}\\otimes_k k_n)$ for the Bousfield localization of the algebraic $K$-theory spectrum $K({\\mathcal A}\\otimes_{\\mathbb{F}_q} \\mathbb{F}_{q^n})$ with respect to topological complex $K$-theory $KU$. Under these notations, Tate's conjecture admits the following noncommutative counterpart:\n\n\\vspace{0.1cm}\n\n{\\bf Conjecture $T^l_{\\mathrm{nc}}({\\mathcal A})$:} The abelian groups \\eqref{eq:abeliangroups} are zero.\n\n\\vspace{0.1cm}\n\nWe now relate the conjectures in \\S\\ref{sub:recollections2} with their noncommutative counterparts:\n\\begin{theorem}\\label{thm:conjectures}\nGiven a smooth proper $k$-scheme $X$, we have the equivalences:\n\\begin{eqnarray}\nC^+(X) & \\Leftrightarrow & C^+_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}(X)) \\label{eq:equiv1} \\\\\nD(X) & \\Leftrightarrow & D_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}(X)) \\label{eq:equiv2} \\\\\nV(X) & \\Leftrightarrow & V_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}(X)) \\label{eq:equiv3} \\\\\nS(X) & \\Leftrightarrow & S_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}(X))\\label{eq:equiv4} \\\\\nT^l(X) & \\Leftrightarrow & T^l_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}(X))\\label{eq:equiv5}\\,.\n\\end{eqnarray}\n\\end{theorem}\nMorally speaking, Theorem \\ref{thm:conjectures} shows that the important conjectures in \\S\\ref{sub:recollections2} belong not only to the realm of algebraic geometry but also to the broad noncommutative setting of smooth proper dg categories. Consult \\cite[\\S5]{book}, and the references therein, for the implications $\\Rightarrow$ in \\eqref{eq:equiv1}-\\eqref{eq:equiv2} and also for the equivalences \\eqref{eq:equiv3}-\\eqref{eq:equiv4}. The converse implications $\\Leftarrow$ in \\eqref{eq:equiv1}-\\eqref{eq:equiv2} were established in \\cite[Thm.~1.1]{note-CD}. Finally, the equivalence \\eqref{eq:equiv5} was proved in \\cite[Thm.~1.2]{Tate_Tabuada}.\n\\subsubsection{Homological projective duality}\\label{sub:HPD}\nFor a survey on homological projective duality (=HPD), we invite the reader to consult Kuznetsov's ICM address \\cite{ICM-Kuznetsov}. Let $X$ be a smooth projective $k$-scheme equipped with a line bundle ${\\mathcal L}_X(1)$; we write $X \\to \\mathbb{P}(W)$ for the associated morphism where $W:=H^0(X,{\\mathcal L}_X(1))^\\ast$. Assume that the category $\\mathrm{perf}(X)$ admits a Lefschetz decomposition $\\langle \\mathbb{A}_0, \\mathbb{A}_1(1), \\ldots, \\mathbb{A}_{i-1}(i-1)\\rangle$ with respect to ${\\mathcal L}_X(1)$ in the sense of \\cite[Def.~4.1]{KuznetsovHPD}. Following \\cite[Def.~6.1]{KuznetsovHPD}, let $Y$ be the HP-dual of $X$, ${\\mathcal L}_Y(1)$ the HP-dual line bundle, and $Y\\to \\mathbb{P}(W^\\ast)$ the morphism associated to ${\\mathcal L}_Y(1)$. Given a linear subspace $L \\subset W^\\ast$, consider the linear sections $X_L:=X\\times_{\\mathbb{P}(W^\\ast)} \\mathbb{P}(L)$ and $Y_L:=Y \\times_{\\mathbb{P}(W)} \\mathbb{P}(L^\\perp)$. The next result, whose proof makes use of Theorem \\ref{thm:conjectures}, is obtained by concatenating \\cite[\\S5.3-5.4]{book} with \\cite[Thm.~1.4]{note-CD}\\cite[Thm.~1.1]{Schur}\\cite[Thm.~1.3]{Tate_Tabuada}:\n\\begin{theorem}[HPD-invariance\\footnote{Consult Theorem \\ref{thm:HPD2} for another HPD-invariance type result.}]\\label{thm:HPD}\nLet $X$ and $Y$ be as above. Assume that $X_L$ and $Y_L$ are smooth, that $\\mathrm{dim}(X_L)=\\mathrm{dim}(X) -\\mathrm{dim}(L)$, that $\\mathrm{dim}(Y_L)=\\mathrm{dim}(Y)- \\mathrm{dim}(L^\\perp)$, and that the following conjectures hold\n\\begin{equation}\\label{eq:conjectures}\nC^+_{\\mathrm{nc}}(\\mathbb{A}_{0, \\mathrm{dg}}) \\quad \\quad D_{\\mathrm{nc}}(\\mathbb{A}_{0, \\mathrm{dg}}) \\quad \\quad V_{\\mathrm{nc}}(\\mathbb{A}_{0, \\mathrm{dg}})\\quad \\quad S_{\\mathrm{nc}}(\\mathbb{A}_{0, \\mathrm{dg}})\\quad \\quad T^l_{\\mathrm{nc}}(\\mathbb{A}_{0, \\mathrm{dg}})\\,,\n\\end{equation}\nwhere $\\mathbb{A}_{0,\\mathrm{dg}}$ stands for the dg enhancement of $\\mathbb{A}_0$ induced by $\\mathrm{perf}_\\mathrm{dg}(X)$. Under these assumptions, we have the following equivalences of conjectures:\n\\begin{eqnarray*}\n?(X_L) \\Leftrightarrow \\,\\,?(Y_L) &\\text{with}& ?\\in \\{C^+, D, V, S, T^l\\}\\,.\n\\end{eqnarray*}\n\\end{theorem}\n\\begin{remark}\nThe conjectures \\eqref{eq:conjectures} hold, for example, whenever the triangulated category $\\mathbb{A}_0$ admits a full exceptional collection (this is the case in all the examples in the literature). Furthermore, Theorem \\ref{thm:HPD} holds more generally when $Y$ (or $X$) is singular. In this case, we need to replace $Y$ by a noncommutative resolution of singularities in the sense of \\cite[\\S2.4]{ICM-Kuznetsov}.\n\\end{remark}\nTheorem \\ref{thm:HPD} shows that the conjectures in \\S\\ref{sub:recollections2} are invariant under homological projective duality. As a consequence, we obtain the following practical result:\n\\begin{corollary}\\label{cor:HPD}\nLet $X_L$ and $Y_L$ be smooth linear sections as in Theorem \\ref{thm:HPD}.\n\\begin{itemize}\n\\item[(a)] If $\\mathrm{dim}(Y_L)\\leq 2$, then the conjectures $C^+(X_L)$ and $V(X_L)$ hold.\n\\item[(b)] If $\\mathrm{dim}(Y_L)\\leq 4$, then the conjecture $D(X_L)$ holds.\n\\item[(c)] If $\\mathrm{dim}(Y_L)\\leq 1$, then the conjectures $S(X_L)$ and $T^l(X_L)$ hold.\n\\end{itemize}\n\\end{corollary}\nBy applying Corollary \\ref{cor:HPD} to the Veronese-Clifford duality, to the spinor duality, to the Grassmannian-Pfaffian duality, to the determinantal duality, and to other (incomplete) HP-dualities (see \\cite[\\S4]{ICM-Kuznetsov}), we obtain a proof of the conjectures in \\S\\ref{sub:recollections2} in several new cases; consult \\cite{Crelle, note-CD,Schur,Tate_Tabuada} for details. In the particular case of the Veronese-Clifford duality, Corollary \\ref{cor:HPD} leads furthermore to an alternative proof of the Tate conjecture for smooth complete intersections of two quadrics (the original (geometric) proof, based on the notion of variety of maximal planes, is due to Reid \\cite{Reid}); consult \\cite[Thm.~1.7]{Tate_Tabuada} for details.\n\\subsubsection{Algebraic stacks}\\label{sub:stacks}\nTheorem \\ref{thm:conjectures} allows us to easily extend the important conjectures in \\S\\ref{sub:recollections2} from smooth proper schemes to smooth proper algebraic stacks ${\\mathcal X}$ by setting $?({\\mathcal X}):=?_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}({\\mathcal X}))$, where $?\\in \\{C^+, D, V, S, T^l\\}$. The next result, obtained by combining \\cite[Thm.~9.2]{Orbifold} with \\cite[Thm.~1.9]{Tate_Tabuada}, proves these conjectures in the case of ``low-dimensional'' orbifolds; consult \\cite{note-CD} for further examples of algebraic stacks satisfying these conjectures.\n\\begin{theorem}\\label{thm:conj-orbifold} Let $G$ be a finite group, $X$ a smooth projective $k$-scheme equipped with a $G$-action, and ${\\mathcal X}:=[X\/G]$ the associated orbifold.\n\\begin{itemize}\n\\item[(a)] The conjectures $C^+({\\mathcal X})$ and $V({\\mathcal X})$ hold when $\\mathrm{dim}(X)\\leq 2$. The conjecture $C^+({\\mathcal X})$ also holds when $G$ acts by group homomorphisms on an abelian variety.\n\\item[(b)] The conjecture $D({\\mathcal X})$ holds when $\\mathrm{dim}(X)\\leq 4$.\n\\item[(c)] The conjectures $S({\\mathcal X})$ and $T^l({\\mathcal X})$ hold when $\\mathrm{dim}(X)\\leq 1$.\n\\end{itemize}\n\\end{theorem}\nRoughly speaking, Theorem \\ref{thm:conj-orbifold} shows that the above conjectures are ``insensitive'' to the $G$-action. Among other ingredients, its proof makes use~of~Theorem~\\ref{thm:orbifold}.\n\\begin{remark}[Generalizations]\nTheorem \\ref{thm:conj-orbifold} holds more generally under the assumption that the conjectures in \\S\\ref{sub:recollections2} are satisfied by the fixed point locus $\\{X^\\sigma\\}_\\sigma$, with $\\sigma \\in {\\varphi\\!}{\/}{\\!\\sim}$. For example, the conjecture $T^l({\\mathcal X})$ also holds when $X$ is an abelian surface and the group $G=C_2$ acts by the involution $a \\mapsto -a$.\n\\end{remark}\n\\section{Noncommutative motivic Galois groups}\\label{sec:Galois}\nLet $F$ be a field of characteristic zero and $\\NNum(k)_F$ the abelian category of numerical motives. The next result was proved in \\cite[Thm.~6.4]{book} and \\cite[Thm.~7.1]{positive}:\n\\begin{theorem}\\label{thm:super}\nThe category $\\NNum(k)_F$ is super-Tannakian in the sense of Deligne \\cite{Deligne}. When $F$ is algebraically closed, $\\NNum(k)_F$ is neutral super-Tannakian.\n\\end{theorem}\nBy combining Theorem \\ref{thm:super} with Deligne's super-Tannakian formalism \\cite{Deligne}, we obtain an affine super-group $F$-scheme $\\mathrm{sGal}(\\NNum(k)_F)$ called the {\\em noncommutative motivic Galois super-group}. The following result relates this super-group with the (classical) motivic Galois super-group $\\mathrm{sGal}(\\Num(k)_F)$:\n\\begin{theorem}[{\\cite[Thm.~7.4]{positive}}]\\label{thm:Galois}\nAssume that $F$ is algebraically closed. Then, there exists a faithfully flat morphism of affine super-group $F$-schemes\n$$ \\mathrm{sGal}(\\NNum(k)_F) \\twoheadrightarrow \\mathrm{Ker} (\\mathrm{sGal}(\\Num(k)_F) \\stackrel{t^\\ast}{\\twoheadrightarrow} \\mathbb{G}_m)\\,,$$\nwhere $\\mathbb{G}_m$ stands for the multiplicative (super-)group scheme and $t$ for the inclusion of the category of Tate motives into numerical motives.\n\\end{theorem}\nTheorem \\ref{thm:Galois} was envisioned by Kontsevich; see his seminal talk \\cite{IAS}. Intuitively speaking, it shows that the ``$\\otimes$-symmetries'' of the commutative world which can be lifted to the noncommutative world are precisely those which become trivial when restricted to Tate motives. Theorem \\ref{thm:Galois} also holds when $F$ is {\\em not} algebraically closed. However, in this case the super-group schemes are only defined over a (very big) commutative $F$-algebra.\n\\begin{remark}[Simplification]\nThe analogue of Theorem \\ref{thm:Galois}, with $k$ of characteristic zero, was proved in \\cite[Thm.~6.7(ii)]{book}. However, therein we assumed the noncommutative counterparts of the standard conjectures of type $C^+$ and $D$ and moreover used Deligne-Milne's theory of Tate-triples. In contrast, Theorem \\ref{thm:Galois} is unconditional and its proof avoids the use of Tate-triples; consult \\cite[\\S7]{positive} for details.\n\\end{remark}\n\\subsection*{Base-change}\nRecall from \\cite[\\S6]{book} the definition of the (conditional\\footnote{We assume the noncommutative counterparts of the standard conjectures of type $C^+$ and $D$.}) noncommutative motivic Galois group $\\mathrm{Gal}(\\NNum^\\dagger(k)_F)$.\n\\begin{theorem}[{\\cite[Thm.~2.2]{rigidity}}]\\label{thm:change}\nGiven a primary field extension $l\/k$, the induced base-change functor $-\\otimes_k l \\colon \\NNum^\\dagger(k)_F \\to \\NNum^\\dagger(l)_F$ gives rise to a faithfully flat morphism of affine group $F$-schemes $\\mathrm{Gal}(\\NNum^\\dagger(l)_F) \\to \\mathrm{Gal}(\\NNum^\\dagger(k)_F)$.\n\\end{theorem}\nRoughly speaking, Theorem \\ref{thm:Galois} shows that every ``$\\otimes$-symmetry'' of the category of noncommutative numerical $k$-linear motives can be extended to a ``$\\otimes$-symmetry'' of the category of noncommutative $l$-linear motives. Among other ingredients, its proof makes use of Theorem \\ref{thm:rigidity}. In the particular case of an extension of algebraically closed fields $l\/k$, the commutative counterpart of Theorem \\ref{thm:change} was established by Deligne-Milne in \\cite[Prop.~6.22(b)]{DM}.\n\\section{Localizing invariants}\\label{sec:localizing}\nRecall from \\cite[\\S8.1]{book} the notion of a {\\em short exact sequence} of dg categories in the sense of Drinfeld\/Keller. In \\S\\ref{sub:ses} we describe a key structural property of these short exact sequences and explain its implications to secondary $K$-theory.\n\nRecall from \\cite[\\S8.5.1]{book} the construction of the {\\em universal localizing $\\mathbb{A}^1$-homotopy invariant} of dg categories $\\mathrm{U}\\colon \\mathsf{dgcat}(k) \\to \\mathrm{NMot}(k)$; in {\\em loc. cit.} we used the explicit notation $\\mathrm{U}^{\\mathbb{A}^1}_\\mathrm{loc}\\colon \\mathsf{dgcat}(k) \\to \\mathrm{NMot}_\\mathrm{loc}^{\\mathbb{A}^1}(k)$. In \\cite[\\S8.5.3]{book} we described the behavior of $\\mathrm{U}$ with respect to dg orbit categories and dg cluster categories. In \\S\\ref{sub:Gysin}-\\ref{sub:NCproj} we describe the behavior of $\\mathrm{U}$ with respect to open\/closed scheme decompositions, corner skew Laurent polynomial algebras, and noncommutative projective schemes. As explained in \\cite[Thm.~8.25]{book}, all the results in \\S\\ref{sub:Gysin}-\\ref{sub:NCproj} are {\\em motivic} in the sense that they hold similarly for every localizing $\\mathbb{A}^1$-homotopy invariant such as mod$\\text{-}n$ algebraic $K$-theory (when $1\/n \\in k$), homotopy $K$-theory, \\'etale $K$-theory, periodic cyclic homology\\footnote{Periodic cyclic homology is not a localizing $\\mathbb{A}^1$-homotopy invariant in the sense of \\cite[\\S8.5]{book} because it does not preserves filtered (homotopy) colimits. Nevertheless, all the results of \\S\\ref{sub:Gysin}-\\ref{sub:NCproj} hold similarly for periodic cyclic homology.} (when $\\mathrm{char}(k)=0$),~etc. The results of \\S\\ref{sub:NCproj} do {\\em not} require $\\mathbb{A}^1$-homotopy invariance and so they hold for every localizing invariant; see~Remark~\\ref{rk:localizing}.\n\n\\begin{notation} Given a $k$-scheme $X$ (or more generally an algebraic stack ${\\mathcal X}$), we will write $\\mathrm{U}(X)$ instead of $\\mathrm{U}(\\mathrm{perf}_\\mathrm{dg}(X))$.\n\\end{notation}\n\\subsection{Short exact sequences}\\label{sub:ses}\nRecall from \\cite[\\S8.4]{book} the notion of a {\\em split short exact sequence} of dg categories $0 \\to {\\mathcal A}\\to {\\mathcal B} \\to {\\mathcal C} \\to 0$. Up to Morita equivalence, this data is equivalent to inclusions of dg categories ${\\mathcal A} , {\\mathcal C} \\subseteq {\\mathcal B}$ yielding a semi-orthogonal decomposition of triangulated categories $\\mathrm{H}^0({\\mathcal B})=\\langle \\mathrm{H}^0({\\mathcal A}), \\mathrm{H}^0({\\mathcal C})\\rangle$ in the sense of Bondal-Orlov \\cite{BO}; by definition, the category $\\mathrm{H}^0({\\mathcal A})$ has the same objects as ${\\mathcal A}$ and morphisms $\\mathrm{H}^0({\\mathcal A})(x,y):=H^0({\\mathcal A}(x,y))$.\n\\begin{theorem}[{\\cite[Thm.~4.4]{Secondary}}]\\label{thm:split}\nLet $0 \\to {\\mathcal A} \\to {\\mathcal B} \\to {\\mathcal C} \\to 0$ be a short exact sequence of dg categories in the sense of Drinfeld\/Keller. If ${\\mathcal A}$ is smooth and proper and ${\\mathcal B}$ is proper, then the short exact sequence is split.\n\\end{theorem}\nMorally speaking, Theorem \\ref{thm:split} shows that the smooth proper dg categories behave as ``injective'' objects. In the setting of triangulated categories, this conceptual idea goes back to the pioneering work of Bondal-Kapranov \\cite{BK}.\n\\subsubsection{Secondary $K$-theory}\\label{sub:secondary}\nTwo decades ago, Bondal-Larsen-Lunts introduced in \\cite{BLL} the {\\em Grothendieck ring of smooth proper dg categories ${\\mathcal P}{\\mathcal T}(k)$}. This ring is defined by generators and relations. The generators are the Morita equivalence classes of smooth proper dg categories\\footnote{Bondal-Larsen-Lunts worked originally with (pretriangulated) dg categories. In this generality, the classical Eilenberg's swindle argument implies that the Grothendieck ring is trivial.} and the relations $[{\\mathcal B}]=[{\\mathcal A}]+[{\\mathcal C}]$ arise from semi-orthogonal decompositions $\\mathrm{H}^0({\\mathcal B})=\\langle \\mathrm{H}^0({\\mathcal A}), \\mathrm{H}^0({\\mathcal C}) \\rangle$. The multiplication law is induced by the tensor product of dg categories. One decade ago, To\\\"en introduced in \\cite{Toen} a ``categorified'' version of the Grothendieck ring named {\\em secondary Grothendieck ring $K_0^{(2)}(k)$}. By definition, $K_0^{(2)}(k)$ is the~quotient of the free abelian group on the Morita equivalence classes of smooth proper dg categories by the relations $[{\\mathcal B}]=[{\\mathcal A}]+[{\\mathcal C}]$ arising from short exact sequences $0 \\to {\\mathcal A} \\to {\\mathcal B} \\to {\\mathcal C}\\to 0$. The multiplication law is also induced by the tensor product of dg categories.\n\nTheorem \\ref{thm:split} directly leads to the following result:\n\\begin{corollary}\\label{cor:secondary}\nThe rings ${\\mathcal P}{\\mathcal T}(k)$ and $K_0^{(2)}(k)$ are isomorphic. \n\\end{corollary}\nMorally speaking, Corollary \\ref{cor:secondary} shows that the secondary Grothendieck ring is {\\em not} a new mathematical notion. \n\nBy construction, the universal additive invariant $U$ (see \\S\\ref{sec:additive}) sends semi-orthogonal decompositions to direct sums. Therefore, it gives rise to a ring homomorphism ${\\mathcal P}{\\mathcal T}(k)\\to K_0(\\NChow(k))$. Making use of Corollary \\ref{cor:new}, we then obtain the result:\n\\begin{corollary}\\label{cor:secondary1}\nThe following map is injective:\n\\begin{eqnarray}\\label{eq:map}\n\\mathrm{Br}(k) \\longrightarrow {\\mathcal P}{\\mathcal T}(k)\\simeq K_0^{(2)}(k) && [A] \\mapsto [A]\\,.\n\\end{eqnarray}\n\\end{corollary}\nThe map \\eqref{eq:map} may be understood as the ``categorification'' of the canonical map from the Picard group $\\mathrm{Pic}(k)$ to the Grothendieck ring $K_0(k)$. In contrast with $\\mathrm{Pic}(k) \\to K_0(k)$, the map \\eqref{eq:map} does not seems to admit a ``determinant'' map in the converse direction. Nevertheless, Corollary \\ref{cor:secondary1} shows that this~map~is~injective.\n\\begin{remark}[Generalizations]\nSimilarly to Remark \\ref{rk:generalization1}, Corollaries \\ref{cor:secondary}-\\ref{cor:secondary1} hold more generally with $k$ replaced by a base $k$-scheme $X$.\n\\end{remark}\n\\subsection{Gysin triangle}\\label{sub:Gysin}\nLet $X$ be a smooth $k$-scheme, $i\\colon Z \\hookrightarrow X$ a smooth closed subscheme, and $j\\colon V \\hookrightarrow X$ the open complement of $Z$.\n\\begin{theorem}[{\\cite[Thm.~1.9]{Gysin}}]\\label{thm:Gysin}\nWe have an induced distinguished ``Gysin'' triangle\n\\begin{equation}\\label{eq:Gysin-mot1}\n\\mathrm{U}(Z) \\stackrel{\\mathrm{U}(i_\\ast)}{\\longrightarrow} \\mathrm{U}(X) \\stackrel{\\mathrm{U}(j^\\ast)}{\\longrightarrow} \\mathrm{U}(V) \\stackrel{\\partial}{\\longrightarrow} \\mathrm{U}(Z)[1]\\,,\n\\end{equation}\nwhere $i_\\ast$, resp. $j^\\ast$, stands for the push-forward, resp. pull-back, dg functor.\n\\end{theorem}\n\\begin{remark}[Generalizations]\nAs explained in \\cite[\\S7]{Gysin}, Theorem \\ref{thm:Gysin} holds not only for smooth schemes but also for smooth algebraic spaces in the sense of Artin.\n\\end{remark}\nRoughly speaking, Theorem \\ref{thm:Gysin} shows that the difference between the localizing $\\mathbb{A}^1$-homotopy invariants of $X$ and of $V$ is completely determined by the closed subscheme $Z$. Consult Remark \\ref{rk:motivic1}, resp. \\ref{rk:motivic2}, for the relation between \\eqref{eq:Gysin-mot1} and the motivic Gysin triangles constructed by Morel-Voevodsky, resp. Voevodsky.\n\\subsubsection{Quillen's localization theorem}\nHomotopy $K$-theory is a localizing $\\mathbb{A}^1$-homotopy invariant which agrees with Quillen's algebraic $K$-theory when restricted to smooth $k$-schemes. Therefore, Theorem \\ref{thm:Gysin} leads to the $K$-theoretical localization theorem\n\\begin{equation}\\label{eq:localization}\nK(Z) \\stackrel{K(i_\\ast)}\\longrightarrow K(X) \\stackrel{K(j^\\ast)}{\\longrightarrow} K(V) \\stackrel{\\partial}{\\longrightarrow} K(Z)[1]\n\\end{equation}\noriginally established by Quillen in \\cite[Chapter 7 \\S3]{Quillen}. Among other ingredients, Quillen's proof makes essential use of \nd\\'evissage. The proof of Theorem \\ref{thm:Gysin}, and hence of \\eqref{eq:localization}, is quite different and avoids the use of d\\'evissage.\n\\subsubsection{Six-term exact sequence in de Rham cohomology}\\label{sub:deRham}\nPeriodic cyclic homology is a localizing $\\mathbb{A}^1$-homotopy invariant (when $\\mathrm{char}(k)=0$). Moreover, thanks to the Hochschild-Kostant-Rosenberg theorem, we have an isomorphism of $\\mathbb{Z}\/2$-graded $k$-vector spaces $HP^\\pm(X) \\simeq (\\bigoplus_{n\\,\\mathrm{even}} H^n_{dR}(X), \\bigoplus_{n\\,\\mathrm{odd}}H^n_{dR}(X))$, where $H^\\ast_{dR}$ stands for de Rham cohomology. Furthermore, the maps $i$ and $j$ give rise to homomorphisms $H^n_{dR}(i_\\ast)\\colon H^n_{dR}(Z) \\to H^{n+2c}_{dR}(X) $ and $H^n_{dR}(j^\\ast)\\colon H^n_{dR}(X) \\to H^n_{dR}(V)$, where $c$ stands for the codimension of $i$. Therefore, Theorem \\ref{thm:Gysin} leads to the following six-term exact sequence in de Rham cohomology:\n$$\n\\xymatrix{\n\\bigoplus_{n\\,\\mathrm{even}}H^n_{dR}(Z) \\ar[rr]^-{\\bigoplus_nH^n_{dR}(i_\\ast)} && \\bigoplus_{n\\,\\mathrm{even}}H^n_{dR}(X) \\ar[rr]^-{\\bigoplus_nH^n_{dR}(j^\\ast)} && \\bigoplus_{n\\,\\,\\mathrm{even}}H^n_{dR}(V) \\ar[d]^-\\partial \\\\\n\\bigoplus_{n\\,\\mathrm{odd}}H^n_{dR}(V) \\ar[u]^-\\partial && \\bigoplus_{n\\,\\mathrm{odd}}H^n_{dR}(X) \\ar[ll]^-{\\bigoplus_n H^n_{dR}(j^\\ast)} && \\bigoplus_{n\\,\\,\\mathrm{odd}}H^n_{dR}(Z) \\ar[ll]^-{\\bigoplus_nH^n_{dR}(i_\\ast)} \\,. \n}\n$$\nThis exact sequence is the ``$2$-periodization'' of the\nGysin long exact sequence on de\nRham~cohomology~originally~constructed~by~Hartshorne~in~\\cite[Chapter\nII~\\S3]{Hartshorne}. \n\\subsubsection{Reduction to projective schemes}\nAs a byproduct of Theorem \\ref{thm:Gysin}, the study of the localizing $\\mathbb{A}^1$-homotopy invariants of smooth $k$-schemes can be reduced to the study of the localizing $\\mathbb{A}^1$-homotopy invariants of smooth {\\em projective} $k$-schemes:\n\\begin{theorem}[{\\cite[Thm.~2.1]{Gysin}}]\\label{thm:reduction}\nLet $X$ a smooth $k$-scheme.\n\\begin{itemize}\n\\item[(i)] If $\\mathrm{char}(k)=0$, then $\\mathrm{U}(X)$ belongs to the smallest triangulated subcategory of $\\mathrm{NMot}(k)$ containing the objects $\\mathrm{U}(Y)$, with $Y$ a smooth projective $k$-scheme.\n\\item[(ii)] If $k$ is a perfect field of characteristic $p>0$, then $\\mathrm{U}(X)_{1\/p}$ belongs to the smallest thick triangulated subcategory of $\\mathrm{NMot}(k)_{1\/p}$ containing the objects $\\mathrm{U}(Y)_{1\/p}$, with $Y$ a smooth projective $k$-scheme.\n\\end{itemize}\n\\end{theorem}\nAmong other ingredients, the proof of item (i), resp. item (ii), of Theorem \\ref{thm:reduction} makes use of resolution of singularities, resp. of Gabber's refined version of de Jong's theory of alterations; consult \\cite[\\S8]{Gysin} for details.\n\\begin{remark}[Dualizable objects]\\label{rk:dualizable}\nGiven a smooth projective $k$-scheme $Y$, the associated dg category $\\mathrm{perf}_\\mathrm{dg}(Y)$ is smooth and proper; see \\cite[Example~1.42]{book}. Therefore, since the universal localizing $\\mathbb{A}^1$-homotopy invariant $\\mathrm{U}$ is symmetric monoidal, it follows from \\cite[Thm.~1.43]{book} that $\\mathrm{U}(Y)$ is a dualizable object of the symmetric monoidal category $\\mathrm{NMot}(k)$. Given a smooth $k$-scheme $X$, the associated dg category $\\mathrm{perf}_\\mathrm{dg}(X)$ is smooth but {\\em not} necessarily proper. Nevertheless, Theorem \\ref{thm:reduction} implies that $\\mathrm{U}(X)$, resp. $\\mathrm{U}(X)_{1\/n}$, is still a dualizable object of the symmetric monoidal category $\\mathrm{NMot}(k)$, resp. $\\mathrm{NMot}(k)_{1\/p}$.\n\\end{remark}\n\\subsection{Corner skew Laurent polynomial algebras}\\label{sub:corner}\nLet $A$ be a unital $k$-algebra, $e$ an idempotent of $A$, and $\\phi\\colon A\\stackrel{\\sim}{\\to} eAe$ a ``corner'' isomorphism. The associated {\\em corner skew Laurent polynomial algebra $A[t_+,t_-;\\phi]$} is defined as follows: the elements are formal expressions $t^m_- a_{-m} + \\cdots + t_- a_{-1} + a_0 + a_1 t_+ \\cdots + a_n t_+^n$\nwith $a_{-i} \\in \\phi^i(1)A$ and $a_i \\in A\\phi^i(1)$ for every $i \\geq 0$; the addition is defined componentwise; the multiplication is determined by the distributive law and by the relations $t_-t_+=1$, $t_+t_-=e$, $at_-=t_-\\phi(a)$ for every $a \\in A$, and $t_+a = \\phi(a)t_+$ for every $a \\in A$. Note that $A[t_+,t_-;\\phi]$ admits a canonical $\\mathbb{Z}$-grading with $\\mathrm{deg}(t_\\pm)=\\pm 1$. As proved in \\cite[Lem.~2.4]{Fractional}, the corner skew Laurent polynomial algebras can be characterized as those $\\mathbb{Z}$-graded algebras $C=\\bigoplus_{n \\in \\mathbb{Z}}C_n$ containing elements $t_+\\in C_1$ and $t_-\\in C_{-1}$ such that $t_-t_+=1$. Concretely, we have $C=A[t_+,t_-;\\phi]$ with $A:=C_0$, $e:=t_+t_-$, and $\\phi\\colon C_0 \\to t_+ t_- C_0 t_+ t_-$ given by $c_0\\mapsto t_+ c_0 t_-$.\n\\begin{example}[Skew Laurent polynomial algebras]\nWhen $e=1$, $A[t_+,t_-;\\phi]$ reduces to the classical skew Laurent polynomial algebra $A \\rtimes_\\phi \\mathbb{Z}$. In the particular case where $\\phi$ is the identity, $A\\rtimes_\\phi \\mathbb{Z}$ reduces furthermore to $A[t,t^{-1}]$.\n\\end{example}\n\\begin{example}[Leavitt algebras]\nFollowing \\cite{Leavitt}, the {\\em Leavitt algebra $L_n$, $n\\geq 0$,} is the $k$-algebra generated by elements $x_0, \\ldots, x_n, y_0, \\ldots, y_n$ subject to the relations $y_i x_j = \\delta_{ij}$ and $\\sum^n_{i=0} x_i y_i =1$. Note that the canonical $\\mathbb{Z}$-grading, with $\\mathrm{deg}(x_i)=1$ and $\\mathrm{deg}(y_i)=-1$, makes $L_n$ into a corner skew Laurent polynomial algebra. Note also that $L_0\\simeq k[t,t^{-1}]$. In the remaining cases $n\\geq 1$, $L_n$ is the universal example of a $k$-algebra of {\\em module type $(1,n+1)$}, \\textsl{i.e.}\\ $L_n\\simeq L_n^{\\oplus (n+1)}$ as right $L_n$-modules.\n\\end{example}\n\\begin{example}[Leavitt path algebras]\\label{ex:Leavitt-path}\nLet $Q=(Q_0,Q_1, s,r)$ be a finite quiver with no sources; $Q_0$ and $Q_1$ stand for the sets of vertices and arrows, respectively, and $s$ and $r$ for the source and target maps, respectively. Consider the double quiver $\\overline{Q}=(Q_0,Q_1\\cup Q_1^\\ast, s,r)$ obtained from $Q$ by adding an arrow $\\alpha^\\ast$, in the converse direction, for each arrow $\\alpha \\in Q_1$. The {\\em Leavitt path algebra $L_Q$ of $Q$} is the quotient of the quiver algebra $k\\overline{Q}$ (which is generated by elements $\\alpha \\in Q_1\\cup Q_1^\\ast$ and $e_i$ with $i \\in Q_0$) by the Cuntz-Krieger's relations: $\\alpha^\\ast \\beta = \\delta_{\\alpha \\beta} e_{r(\\alpha)}$ for every $\\alpha, \\beta \\in Q_1$ and $\\sum_{\\{\\alpha \\in Q_1|s(\\alpha)=i\\}} \\alpha \\alpha^\\ast =e_i$ for every non-sink $i \\in Q_0$. Note that $L_Q$ admits a canonical $\\mathbb{Z}$-grading with $\\mathrm{deg}(\\alpha)=1$ and $\\mathrm{deg}(\\alpha^\\ast)=-1$. For every vertex $i \\in Q_0$ choose an arrow $\\alpha_i$ such that $r(\\alpha_i)=i$ and consider the associated elements $t_+:=\\sum_{i \\in Q_0}\\alpha_i$ and $t_-:=t_+^\\ast$. Since $\\mathrm{deg}(t_\\pm)=\\pm1$ and $t_-t_+=1$, $L_Q$ is an example of a corner skew Laurent polynomial algebra. In the particular case where $Q$ is the quiver with one vertex and $n+1$ arrows, $L_Q$ reduces to the Leavitt algebra $L_n$. \n\\end{example}\n\\begin{theorem}[{\\cite[Thm.~3.1]{Corner}}]\\label{thm:corner}\nWe have an induced distinguished triangle\n\\begin{equation*\n\\mathrm{U}(A) \\stackrel{\\id - \\mathrm{U}({}_\\phi A)}{\\longrightarrow} \\mathrm{U}(A) \\longrightarrow \\mathrm{U}(A[t_+, t_-; \\phi]) \\stackrel{\\partial}{\\longrightarrow} \\mathrm{U}(A)[1]\\,,\n\\end{equation*}\nwhere ${}_\\phi A$ stands for the $A\\text{-}A$-bimodule associated to $\\phi$.\n\\end{theorem}\nRoughly speaking, Theorem \\ref{thm:corner} shows that $A[t_+, t_-; \\phi]$ may be considered as a model for the orbits of the $\\mathbb{N}$-action on $\\mathrm{U}(A)$ induced by the endomorphism $\\mathrm{U}({}_\\phi A)$.\n\\subsubsection{Leavitt path algebras}\nLet $Q=(Q_0,Q_1,s,r)$ be a quiver as in Example \\ref{ex:Leavitt-path}, with $v$ vertices and $v'$ sinks. Assume that the set $Q_0$ is ordered with the first $v'$ elements corresponding to the sinks. Let $I'_Q$ be the incidence matrix of $Q$, $I_Q$ the matrix obtained from $I'_Q$ by removing the first $v'$ rows (which are zero), and $I_Q^t$ the transpose of $I_Q$. Under these notations, Theorem \\ref{thm:corner} (concerning the Leavitt path algebra $L_Q$) admits the following refinement:\n\\begin{theorem}[{\\cite[Thm.~3.7]{Corner}}]\\label{thm:corner2}\nWe have an induced distinguished triangle:\n\\begin{equation}\\label{eq:triangle2}\n\\mathrm{U}(k)^{(v-v') \\oplus} \\stackrel{\\binom{0}{\\id} - I^t_Q}{\\longrightarrow} \\mathrm{U}(k)^{v \\oplus} \\longrightarrow \\mathrm{U}(L_Q) \\stackrel{\\partial}{\\longrightarrow} \\mathrm{U}(k)[1]^{(v-v') \\oplus}\\,.\n\\end{equation}\n\\end{theorem}\nRoughly speaking, Theorem \\ref{thm:corner2} shows that all the information about the localizing $\\mathbb{A}^1$-homotopy invariants of Leavitt path algebras $L_Q$ is encoded in the incidence matrix of the quiver $Q$. As an application, Theorem \\ref{thm:corner2} directly leads to the following explicit model for the mod$\\text{-}n$ Moore construction\\footnote{Explicit models for the suspension construction, namely the Waldhausen's $S_\\bullet$-construction and the Calkin algebra, are described in \\cite[\\S8.3.2 and \\S8.4.4]{book}.}:\n\\begin{example}[mod$\\text{-}n$ Moore construction]\nLet $Q$ be the quiver with one vertex and $n+1$ arrows. In this particular case, \\eqref{eq:triangle2} reduces to the distinguished triangle\n\\begin{equation*\n\\mathrm{U}(k) \\stackrel{n\\cdot \\id}{\\longrightarrow} \\mathrm{U}(k) \\longrightarrow \\mathrm{U}(L_n) \\stackrel{\\partial}{\\longrightarrow} \\mathrm{U}(k)[1]\\,.\n\\end{equation*}\nThis shows that the Leavitt algebra $L_n, n\\geq 2$, is a model for the mod$\\text{-}n$ Moore object of $\\mathrm{U}(k)$. Therefore, since the universal localizing $\\mathbb{A}^1$-homotopy invariant $\\mathrm{U}$ is symmetric monoidal, given a small dg category ${\\mathcal A}$, we conclude that the tensor product ${\\mathcal A}\\otimes L_n$ is a model for the mod$\\text{-}n$ Moore object of $\\mathrm{U}({\\mathcal A})$. \n\\end{example}\n\\subsection{Noncommutative projective schemes}\\label{sub:NCproj}\nLet $A=\\bigoplus_{n\\geq 0} A_n$ be a $\\mathbb{N}$-graded Noetherian $k$-algebra. In what follows, we assume that $A$ is {\\em connected}, \\textsl{i.e.}\\ $A_0=k$, and {\\em locally finite-dimensional}, \\textsl{i.e.}\\ $\\mathrm{dim}_k(A_n)<\\infty$ for every $n$. Following Manin \\cite{Manin}, Gabriel \\cite{Gabriel}, Artin-Zhang \\cite{Artin-Zhang}, and others, the {\\em noncommutative projective scheme $\\mathrm{qgr}(A)$} associated to $A$ is defined as the quotient abelian category $\\mathrm{gr}(A)\/\\mathrm{tors}(A)$, where $\\mathrm{gr}(A)$ stands for the abelian category of finitely generated $\\mathbb{Z}$-graded (right) $A$-modules and $\\mathrm{tors}(A)$ for the Serre subcategory of torsion $A$-modules. This definition was motivated by Serre's celebrated result \\cite[Prop.~7.8]{Serre}, which asserts that in the particular case where $A$ is commutative and generated by elements of degree $1$ the quotient category $\\mathrm{qgr}(A)$ is equivalent to the abelian category of coherent ${\\mathcal O}_{\\mathrm{Proj}(A)}$-modules $\\mathrm{coh}(\\mathrm{Proj}(A))$. For example, when $A$ is the polynomial $k$-algebra $k[x_1, \\ldots, x_d]$, with $\\mathrm{deg}(x_i)=1$, we have the equivalence of categories $\\mathrm{qgr}(k[x_1, \\ldots, x_d])\\simeq \\mathrm{coh}(\\mathbb{P}^{d-1})$. For a survey on noncommutative projective geometry, we invite the reader to consult Stafford's ICM address \\cite{Stafford}.\n\nAssume that $A$ is Koszul and has finite global dimension $d$. Under these assumptions, the Hilbert series $h_A(t):=\\sum_{n\\geq 0} \\mathrm{dim}_k(A_n)t^n \\in \\mathbb{Z}[\\![t]\\!]$ is invertible and its inverse $h_A(t)^{-1}$ is a polynomial $1-\\beta_1t + \\beta_2t^2 - \\cdots + (-1)^d\\beta_d t^d$ of degree $d$, with $\\beta_i$ the dimension of the $k$-vector space $\\mathrm{Tor}^A_i(k,k)$ (or $\\mathrm{Ext}_A^i(k,k)$).\n\\begin{example}[Quantum polynomial algebras]\nChoose constant elements $q_{ij} \\in k^\\times$ with $1 \\leq i < j \\leq d$. Following Manin~\\cite[\\S1]{Manin-Fourier}, the $\\mathbb{N}$-graded Noetherian $k$-algebra\n$$ A:=k\\langle x_1, \\ldots, x_d\\rangle\/\\langle x_j x_i - q_{ij} x_i x_j\\,|\\, 1 \\leq i < j \\leq d\\rangle\\,,$$\nwith $\\mathrm{deg}(x_i)=1$, is called the {\\em quantum polynomial algebra} associated to $q_{ij}$. This algebra is Koszul, has global dimension $d$, and $h_A(t)^{-1}=(1-t)^d$.\n\\end{example}\n\\begin{example}[Quantum matrix algebras]\nChoose a constant element $q \\in k^\\times$. Following Manin \\cite[\\S1]{Manin-Fourier}, the $\\mathbb{N}$-graded Noetherian $k$-algebra $A$ defined as the quotient of $k\\langle x_1, x_2, x_3, x_4\\rangle$ by the following relations\n\\begin{eqnarray*}\nx_1x_2 = q x_2 x_1 & x_1x_3=qx_3 x_1 & x_1 x_4 - x_4 x_1 = (q-q^{-1}) x_2 x_3 \\\\\nx_2 x_3 = x_3 x_2 & x_2 x_4=q x_4 x_2 & x_1 x_4 = q x_4 x_3\\,,\n\\end{eqnarray*}\nwith $\\mathrm{deg}(x_i)=1$, is called the {\\em quantum matrix algebra} associated to $q$. This algebra is Koszul, has global dimension $4$, and $h_A(t)^{-1}=(1-t)^4$.\n\\end{example}\n\\begin{example}[Sklyanin algebras]\nLet $C$ be a smooth elliptic $k$-curve, $\\sigma$ an automorphism of $C$ given by translation under the group law, and ${\\mathcal L}$ a line bundle on $C$ of degree $d\\geq 3$. We write $\\Gamma_\\sigma \\subset C\\times C$ for the graph of $\\sigma$ and $W$ for the $d$-dimensional $k$-vector space $H^0(C,{\\mathcal L})$. Following Feigin-Odesskii \\cite{Feigin} and Tate-Van den Bergh \\cite[\\S1]{TateVdb}, the $\\mathbb{N}$-graded Noetherian $k$-algebra $A:=T(W)\/R$, where\n$$R:=H^0(C\\times C, ({\\mathcal L}\\boxtimes {\\mathcal L})(-\\Gamma_\\sigma))\\subset H^0(C\\times C, {\\mathcal L} \\boxtimes {\\mathcal L})=W\\otimes W\\,,$$\nis called the {\\em Sklyanin algebra} associated to the triple $(C,\\sigma, {\\mathcal L})$. This algebra is Koszul, has global dimension $d$, and $h_A(t)^{-1}=(1-t)^d$.\n\\end{example}\n\\begin{example}[Homogenized enveloping algebras]\nLet $\\mathfrak{g}$ be a finite-dimensional Lie algebra. Following Smith \\cite[\\S12]{Smith}, the $\\mathbb{N}$-graded Noetherian $k$-algebra\n$$ A:= T(\\mathfrak{g} \\oplus kz)\/ \\langle\\{z\\otimes x - x\\otimes z \\,|\\, x \\in \\mathfrak{g}\\}\\cup \\{x\\otimes y - y\\otimes x - [x,y]\\otimes z \\,|\\, x, y \\in \\mathfrak{g}\\}\\rangle\\,,$$\nis called the {\\em homogenized enveloping algebra} of $\\mathfrak{g}$. This algebra is Koszul, has global dimension $d:=\\mathrm{dim}(\\mathfrak{g})+1$, and $h_A(t)^{-1}=(1-t)^{d}$.\n\\end{example}\nGiven a $\\mathbb{N}$-graded $k$-algebra $A$ as above, let us write ${\\mathcal D}^b_{\\mathrm{dg}}(\\mathrm{qgr}(A))$ for the canonical dg enhancement of the bounded derived category of $\\mathrm{qgr}(A)$. This dg category is, in general, {\\em not} proper; see \\cite[\\S1]{NCProj}. The following result is contained \\cite[Thm.~1.2]{NCProj}:\n\\begin{theorem}\\label{thm:NCProj}\nWe have an induced distinguished triangle\n\\begin{equation}\\label{eq:triangle-NCProj}\n\\bigoplus^{+\\infty}_{-\\infty} \\mathrm{U}(k) \\stackrel{\\mathrm{M}}{\\longrightarrow} \\bigoplus^{+\\infty}_{-\\infty} \\mathrm{U}(k) \\longrightarrow \\mathrm{U}({\\mathcal D}^b_{\\mathrm{dg}}(\\mathrm{qgr}(A))) \\stackrel{\\partial}{\\longrightarrow} \\bigoplus^{+\\infty}_{-\\infty} \\mathrm{U}(k)[1]\\,,\n\\end{equation}\nwhere $\\mathrm{M}$ stands for the (infinite) matrix $\\mathrm{M}_{ij}:=(-1)^j (-1)^{(i-j)} \\beta_{i-j}$. Moreover, when $\\beta_d=1$, the triangle \\eqref{eq:triangle-NCProj} induces an isomorphism $\\mathrm{U}({\\mathcal D}^b_{\\mathrm{dg}}(\\mathrm{qgr}(A)))\\simeq \\mathrm{U}(k)^{\\oplus d}$.\n\\end{theorem}\nAs proved in \\cite[Cor.~0.2]{Zhang}, we have $h_A(t)^{-1}=(1-t)^3$ whenever $d=3$.\n\\begin{remark}[Localizing invariants]\\label{rk:localizing}\nThe proof of Theorem \\ref{thm:NCProj} does {\\em not} makes use of $\\mathbb{A}^1$-homotopy invariance. Consequently, as explained in \\cite[Thm.~8.5]{book}, Theorem \\ref{thm:NCProj} holds similarly for every localizing invariant in the sense of \\cite[Def.~8.3]{book}. Examples of localizing invariants which are {\\em not} $\\mathbb{A}^1$-homotopy invariant include nonconnective algebraic $K$-theory, Hochschild homology, cyclic homology, negative cyclic homology, periodic cyclic homology (when $\\mathrm{char}(k)=p>0$), topological Hochschild homology, topological cyclic homology, topological periodic cyclic homology, etc.\n\\end{remark}\nRoughly speaking, Theorem \\ref{thm:NCProj} (and Remark \\ref{rk:localizing}) shows that the localizing invariants of a noncommutative projective scheme $\\mathrm{qgr}(A)$ are completely determined by the Hilbert series $h_A(t)$.\n\\section{Noncommutative mixed motives}\\label{sec:NCmixed}\nIn this section we assume that the base field $k$ is perfect. Kontsevich introduced in \\cite{Miami,finMot,IAS} a certain rigid symmetric monoidal triangulated category of noncommutative mixed motives $\\mathrm{NMix}(k)$. As explained in \\cite[\\S9.1.1]{book}, this category can be (conceptually) described as the smallest thick triangulated subcategory of $\\mathrm{NMot}(k)$ (see \\S\\ref{sec:localizing}) containing the objects $\\mathrm{U}({\\mathcal A})$, with ${\\mathcal A}$ smooth and proper. \n\nIn \\S\\ref{sub:Picard} we compute the Picard group of the thick triangulated subcategory of $\\mathrm{NMix}(k)$ generated by the noncommutative mixed motives of central simple $k$-algebras. Subsections \\S\\ref{sub:MV}-\\ref{sub:Levine} are devoted to the precise relation between the category $\\mathrm{NMix}(k)$ and Morel-Voevodsky's stable $\\mathbb{A}^1$-homotopy category, Voevodsky's triangulated category of geometric mixed motives, and Levine's triangulated category of mixed motives, respectively. In \\S\\ref{sec:Schur} we address the Schur-finiteness conjecture in the case of quadric fibrations. Finally, subsection \\S\\ref{sub:rigidity2} is devoted to the rigidity property of the category of mod-$n$ noncommutative mixed motives.\n\\subsection{Picard group}\\label{sub:Picard}\nThe computation of the Picard group of the category of noncommutative mixed motives is a major challenge which seems completely out of reach at the present time. However, this major challenge can be met if we restrict ourselves to central simple $k$-algebras. Let $\\mathrm{NMix}_{\\mathrm{csa}}(k)$ be the thick triangulated subcategory of $\\mathrm{NMix}(k)$ generated by the noncommutative mixed motives $\\mathrm{U}(A)$ of central simple $k$-algebras $A$. Similarly to \\S\\ref{sub:Brauer}, the equivalence $[A]=[B]\\Leftrightarrow \\mathrm{U}(A)\\simeq \\mathrm{U}(B)$ holds for any two central simple $k$-algebras $A$ and $B$. Moreover, following \\cite[Thm.~8.28]{book}, we have non-trivial Ext-groups:\n\\begin{equation}\\label{eq:equiv-last1}\n\\mathrm{Hom}_{\\mathrm{NMix}(k)}(\\mathrm{U}(A), \\mathrm{U}(B)[-n]) \\simeq K_n(A^\\mathrm{op} \\otimes B) \\quad \\quad n\\in \\mathbb{Z}\\,.\n\\end{equation}\nThis shows that $\\mathrm{NMix}_{\\mathrm{csa}}(k)$ contains information not only about the Brauer group $\\mathrm{Br}(k)$ but also about all the higher algebraic $K$-theory of central simple $k$-algebras.\n\\begin{theorem}[{\\cite[Thm.~2.22]{Picard}}]\\label{thm:Picard}\nWe have the following isomorphism:\n\\begin{eqnarray*}\n\\mathrm{Br}(k) \\times \\mathbb{Z} \\stackrel{\\sim}{\\longrightarrow} \\mathrm{Pic}(\\mathrm{NMix}_{\\mathrm{csa}}(k)) && ([A],n) \\mapsto \\mathrm{U}(A)[n]\\,.\n\\end{eqnarray*}\n\\end{theorem}\nTheorem \\ref{thm:Picard} shows that, although $\\mathrm{NMix}_{\\mathrm{csa}}(k)$ contains information about all the higher algebraic $K$-theory of central simple $k$-algebras, none of the noncommutative mixed motives which are built using the non-trivial Ext-groups \\eqref{eq:equiv-last1} is $\\otimes$-invertible.\n\\subsection{Morel-Voevodsky's motivic category}\\label{sub:MV}\nMorel-Voevodsky introduced in \\cite{MV,Voevodsky-ICM} the stable $\\mathbb{A}^1$-homotopy category of $(\\mathbb{P}^1,\\infty)$-spectra $\\mathrm{SH}(k)$. By construction, we have a symmetric monoidal functor $\\Sigma^\\infty(-_+)\\colon \\mathrm{Sm}(k) \\to \\mathrm{SH}(k)$ defined on smooth $k$-schemes. Let $\\mathrm{KGL} \\in \\mathrm{SH}(k)$ be the ring $(\\mathbb{P}^1,\\infty)$-spectrum representing homotopy $K$-theory and $\\mathrm{Mod}(\\mathrm{KGL})$ the homotopy category of $\\mathrm{KGL}$-modules.\n\\begin{theorem}\\label{thm:bridge1}\n\\begin{itemize}\n\\item[(i)] If $\\mathrm{char}(k)=0$, then there exists a fully-faithful, symmetric monoidal, triangulated functor $\\Psi$ making the following diagram commute\n\\begin{equation}\\label{eq:diagram-1}\n\\xymatrix{\n\\mathrm{Sm}(k) \\ar[d]_-{\\Sigma^\\infty(-_+)} \\ar[rrr]^-{X \\mapsto \\mathrm{perf}_\\mathrm{dg}(X)} \\ar[drr] &&& \\mathsf{dgcat}(k) \\ar[d]^-{\\mathrm{U}} \\\\\n\\mathrm{SH}(k) \\ar[d]_-{-\\wedge \\mathrm{KGL}} && \\mathrm{NMix}(k) \\ar[d]_-{(-)^\\vee} \\ar[r] & \\mathrm{NMot}(k) \\ar[d]^-{\\underline{\\mathrm{Hom}}(-,\\mathrm{U}(k))} \\\\\n\\mathrm{Mod}(\\mathrm{KGL}) \\ar[rr]_-\\Psi && \\mathrm{NMix}(k)^\\oplus \\ar[r] & \\mathrm{NMot}(k) \\,,\n}\n\\end{equation}\nwhere $\\underline{\\mathrm{Hom}}(-,-)$ stands for the internal-Hom of the closed symmetric monoidal category $\\mathrm{NMot}(k)$, $(-)^\\vee$ for the (contravariant) duality functor, and $\\mathrm{NMix}(k)^\\oplus$ for the smallest triangulated subcategory of $\\mathrm{NMot}(k)$ which contains $\\mathrm{NMix}(k)$ and is stable under arbitrary direct sums.\n\\item[(ii)] If $\\mathrm{char}(k)=p>0$, then there exists a\n $\\mathbb{Z}[1\/p]$-linear, fully-faithful, symmetric monoidal,\n triangulated functor $\\Psi_{1\/p}$ making the following diagram commute:\n$$\n\\xymatrix{\n\\mathrm{Sm}(k) \\ar[d]_-{\\Sigma^\\infty(-_+)_{1\/p}} \\ar[rrr]^-{X \\mapsto \\mathrm{perf}_\\mathrm{dg}(X)} \\ar[drr] &&& \\mathsf{dgcat}(k) \\ar[d]^-{\\mathrm{U}(-)_{1\/p}} \\\\\n\\mathrm{SH}(k)_{1\/p} \\ar[d]_-{-\\wedge \\mathrm{KGL}_{1\/p}} && \\mathrm{NMix}(k)_{1\/p} \\ar[d]_-{(-)^\\vee} \\ar[r] & \\mathrm{NMot}(k)_{1\/p} \\ar[d]^-{\\underline{\\mathrm{Hom}}(-,\\mathrm{U}(k)_{1\/p})} \\\\\n\\mathrm{Mod}(\\mathrm{KGL}_{1\/p}) \\ar[rr]_-{\\Psi_{1\/p}} && \\mathrm{NMix}(k)_{1\/p}^\\oplus \\ar[r] & \\mathrm{NMot}(k)_{1\/p} \\,.\n}\n$$\n\\end{itemize}\n\\end{theorem}\nIntuitively speaking, Theorem \\ref{thm:bridge1} shows that as soon as we pass to $\\mathrm{KGL}$-modules, the commutative world embeds fully-faithfully into the noncommutative world. Consult \\cite[\\S9.4]{book}, and the references therein, for the construction of the two outer commutative diagrams. The inner commutative squares follow from the combination of Theorem \\ref{thm:reduction} with Remark \\ref{rk:dualizable}; consult \\cite[Thm.~3.1]{Gysin} for details.\n\\begin{remark}[Morel-Voevodsky's motivic Gysin triangle]\\label{rk:motivic1}\nLet $X$ be a smooth $k$-scheme, $Z \\hookrightarrow X$ a smooth closed subscheme with normal vector bundle $N$, and $j\\colon V \\hookrightarrow X$ the open complement of $Z$. Making use of homotopy purity, Morel-Voevodsky constructed in \\cite[\\S3.2]{MV}\\cite[\\S4]{Voevodsky-ICM} a motivic Gysin triangle\n\\begin{equation}\\label{eq:Gysin-mot2}\n\\Sigma^\\infty(V_+) \\stackrel{\\Sigma^\\infty(j_+)}{\\longrightarrow} \\Sigma^\\infty(X_+) \\longrightarrow \\Sigma^\\infty(\\mathrm{Th}(N))\\stackrel{\\partial}{\\longrightarrow} \\Sigma^\\infty(V_+)[1]\n\\end{equation}\nin $\\mathrm{SH}(k)$, where $\\mathrm{Th}(N)$ stands for the Thom space of $N$. Since homotopy $K$-theory is an orientable and periodic cohomology theory, $\\Sigma^\\infty(\\mathrm{Th}(N))\\wedge \\mathrm{KGL}$ is isomorphic to $\\Sigma^\\infty(Z_+)\\wedge \\mathrm{KGL}$. Using the commutative diagram \\eqref{eq:diagram-1}, we hence observe that the image of \\eqref{eq:Gysin-mot2} under the composed functor $\\Psi \\circ (-\\wedge \\mathrm{KGL})\\colon \\mathrm{SH}(k) \\to \\mathrm{NMix}(k)^\\oplus$ agrees with the dual of the Gysin triangle \\eqref{eq:Gysin-mot1}. In other words, the Gysin triangle \\eqref{eq:Gysin-mot1} is the dual of the ``$\\mathrm{KGL}$-linearization'' of \\eqref{eq:Gysin-mot2}.\n\\end{remark}\n\\subsection{Voevodsky's motivic category}\\label{sub:Voevodsky}\nVoevodsky introduced in \\cite[\\S2]{Voevodsky} the triangulated category of geometric mixed motives $\\mathrm{DM}_{\\mathrm{gm}}(k)$. By construction, this category comes equipped with a symmetric monoidal functor $M\\colon \\mathrm{Sm}(k) \\to \\mathrm{DM}_{\\mathrm{gm}}(k)$ and is the natural setting for the study of algebraic cycle (co)homology theories such as higher Chow groups, Suslin homology, motivic cohomology, etc.\n\\begin{theorem}\\label{thm:bridge2}\nThere exists a $\\mathbb{Q}$-linear, fully-faithful, symmetric monoidal functor $\\Phi_\\mathbb{Q}$ making~the~following~diagram~commute:\n\\begin{equation}\\label{eq:diagram-3}\n\\xymatrix{\n\\mathrm{Sm}(k) \\ar[d]_-{M(-)_\\mathbb{Q}} \\ar[rrr]^-{X \\mapsto \\mathrm{perf}_\\mathrm{dg}(X)} \\ar[drr] &&& \\mathsf{dgcat}(k) \\ar[d]^-{\\mathrm{U}(-)_\\mathbb{Q}} \\\\\n\\mathrm{DM}_{\\mathrm{gm}}(k)_\\mathbb{Q} \\ar[d] && \\mathrm{NMix}(k)_\\mathbb{Q} \\ar[d]_-{(-)^\\vee} \\ar[r] & \\mathrm{NMot}(k)_\\mathbb{Q} \\ar[d]^-{\\underline{\\mathrm{Hom}}(-,\\mathrm{U}(k)_\\mathbb{Q})} \\\\\n\\mathrm{DM}_{\\mathrm{gm}}(k)_\\mathbb{Q}\/_{\\!\\!-\\otimes \\mathbb{Q}(1)[2]} \\ar[rr]_-{\\Phi_\\mathbb{Q}} && \\mathrm{NMix}(k)_\\mathbb{Q} \\ar[r] & \\mathrm{NMot}(k)_\\mathbb{Q} \\,.\n}\n\\end{equation}\n\\end{theorem}\nIntuitively speaking, Theorem \\ref{thm:bridge2} shows that as soon as we ``$\\otimes$-trivialize'' the Tate motive $\\mathbb{Q}(1)[2]$, the commutative world embedds fully-faithfully into the noncommutative world. Consult \\cite[\\S9.5]{book}, and the references therein, for the construction of the outer commutative diagram. The inner commutative square follows from the combination of Theorem \\ref{thm:reduction} with Remark \\ref{rk:dualizable}; consult \\cite[Thm.~3.7]{Gysin}\n\\begin{remark}[Voevodsky's motivic Gysin triangle]\\label{rk:motivic2}\nLet $X$ be a smooth $k$-scheme, $Z \\hookrightarrow X$ a smooth closed subscheme of codimension $c$, and $j\\colon V \\hookrightarrow X$ the open complement of $Z$. \nMaking use of deformation to the normal cone, Voevodsky constructed in \\cite[\\S2]{Voevodsky} a motivic Gysin triangle\n\\begin{equation}\\label{eq:Gysin-mot3}\nM(V)_\\mathbb{Q} \\stackrel{M(j)_\\mathbb{Q}}{\\longrightarrow} M(X)_\\mathbb{Q} \\longrightarrow M(Z)_\\mathbb{Q}(c)[2c] \\stackrel{\\partial}{\\longrightarrow} M(V)_\\mathbb{Q}[1]\n\\end{equation}\nin $\\mathrm{DM}_{\\mathrm{gm}}(k)_\\mathbb{Q}$. Using the commutative diagram \\eqref{eq:diagram-3}, we observe that the image of \\eqref{eq:Gysin-mot3} under the (composed) functor $\\Phi_\\mathbb{Q} \\colon \\mathrm{DM}_{\\mathrm{gm}}(k)_\\mathbb{Q} \\to \\mathrm{NMix}(k)_\\mathbb{Q}$ agrees with the dual of the rationalized Gysin triangle \\eqref{eq:Gysin-mot1}. In other words, the rationalized Gysin triangle \\eqref{eq:Gysin-mot1} is the dual of the ``Tate $\\otimes$-trivialization'' of \\eqref{eq:Gysin-mot3}.\n\\end{remark}\nLet $\\mathrm{DM}_{\\mathrm{gm}}^{\\mathrm{et}}(k)$ be the \\'etale variant of $\\mathrm{DM}_{\\mathrm{gm}}(k)$ introduced by Voevodsky in \\cite[\\S3.3]{Voevodsky}. As proved in {\\em loc. cit.}, $\\mathrm{DM}_{\\mathrm{gm}}(k)_\\mathbb{Q}$ is equivalent to $\\mathrm{DM}_{\\mathrm{gm}}^{\\mathrm{et}}(k)_\\mathbb{Q}$. Consequently, Theorem \\ref{thm:bridge2} leads to the following result (see \\cite[Thm.~3.13]{Gysin}):\n\\begin{corollary}[\\'Etale descent]\nThe presheaf of noncommutative mixed motives $\\mathrm{Sm}(k)^\\mathrm{op} \\to \\mathrm{NMot}(k)_\\mathbb{Q}, X \\mapsto \\mathrm{U}(X)_\\mathbb{Q}$, satisfies {\\em \\'etale descent}, \\textsl{i.e.}\\ for every {\\'e}tale cover ${\\mathcal V}=\\{V_i \\to X\\}_{i \\in I}$ of $X$, we have an isomorphism $\\mathrm{U}(X)_\\mathbb{Q} \\simeq \\mathrm{holim}_{n\\geq 0} \\mathrm{U}(\\text{\\v{C}}_n{\\mathcal V})_\\mathbb{Q}$, where $\\text{\\v{C}}_\\bullet{\\mathcal V}$ stands for the \\v{C}ech simplicial $k$-scheme associated to the cover ${\\mathcal V}$.\n\\end{corollary}\n\\subsection{Levine's motivic category}\\label{sub:Levine}\nLevine introduced in \\cite[Part I]{Levine} a triangulated category of mixed motives ${\\mathcal D}{\\mathcal M}(k)$ and a (contravariant) symmetric monoidal functor $h\\colon \\mathrm{Sm}(k) \\to {\\mathcal D}{\\mathcal M}(k)$. As proved in \\cite[Thm.~4.2]{Ivorra}, the following assignment $h(X)_\\mathbb{Q}(n) \\mapsto \\underline{\\mathrm{Hom}}(M(X),\\mathbb{Q}(n))$ gives rise to an equivalence of categories ${\\mathcal D}{\\mathcal M}(k)_\\mathbb{Q} \\to \\mathrm{DM}_{\\mathrm{gm}}(k)_\\mathbb{Q}$ whose precomposition with $h(-)_\\mathbb{Q}$ is $X \\mapsto M(X)^\\vee_\\mathbb{Q}$. Consequently, thanks to Theorem \\ref{thm:bridge2}, there exists a $\\mathbb{Q}$-linear, fully-faithful, symmetric monoidal functor $\\Phi_\\mathbb{Q}$ making the following diagram commute:\n\\begin{equation}\\label{eq:Levine}\n\\xymatrix{\n\\mathrm{Sm}(k) \\ar[d]_-{h(-)_\\mathbb{Q}} \\ar[rrr]^-{X \\mapsto \\mathrm{perf}_\\mathrm{dg}(X)}&&& \\mathsf{dgcat}(k) \\ar[dd]^-{\\mathrm{U}(-)_\\mathbb{Q}} \\\\\n{\\mathcal D}{\\mathcal M}(k)_\\mathbb{Q} \\ar[d] &&& \\\\\n{\\mathcal D}{\\mathcal M}(k)_\\mathbb{Q}\/_{\\!\\!-\\otimes \\mathbb{Q}(1)[2]} \\ar[rr]_-{\\Phi_\\mathbb{Q}} && \\mathrm{NMix}(k)_\\mathbb{Q} \\ar[r] & \\mathrm{NMot}(k)_\\mathbb{Q} \\,.\\quad \\quad \n}\n\\end{equation}\nNote that in contrast with the diagrams of Theorems \\ref{thm:bridge1} and \\ref{thm:bridge2}, the commutative diagram \\eqref{eq:Levine} does {\\em not} uses any kind of duality functor.\n\\subsection{Schur-finiteness conjecture}\\label{sec:Schur}\nGiven a smooth $k$-scheme $X$, the Schur-finiteness conjecture, denoted by $S(X)$, asserts that the mixed motive $M(X)_\\mathbb{Q}$ is Schur-finite in the sense of Deligne \\cite[\\S1]{Deligne}. Thanks to the (independent) work of Guletskii \\cite{Guletskii} and Mazza \\cite{Mazza}, the conjecture $\\mathrm{S}(X)$ holds when $\\mathrm{dim}(X)\\leq 1$, and also for abelian varieties. In addition to these cases, it remains wide open.\n\\begin{theorem}[{\\cite[Thm.~1.1]{Schur}}]\\label{thm:fibrations}\nLet $q\\colon Q \\to B$ a flat quadric fibration of relative dimension $d-2$. Assume that $B$ and $Q$ are $k$-smooth and that $q$ has only {\\em simple degenerations}, \\textsl{i.e.}\\ that all the fibers of $q$ have corank $\\leq 1$ and that the locus $D \\subset B$ of the critical values of $q$ is $k$-smooth. Under these assumptions, the following holds:\n\\begin{itemize}\n\\item[(i)] If $d$ is even, then we have $S(B) + S(\\widetilde{B}) \\Leftrightarrow S(Q)$, where $\\widetilde{B}$ stands for the discriminant $2$-fold cover of $B$ (ramified over $D$).\n\\item[(ii)] If $d$ is odd and $\\mathrm{char}(k)\\neq 2$, then we have $\\{S(V_i)\\} + \\{S(\\widetilde{D}_i)\\} \\Rightarrow S(Q)$, where $V_i$ is any affine open of $B$ and $\\widetilde{D}_i$ is any Galois $2$-fold cover of $D_i:=D\\cap V_i$.\n\\end{itemize}\n\\end{theorem}\nRoughly speaking, Theorem \\ref{thm:fibrations} relates the Schur-finiteness conjecture for the total space $Q$ with the Schur-finiteness conjecture for certain coverings\/subschemes of the base $B$. Among other ingredients, its proof makes use of Theorem \\ref{thm:bridge2} and of the twisted analogue of Theorem \\ref{thm:orbifold} (see \\S\\ref{sec:twisted}). Theorem \\ref{thm:fibrations} enables the proof of the Schur-finiteness conjecture in the following new cases:\n\\begin{corollary}[{\\cite[Cor.~1.3 and 1.5]{Fibrations}}]\\label{cor:fibrations}\nLet $q\\colon Q \\to B$ be as in Theorem \\ref{thm:fibrations}.\n\\begin{itemize}\n\\item[(i)] Assume that $B$ is a curve, and that $\\mathrm{char}(k)\\neq 2$ when $d$ is odd. Under these assumptions, the conjecture $S(Q)$ holds.\n\\item[(ii)] Assume that $B$ is a surface, that $d$ is odd, that $\\mathrm{char}(k)\\neq 2$, and that the conjecture $S(B)$ holds. Under these assumptions, the conjecture~$S(Q)$~also~holds.\n\\end{itemize}\n\\end{corollary}\nCorollary \\ref{cor:fibrations}(ii) can be applied, for example, to the case where $B$ is an open subscheme of an abelian surface or smooth projective surface with $p_g=0$ satisfying Bloch's conjecture (see Guletskii-Pedrini \\cite[\\S4 Thm.~7]{GP}). Recall that Bloch's conjecture holds for surfaces not of general type (see Bloch-Kas-Leiberman \\cite{BKL}), for surfaces which are rationally dominated by a product of curves (see Kimura \\cite{Kimura}), for Godeaux, Catanese and Barlow surfaces (see Voisin \\cite{Voisin2, Voisin}), etc.\n\\begin{remark}[Bass-finiteness conjecture]\nLet $k=\\mathbb{F}_q$ be a finite field and $X$ a smooth $k$-scheme. The Bass-finiteness conjecture (see \\cite[\\S9]{Bass}) asserts that the algebraic $K$-theory groups $K_n(X), n \\geq 0$, are finitely generated. Thanks to the work of Quillen \\cite{Grayson,Quillen2,Quillen1}, the Bass-finiteness conjecture holds when $\\mathrm{dim}(X)\\leq 1$.\n\nIn the same vein, we can consider the {\\em mod $2$-torsion} Bass-finiteness conjecture, where $K_n(X)$ is replaced by $K_n(X)_{1\/2}$. As proved in \\cite{Fibrations}, Theorem \\ref{thm:fibrations} and Corollary \\ref{cor:fibrations} hold similarly with the Schur-finiteness conjecture replaced by the mod $2$-torsion Bass-finiteness conjecture. As a consequence, we obtain a proof of the (mod $2$-torsion) Bass-finiteness conjecture in new cases.\n\\end{remark}\n\\subsection{Rigidity}\\label{sub:rigidity2}\nGiven an integer $n \\geq 2$, recall from \\cite[\\S9.9]{book} the definition of the category of mod-$n$ noncommutative mixed motives $\\mathrm{NMix}(k;\\mathbb{Z}\/n)$. By construction, given smooth proper dg categories ${\\mathcal A}$ and ${\\mathcal B}$, we have isomorphisms\n\\begin{eqnarray}\\label{eq:mod-n}\n& \\mathrm{Hom}_{\\mathrm{NMix}(k;\\mathbb{Z}\/n)}(\\mathrm{U}({\\mathcal A}), \\mathrm{U}({\\mathcal B})[-n])\\simeq K_n({\\mathcal A}^\\mathrm{op} \\otimes {\\mathcal B}; \\mathbb{Z}\/n) & n \\in \\mathbb{Z}\\,,\n\\end{eqnarray}\nwhere the right-hand side stands for mod-$n$ algebraic $K$-theory.\n\\begin{theorem}[{\\cite[Thm.~2.1(ii)]{rigidity}}]\\label{thm:rigidity2}\nGiven an extension of separably closed fields $l\/k$, the base-change functor $-\\otimes_k l \\colon \\mathrm{NMix}(k;\\mathbb{Z}\/n) \\to \\mathrm{NMix}(l;\\mathbb{Z}\/n)$ is fully-faithful whenever $n$ is coprime to the characteristic of $k$.\n\\end{theorem}\nTheorem \\ref{thm:rigidity2} is the mixed analogue of Theorem \\ref{thm:rigidity}. Intuitively speaking, it shows that the theory of mod-$n$ noncommutative mixed motives is ``rigid'' under extensions of separably closed fields. Alternatively, thanks to the isomorphisms \\eqref{eq:mod-n}, Theorem \\ref{thm:rigidity2} shows that mod-$n$ algebraic $K$-theory is ``rigid'' under extensions of separably closed fields. This is a far-reaching noncommutative generalization of Suslin's celebrated rigidity theorem \\cite{Suslin}; consult \\cite[\\S2]{rigidity} for details and also for applications to equivariant and twisted algebraic $K$-theory. In the particular case of an extension of algebraically closed fields, the commutative counterpart of Theorem \\ref{thm:rigidity2} was established by Haesemeyer-Hornbostel in \\cite[Thm.~30]{HH}. \n\\section{Noncommutative realizations and periods}\\label{sec:periods}\nIn this section we assume that the base field $k$ is perfect. Subsection \\S\\ref{sec:NCrealizations} is devoted to the noncommutative realizations associated to the (classical) cohomology theories. In \\S\\ref{sec:periods1}, making use of the noncommutative realization associated to de Rham-Betti cohomology, we extend Grothendieck's theory of periods to the broad noncommutative setting of dg categories. As an application, we prove that (modulo $2\\pi i$) Grothendieck's theory of periods is HPD-invariant.\n\\subsection{Noncommutative realizations}\\label{sec:NCrealizations}\nLet $F$ be a field of characteristic zero and $({\\mathcal C},\\otimes, {\\bf 1})$ an $F$-linear neutral Tannakian category equipped with a $\\otimes$-invertible ``Tate'' object ${\\bf 1}(1)$. In what follows, we write $\\mathrm{Gal}({\\mathcal C})$ for the Tannakian Galois group of ${\\mathcal C}$ and $\\mathrm{Gal}_0({\\mathcal C})$ for the kernel of the homomorphism $\\mathrm{Gal}({\\mathcal C}) \\twoheadrightarrow \\mathbb{G}_m$, where $\\mathbb{G}_m$ agrees with the Tannakian Galois group of the smallest Tannakian subcategory of ${\\mathcal C}$ containing ${\\bf 1}(1)$. As explained in \\cite[\\S1-2]{IMRN}, given a cohomology theory $H^\\ast\\colon \\mathrm{Sm}(k) \\to \\mathrm{Gr}^b_\\mathbb{Z}({\\mathcal C})$, we can consider the associated {\\em modified} cohomology theory:\n\\begin{eqnarray*}\nH^{\\frac{\\ast}{2}}\\colon \\mathrm{Sm}(k) \\longrightarrow \\mathrm{Rep}_{\\mathbb{Z}\/2}(\\mathrm{Gal}_0({\\mathcal C})) && X\\mapsto (\\bigoplus_{n\\,\\,\\text{even}} H^n(X), \\bigoplus_{n\\,\\,\\text{odd}} H^n(X))\n\\end{eqnarray*}\nwith values in the category of finite-dimensional $\\mathbb{Z}\/2$-graded continuous representations of $\\mathrm{Gal}_0({\\mathcal C})$. Examples of cohomology theories include Nori's cohomology theory $H^\\ast_N$ (with values in Nori's Tannakian category of mixed motives \\cite[\\S8]{Huber}), Jannsen's cohomology theory $H^\\ast_J$ (with values in Jannsen's Tannakian category of mixed motives \\cite[Part I]{Jannsen1}), de Rham cohomology theory $H^\\ast_{dR}$ (with values in the Tannakian category of finite-dimensional $k$-vector spaces), Betti cohomology theory $H^\\ast_B$ (with values in the Tannakian category finite-dimensional $\\mathbb{Q}$-vector spaces), de Rham-Betti cohomology theory $H^\\ast_{dRB}$ (with values in the Tannakian category $\\mathrm{Vect}(k,\\mathbb{Q})$ of triples $(V,W,\\omega)$, where $V$ is a finite-dimensional $k$-vector space, $W$ is a finite-dimensional $\\mathbb{Q}$-vector space, and $\\omega$ is an isomorphism $V\\otimes_k \\mathbb{C}\\simeq W\\otimes_\\mathbb{Q} \\mathbb{C}$), \\'etale $l$-adic cohomology theory $H^\\ast_{l\\text{-}\\mathrm{adic}}$ (with values in the Tannakian category of finite-dimensional $l$-adic representations of the absolute Galois group of $k$), Hodge cohomology theory $H^\\ast_{\\mathrm{Hod}}$ (with values in the Tannakian category of mixed $\\mathbb{Q}$-Hodge structures \\cite[\\S1]{Steenbrink}), etc; consult \\cite[\\S2]{IMRN} for details and for further examples. Each one of these cohomology theories gives rise to a modified cohomology theory. \n\nThe (proof of the) next result is contained in \\cite[Thms.~1.2, 2.2 and Prop.~3.1]{IMRN}:\n\\begin{theorem}\\label{thm:realizations}\nGiven a cohomology theory $H^\\ast$, there exists an additive invariant\n\\begin{equation}\\label{eq:nc-realization}\nH^{\\frac{\\ast}{2}}_{\\mathrm{nc}}\\colon \\mathsf{dgcat}(k) \\longrightarrow \\mathrm{Ind}(\\mathrm{Rep}_{\\mathbb{Z}\/2}(\\mathrm{Gal}_0({\\mathcal C})))\\,,\n\\end{equation}\nwith values in the category of ind-objects, such that $H^{\\frac{\\ast}{2}}_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}(X))\\simeq H^{\\frac{\\ast}{2}}(X)$ for every smooth $k$-scheme $X$.\n\\end{theorem}\nThe additive invariant \\eqref{eq:nc-realization} is called the {\\em noncommutative realization} associated to the cohomology theory $H^\\ast$. Morally speaking, Theorem \\ref{thm:realizations} shows that the modified cohomology theories belong not only to the realm of algebraic geometry but also to the broad noncommutative setting of dg categories. This insight goes back to Kontsevich's definition of noncommutative \\'etale cohomology theory; see \\cite{Kontsevich-talk}. Among other ingredients, the proof of Theorem \\ref{thm:realizations} makes use~of~Theorem~\\ref{thm:bridge2}.\n\\begin{remark}[Generalizations]\n\\begin{itemize}\n\\item[(i)] In the case where $k$ is of characteristic zero, $\\mathrm{Sm}(k)$ can be replaced by the category of $k$-schemes.\n\\item[(ii)] By construction, \\eqref{eq:nc-realization} can be promoted to a localizing invariant. \n\\end{itemize}\n\\end{remark}\nThe following result describes the behavior of the noncommutative realizations with respect to sheaves of differential operators in characteristic zero\\footnote{Consult \\cite[Example~2.20]{book} for the description of the behavior of {\\em all} additive invariants with respect to sheaves of differential operators in positive characteristic.}:\n\\begin{theorem}[{\\cite[Thm.~3.4]{IMRN}}]\\label{thm:D}\nLet $k$ be a field of characteristic zero, $X$ a smooth $k$-scheme, and ${\\mathcal D}_X$ the sheaf of differential operators on $X$. Given a cohomology theory $H^\\ast$, we have an isomorphism $H^{\\frac{\\ast}{2}}_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}({\\mathcal D}_X))\\simeq H^{\\frac{\\ast}{2}}(X)$.\n\\end{theorem}\n\\begin{example}[Lie algebras]\nLet $G$ be a connected semisimple algebraic $\\mathbb{C}$-group, $B$ a Borel subgroup of $G$, $\\mathfrak{g}$ the Lie algebra of $G$, and $U_{\\mathrm{ev}}(\\mathfrak{g})\/I$ the quotient of the universal enveloping algebra of $\\mathfrak{g}$ by the kernel of the trivial character. Thanks to Beilinson-Bernstein's celebrated ``localisation'' theorem \\cite{BeiBer}, it follows from Theorem \\ref{thm:D} that $H^{\\frac{\\ast}{2}}_{\\mathrm{nc}}(U_{\\mathrm{ev}}(\\mathfrak{g})\/I) \\simeq H^{\\frac{\\ast}{2}}_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}({\\mathcal D}_{G\/B})) \\simeq H^{\\frac{\\ast}{2}}(G\/B)$.\n\\end{example}\n\\begin{remark}\nTheorem \\ref{thm:D} does {\\em not} holds for every additive invariant. For example, in the case of Hochschild homology we have $HH_\\ast(\\mathrm{perf}_\\mathrm{dg}({\\mathcal D}_X))\\simeq H_{dR}^{2d-\\ast}(X)$ for every smooth affine $k$-scheme $X$ of dimension $d$; see Wodzicki \\cite[Thm.~2]{Wodzicki}. Since $H_{dR}^{2d}(X)=0$, this implies that $HH(\\mathrm{perf}_\\mathrm{dg}({\\mathcal D}_X))\\not\\simeq HH(X)$. More generally, we have $HH(\\mathrm{perf}_\\mathrm{dg}({\\mathcal D}_X))\\not\\simeq HH(A)$ for every {\\em commutative} $k$-algebra $A$.\n\\end{remark}\n\\subsection{Periods}\\label{sec:periods1}\nIn this subsection we assume that the base field is endowed with an embedding $k \\hookrightarrow \\mathbb{C}$. Consider the $\\mathbb{Z}$-graded $\\mathbb{C}$-algebra of Laurent polynomials $\\mathbb{C}[t,t^{-1}]$ with $t$ of degree $1$. Given a triple $(V,W,\\omega) \\in \\mathrm{Vect}(k,\\mathbb{Q})$, let us write ${\\mathcal P}(V,W,\\omega) \\subseteq \\mathbb{C}$ for the subset of entries of the matrix representations of $\\omega$ (with respect to basis of $V$ and $W$). In the same vein, given an object $\\{(V_n, W_n, \\omega_n)\\}_{n \\in \\mathbb{Z}}$ of the category $\\mathrm{Gr}_\\mathbb{Z}^b(\\mathrm{Vect}(k,\\mathbb{Q}))$, let us write ${\\mathcal P}(\\{(V_n,W_n,\\omega_n)\\}_{n \\in \\mathbb{Z}})$ for the $\\mathbb{Z}$-graded $k$-subalgebra of $\\mathbb{C}[t,t^{-1}]$ generated in degree $n$ by the elements of the set $P(W_n,W_n, \\omega_n)$. In the case of a smooth $k$-scheme $X$, ${\\mathcal P}(X):={\\mathcal P}(H^\\ast_{dRB}(X))$ is called the ($\\mathbb{Z}$-graded) {\\em algebra of periods of $X$}. This algebra, originally introduced by Grothendieck in the sixties, plays a key role in the study of transcendental numbers; consult, for example, the work of Kontsevich-Zagier \\cite{KZ}.\n\nConsider the quotient homomorphism $\\phi\\colon \\mathbb{C}[t,t^{-1}] \\twoheadrightarrow \\mathbb{C}[t,t^{-1}]\/\\langle 1 - (2\\pi i)t^2\\rangle$. The next result extends Grothendieck's theory of periods from schemes to dg categories:\n\\begin{theorem}[{\\cite[Thm.~4.1]{IMRN}}]\\label{thm:periods}\nThere exists an assignment ${\\mathcal A} \\mapsto {\\mathcal P}_{\\mathrm{nc}}({\\mathcal A})$, with ${\\mathcal P}_{\\mathrm{nc}}({\\mathcal A})$ a $\\mathbb{Z}\/2$-graded $k$-subalgebra of $\\mathbb{C}[t,t^{-1}]\/\\langle 1 - (2\\pi i)t^2\\rangle$, such that ${\\mathcal P}_{\\mathrm{nc}}(\\mathrm{perf}_\\mathrm{dg}(X))$ is isomorphic to $\\phi({\\mathcal P}(X))$ for every smooth $k$-scheme $X$.\n\\end{theorem}\nMorally speaking, Theorem \\ref{thm:periods} shows that Grothendieck's theory of periods can be extended from schemes to the broad noncommutative setting of dg categories as long as we work modulo $2\\pi i$. Among other ingredients, its proof makes use of the noncommutative realization associated to de Rham-Betti cohomology theory.\n\\subsubsection{Homological projective duality}\nLet $X$ and $Y$ be two HP-dual smooth projective $k$-schemes as in \\S\\ref{sub:HPD}. Recall from {\\em loc. cit.} that the category $\\mathrm{perf}(X)$ admits, in particular, a Lefschetz decomposition $\\langle \\mathbb{A}_0, \\mathbb{A}_1(1), \\ldots, \\mathbb{A}_{i-1}(i-1)\\rangle$. Given a linear subspace $L \\subset W^\\ast$, consider the linear sections $X_L:=X\\times_{\\mathbb{P}(W^\\ast)} \\mathbb{P}(L)$ and $Y_L:=Y \\times_{\\mathbb{P}(W)} \\mathbb{P}(L^\\perp)$. The next result, proved in \\cite[Thm.~4.6]{IMRN}, relates the algebra of periods of $X_L$ with the algebra of periods of $Y_L$:\n\\begin{theorem}[HPD-invariance]\\label{thm:HPD2}\nLet $X$ and $Y$ be as above. Assume that $X_L$ and $Y_L$ are smooth, that $\\mathrm{dim}(X_L)=\\mathrm{dim}(X) -\\mathrm{dim}(L)$, that $\\mathrm{dim}(Y_L)=\\mathrm{dim}(Y)- \\mathrm{dim}(L^\\perp)$, and that the category $\\mathbb{A}_0$ admits a full exceptional collection. Under these assumptions, the $\\mathbb{Z}\/2$-graded $k$-algebras $\\phi({\\mathcal P}(X_L))$ and $\\phi({\\mathcal P}(Y_L))$ are isomorphic.\n\\end{theorem}\nRoughly speaking, Theorem \\ref{thm:HPD2} shows that (modulo $2\\pi i$) Grothendieck's theory of periods is invariant under homological projective duality. This result can be applied, for example, to the Veronese-Clifford duality, to the Spinor duality, to the Grassmannian-Pfaffian duality, to the Determinantal duality, etc.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe neutron-capture (n-capture) process which dominantly creates\nheavy elements consists of the slow n-capture process (s-process) and\nthe rapid n-capture process (r-process) \\citep{burbidge1957}. The\ns-process consists of the weak s-process and the main s-process. Massive\nstars are the astrophysical sites of the weak s-process\nwhich mainly produces the lighter n-capture elements (e.g., Sr, Y\nand Zr) \\citep{lamb1977,raiteri1991,raiteri1993}. Asymptotic\ngiant branch stars are the astrophysical sites of the main\ns-process which produces both the lighter and heavier n-capture\nelements \\citep{busso1999}. The r-process consists of the weak\nr-process (or ``lighter element primary process'' (LEPP),\n\\citealt{travaglio2004}) and main r-process \\citep{cowan1991}.\nBecause the yields of both the weak r-process (or LEPP) elements\nand the light elements have primary nature, the weak r-process\nis suggested to occur in SNe II with\nthe progenitor mass $M > 10 M_{\\odot}$ \\citep{travaglio2004,montes2007}.\nThe main r-process is considered to associate with the final stages of\nmassive star evolution, yet the actual astrophysical sites have\nnot been confirmed \\citep{sneden2008,ishimaru2015,goriely2016}.\nTwo astrophysical sites of the main r-process are paid attention:\n(1) the SNe II with the progenitor mass $M \\approx 8-10 M_{\\odot}$\n\\citep{travaglio1999,wanajo2003,qian2007}\nand (2) neutron star mergers (NSMs)\n\\citep{lattimer1974,eichler1989,tsujimoto2014a,tsujimoto2014b}.\nThe r-process abundances of the solar system have been obtained\nby \\citet{kappeler1989} and \\citet{arlandini1999} using the\nresidual approach. The two components of n-capture process have\nbeen found in the elemental abundances of metal-poor stars\n\\citep{wasserburg1996,qian1998}. Based on the elemental abundances\nof the main r-process stars and the weak r-process stars, adopting\nthe iterative method, \\citet{li2013b} and \\citet{hansen2014} derived\nthe abundances of the main r-process and the weak r-process.\n\nElemental abundances of metal-poor stars reflect the chemical\ncomposition of the natal clouds polluted by different\nnucleosynthetic processes. \\citet{truran1981} studied the\nabundances of the r-process elements in the Galactic halo stars\nand suggested that the r-process enrichment in metal-poor stars\noriginates from the early massive stars. Because of the secondary\nnature (i.e., the yields are correlated with the initial stellar\nmetallicity), the s-process contributions to the n-capture\nelements in interstellar medium (ISM) are negligible at low\nmetallicities ([Fe\/H] $\\leq -1.5$). In\nthis case, the n-capture elements of metal-poor stars dominantly\ncome from the r-process \\citep{travaglio1999,burris2000}. Study of the\nabundance characteristics of the n-capture elements in metal-poor\nstars is significant for understanding the r-process nucleosynthesis\nand the chemical enrichment history of the early Galaxy.\n\\citet{gilroy1988} reported that there exists a large [r\/Fe] scatter\nfor the heavier n-capture elements at low\nmetallicity. They proposed that the scatter reflects the\ninhomogeneous mixing of products from different nucleosynthetic\nprocesses. The large [r\/Fe] scatter is also shown in more observational\nwork \\citep[e.g.,][]{mcwilliam1995,ryan1996,mcwilliam1998,\nsneden1998,burris2000,honda2004,barklem2005,francois2007,roederer2010}.\n\\citet{mathews1992} and \\citet{travaglio1999} calculated the\nGalactic evolution of the n-capture elements and suggested that the\nabundances of the heavier r-process elements should be produced by\nthe lower-mass SNe II. \\citet{sneden1994} studied the abundances of\nthe main r-process star CS 22892-052 and ascribed the Eu enrichment\nin this star to the unmixed chemical composition of the cloud swept\nup by the SN event. \\citet{mcwilliam1995} and \\citet{mcwilliam1998}\nanalyzed the chemical abundances of extremely metal-poor stars.\nThey proposed that the observed abundance\ndispersions of the heavier n-capture elements indicate the chemical\ninhomogeneities of the ISM. However, \\citet{ryan1996} suggested\nthat the different gas clouds would be entirely inhomogeneous and\nthe r-process element enrichment in a gas cloud is only due\nto the pollution from a main r-process event. \\citet{tsujimoto1999}\nstudied the [Eu\/Fe] scatter in the Galactic halo and suggested\nthat the [Eu\/Fe] scatter in metal-poor stars is the integrated\nresults of remnants of the SNe with different progenitor mass.\n\\citet{argast2000} analyzed the [Eu\/Fe] pattern of metal-poor stars using\nthe three-dimensional stochastic evolution model. They proposed that\n(1) the large [Eu\/Fe] scatter for [Fe\/H] $< -3.0$ should result from\nthe inhomogeneities of the clouds swept up by the\nSNe II with different initial mass and (2) the [Eu\/Fe] scatter\ndecreases with increasing metallicity because of the chemical mixing of\nthe ISM. \\citet{travaglio2001} calculated the Galactic halo evolution\nconsidering the chemical mixing of the clouds\nand the products of SNe. They found that the [r\/Fe] scatter at low\nmetallicity is the results of inhomogeneous mixing of the ISM and\nthe chemical enrichment of the gas clouds is dominated by the star\nformation episodes. \\citet{fields2002} presented a simple model to\nexplain the [Eu\/Fe] scatter for metal-poor stars and suggested that\nthe decreasing [Eu\/Fe] scatter with increasing metallicity is due to\nthe mixing of the sources produced by different nucleosynthetic events.\n\\citet{cescutti2008} used the stochastic evolution model to\ninvestigate the formation of the [r\/Fe] scatter in metal-poor stars\nand suggested that the different [r\/Fe] dispersions are caused by\nthe massive stars with different mass ($12 \\sim 30 M_{\\odot}$).\n\nOver the years, NSM is also argued as a plausible astrophysical site\nof main r-process which leads to the r-process enrichment in\nmetal-poor stars. Based on the observed abundances of lighter\nr-process elements (i.e., Y and Zr) and heavier r-process element Eu\nof metal-poor stars, \\citet{tsujimoto2014a} suggested that (1) the\ncore-collapse supernovae produce the lighter r-process\nelements and (2) there exist two types of NSMs: one type only\nproduces the heavier r-process elements and the other produces both\nthe lighter and heavier r-process elements. Considering the\naccretion of ISM and the chemical enrichment of intergalactic medium,\n\\citet{komiya2014} calculated the chemical evolution of the\nheavier r-process elements Ba and Eu. They found that both the SNe\nII and NSMs can reproduce the abundance patterns of the heavier\nr-process elements of extremely metal-poor stars.\n\\citet{tsujimoto2014b} studied the Eu enrichment in the Galactic\nhalo and proposed that the [Eu\/Fe] scatter in the Galactic halo\nstars can be explained by the hierarchical galaxy formation model.\nFurthermore, they suggested that NSMs should be the dominant\nastrophysical site of the main r-process. Through simulating the\nchemical and dynamic evolution of NSMs, \\citet{rosswog2014}\nsuggested that the dynamic ejecta of NSMs could produce the heavier\nr-process elements with $A > 130$.\nIn order to distinguish the products of the SN II and\nNSM events, \\citet{hotokezaka2015} calculated the abundances of the\nshort-lived element $^{244}$Pu and compared the theoretical values\nwith the observed abundances. They proposed that the NSM events can\nnaturally explain the $^{244}$Pu abundances and are an excellent\ncandidate of the astrophysical origins of the heavier r-process\nelements. By comparing the calculated abundances with the observed\nabundances of CS 22892-052, \\citet{ramirez2015} concluded that\nNSMs are a favorable source of the heavier r-process elements in\nCEMP-r stars. They also found that the NSM events could naturally\nexplain the scatter of observed\nEu abundances in CEMP-r stars. Based on the discovery of strong\nr-process enhanced stars in the ultra-faint dwarf galaxy Reticulum\nII, \\citet{ji2016} suggested that the r-process material in\nReticulum II should be produced by the single NSM\nevent. For examining the NSMs as astrophysical origins of the\nr-process elements in dwarf galaxies, \\citet{beniamini2016a}\ncalculated the proper motion and the time until merger of neutron\nstar binaries. They concluded that (1) more than 50$\\%$ of neutron\nstar binaries have small proper motion to avoid the pollution of SN\nejecta and (2) more than 90$\\%$ of neutron star binaries merge\nwithin 300 Myr. These results indicate that NSMs could be\nnaturally responsible for the observed r-process abundances of the\nmetal-poor stars of the early Galaxy. Based on the hierarchical\nchemical evolution model, \\citet{komiya2016} simulated\nthe abundances of the r-process elements for metal-poor stars after\nconsidering the pollution of NSM ejecta. They suggested that the NSM\nscenario can successfully reproduce the [r\/Fe] scatter for\nmetal-poor stars. Through calculating the rate and yields for the\nr-process event in dwarf galaxies, \\citet{beniamini2016b} found\nthat the dwarf galaxies and the Milky Way share the same r-process\nmechanism.\n\nFor metal-poor stars, the $\\alpha$ elements (e.g., Mg, Si, Ca and\nTi) and Fe originate from massive stars\n\\citep{woosley1995,kobayashi2006, heger2010,mishenina2013}. To\nexplore the abundance correlations between $\\alpha$ elements and\nmain r-process elements, \\citet{li2014} plotted the observed\n[$\\alpha$\/Eu] as a function of [Eu\/Fe] for metal-poor stars and\nfound that the observed [$\\alpha$\/Eu] ratios decrease linearly with\nincreasing [Eu\/Fe] and the slope is close to $-1$. The results mean\nthat the abundances of $\\alpha$ elements and Fe have no correlation\nwith those of Eu, which indicates that the astrophysical site\nproducing main r-process elements does not produce $\\alpha$ elements\nand Fe. The observational and theoretical results of the previous\nwork imply that the [r\/Fe] scatter is dominated by more than one\nfactor. Obviously, the upper and lower boundaries of [r\/Fe] could\nprovide useful clues about the extreme situations of the chemical\nenrichment of the gas clouds. So the quantitative study of the\n[r\/Fe] boundaries in metal-poor stars is important. In this paper,\nwe investigate the formation of the [r\/Fe] boundaries for the\nheavier n-capture elements ($Z \\geq 56$) of metal-poor stars in\n\\S\\,2. Conclusions are presented in \\S\\,3.\n\n\\section{Abundance Boundaries of [r\/Fe] for the Heavier n-capture\nElements in Metal-poor Stars}\n\nGenerally, Eu is deemed as a typical main r-process element in the\nsolar system. The left panel of Figure \\ref{f1.eps} shows the\nobserved [Eu\/Fe] as a function of [Fe\/H] for the metal-poor stars\n\\citep{westin2000,cowan2002,hill2002,johnson2002,sneden2003,christlieb2004,\nhonda2004,barklem2005,ivans2006,honda2006,honda2007,francois2007,lai2008,hayek2009,\naoki2010,mashonkina2010,roederer2010,roederer2014,hollek2011,hansen2012,hansen2015,\nsiqueira2012,cohen2013,yong2013,mashonkina2014,jacobson2015,li2015a,li2015b,siqueira2016}.\nIn order to avoid the effect of s-process, we select the sample\nstars with [Fe\/H] $\\leqslant -1.5$ and [Ba\/Eu] $< 0$. The dotted line\nwith $-1$ slope in the panel represents the sensitivity limit for\nthe observing Eu, which is 0.8 dex lower than that adopted by\n\\citet{travaglio2001}. From the panel we can see that the [Eu\/Fe]\nscatter is larger than 2 dex at [Fe\/H] $= -3.0$, whereas the scatter\ndecreases with increasing metallicity. The observed upper boundary\nof [Eu\/Fe] is explicit: it is close to a straight line with the\nslope $\\thicksim -1$ for [Fe\/H] $\\lesssim -2.5$, while the\ndecreasing trend flattens for [Fe\/H] $> -2.5$. On the other hand,\nthe observed lower boundary shows a sharp increasing trend for\n[Fe\/H] $\\lesssim -2.5$, while the increasing trend flattens for\n[Fe\/H] $> -2.5$.\n\nIn order to explain the upper and lower boundaries of [Eu\/Fe] for\nthe metal-poor stars, we consider the initial chemical composition\nof the gas clouds and the pollution from the two nucleosynthetic\nevents: (1) the main r-process event and (2) the SN II event that\nejects primary Fe (hereafter simply the SN II-Fe event). The main\nr-process event (e.g., the SN II with the progenitor mass $M \\approx\n8-10 M_{\\odot}$ or NSM) produces and ejects the main r-process\nelements \\citep{travaglio1999,qian2007,tsujimoto2014b}. Whereas\nthe SN II-Fe event (i.e., the SN II with the progenitor mass\n$M > 10 M_{\\odot}$) mainly produces the weak r-process (or LEPP)\nelements and ejects light elements and Fe group elements\nsimultaneously \\citep{travaglio2004,montes2007,li2013a}.\n\nFor the main r-process events, although two possible sites\n(i.e., the SNe II with the progenitor mass $M \\approx 8-10 M_{\\odot}$\nor NSMs) attract common attention, the actual main r-process\nsites and corresponding main r-process yields have not been determined.\nFor a gas cloud swept up by a single main r-process event,\nthe relationship between the Eu abundance $N_{Eu}$ and the Fe\nabundance $N_{Fe}$ is\n\\begin{equation}\n\\frac{N_{Eu}}{N_{Fe}}=(\\frac{A_{Fe}}{A_{Eu}})(\\frac{M_{SW,rm}X_{Eu}+Y_{Eu}}{M_{SW,rm}X_{Fe}}),\n\\end{equation}\nwhere $M_{SW,rm}$ is the mass of the cloud. $X_{Eu}$ and $X_{Fe}$\nare the initial mass fractions of Eu and Fe in the cloud. $Y_{Eu}$\nis the Eu yield of the main r-process event. $A_{Eu}$ and $A_{Fe}$\nare the atomic weights of Eu and Fe, respectively. Based on the\nobserved abundances of Ba and Eu, \\citet{cescutti2006} computed the\nmean [Ba\/Fe] and [Eu\/Fe] ratios in different [Fe\/H] bins. In this\nwork, we adopt the mean [Eu\/Fe] ratios presented by\n\\citet{cescutti2006} as the initial abundance ratios ([Eu\/Fe]$_{ini}$)\nof the gas clouds, which are plotted in the left panel of\nFigure \\ref{f1.eps} by the dashed line. The upper limits of [Eu\/Fe]\nof the clouds are calculated for two cases.\n\nCase A: the pollution from the SNe II with the progenitor mass $M\n\\approx 8-10 M_{\\odot}$. The core-collapse SNe should consist of two\nroutes \\citep[e.g.,][]{qian2007}: (1) the Fe core-collapse SNe with\nthe progenitor mass $M \\gtrsim 11 M_{\\odot}$ from which the light\nelements and Fe group elements are ejected and (2) the O-Ne-Mg\ncore-collapse SNe with the progenitor mass $M \\approx 8-10\nM_{\\odot}$ in which the light elements and Fe group elements are not\nproduced. The main r-process may take place in O-Ne-Mg core-collapse\nSNe. In this case, the astrophysical sites ejecting main r-process\nelements do not eject light elements and Fe group elements. Owing to\nthe high Eu abundance ([Eu\/Fe] $= 1.92$) and low metallicity ([Fe\/H] $=\n-3.36$), the sample star SDSS J2357-0052 \\citep{aoki2010} is taken as\nthe representative star and [Eu\/Fe] $= 1.92$ is taken as the upper\nlimit at [Fe\/H] $= -3.36$. Adopting the Eu yield of the main\nr-process event $Y_{Eu} = 6.4\\times10^{-7} M_{\\odot}$\n\\citep{travaglio2001}, from equation (1) we can derive $M_{SW,rm} =\n4.5 \\times 10^{4} M_{\\odot}$. Using the derived cloud mass, from\nequation (1) we can derive the upper limits ([Eu\/Fe]$_{up}$) for the\nclouds with different metallicities which were swept up by the SNe\nII with the progenitor mass $M \\approx 8-10 M_{\\odot}$. The\ncalculated results are shown in the left panel of Figure\n\\ref{f1.eps} by the solid line. We can see that the calculated upper\nboundary is close to a straight line with the slope $\\thicksim -1$\nfor [Fe\/H] $\\lesssim -2.5$, while the decreasing trend flattens for\n[Fe\/H] $> -2.5$. Obviously, the calculated results are consistent\nwith the observed upper boundary of [Eu\/Fe] of the sample stars.\n\nCase B: the pollution from NSMs. Based on the NSM scenario,\n\\citet{komiya2016} explored the [r\/Fe] scatter of metal-poor stars\nusing the hierarchical galaxy formation model. They suggested that\nthe Eu yield of a NSM event is about $1.5\\times10^{-4} M_{\\odot}$\nand the mass of the cloud swept up by the single main r-process\nevent is about $10^{7} M_{\\odot}$. For the NSM scenario, the Eu\nyield is about 3 orders of magnitude higher than what is adopted for\nCase A and the gas mass that Eu is diluted into is also about 3\norders of magnitude higher than what is adopted for Case A. Adopting\nthe Eu yield $Y_{Eu} = 1.5\\times10^{-4} M_{\\odot}$ and the polluted\ncloud mass $M_{SW,rm} = 10^{7} M_{\\odot}$, we calculate the upper\nlimits of [Eu\/Fe] for the clouds swept up by NSMs. The calculated\nresults are plotted in the left panel of Figure \\ref{f1.eps} by the\ndotted-dashed line. We can see that the upper boundaries of [Eu\/Fe] for\nthe two cases are very similar.\n\nFor illustrating the formation of the upper boundary of [Eu\/Fe]\nclearly, in the right panel we use the up arrows to show the jumps\nof [Eu\/Fe]. For the cloud with the initial ratios [Eu\/Fe]$_{ini} = -0.59$\nand [Fe\/H] $= -3.5$, the final ratio [Eu\/Fe]$_{up} \\approx 2.06$ means\nthat the [Eu\/Fe] ratio jumps from $-0.59$ to 2.06 because of the\npollution from the single main r-process event. In this case, the\nincreasing value of [Eu\/Fe] reaches about 2.65 dex, since (1) the initial Fe\nmass fraction $X_{Fe}$ of the cloud is small and (2) the initial Eu\nmass in the cloud is much lower than the Eu yield of the main\nr-process event (i.e., $M_{SW,rm}X_{Eu} \\ll Y_{Eu}$).\nOn the other hand, for the cloud with the initial ratios\n[Eu\/Fe]$_{ini} = 0.44$ and [Fe\/H] $= -1.5$, the final ratio\n[Eu\/Fe]$_{up} \\approx 0.60$ means that the increasing value of [Eu\/Fe] is\nonly about 0.16 dex. The two reasons lead to the flat of the upper boundary\nof [Eu\/Fe]: (1) the cloud contains more Fe and (2) the initial Eu mass\nin the cloud is higher than the Eu yield of the main r-process event\n(i.e., $M_{SW,rm}X_{Eu} > Y_{Eu}$). The derived results mean that,\nfor a cloud from which metal-poor stars formed, the formation of the\nupper limit of [Eu\/Fe] is mainly due to the pollution from a single\nmain r-process event.\n\nFor a gas cloud swept up by a single SN II-Fe event, the\nrelationship between the Eu abundance $N_{Eu}$ and the Fe abundance\n$N_{Fe}$ is\n\\begin{equation}\n\\frac{N_{Eu}}{N_{Fe}}=(\\frac{A_{Fe}}{A_{Eu}})(\\frac{M_{SW,pri}X_{Eu}}{M_{SW,pri}X_{Fe}+Y_{Fe}}),\n\\end{equation}\nwhere $M_{SW,pri}$ is the mass of the cloud. $Y_{Fe}$ is the Fe\nyield of the SN II-Fe event. We take the typical weak r-process star\nHD 122563 ([Fe\/H] $= -2.77$, [Eu\/Fe] $= -0.52$, \\citealt{honda2006})\nas the representative star and take [Eu\/Fe] $= -0.52$ as the lower\nlimit at [Fe\/H] $= -2.77$. Adopting the polluted cloud mass\n$M_{SW,pri} = 6.5\\times10^{4} M_{\\odot}$ \\citep{tsujimoto1999},\nfrom equation (2) we\ncan derive $Y_{Fe} = 0.1 M_{\\odot}$. Using the derived Fe yield,\nfrom equation (2) we can derive the lower limits ([Eu\/Fe]$_{lw}$) for\nthe clouds with different metallicities, which are shown in the left\npanel of Figure \\ref{f1.eps} by the solid line. We can see that the\ncalculated lower boundary shows a sharp increasing trend for [Fe\/H]\n$\\lesssim -2.5$ and a mild increasing trend for [Fe\/H] $> -2.5$.\nThere are some observed [Eu\/Fe] ratios that are significantly lower\nthan the calculated lower boundary in the range $-2.5 <$ [Fe\/H] $<\n-2.2$. Note that the initial [Eu\/Fe] ratio of a cloud takes some\neffect on the lower limit of [Eu\/Fe]. In this work, we take the\nobserved mean [Eu\/Fe] ratios as the initial [Eu\/Fe] ratios of the\nclouds. Obviously, the actual initial ratios of the clouds could\ndeviate from the mean ratios. If the actual initial [Eu\/Fe] ratio\nof a cloud swept up by a SN II-Fe event is lower than the mean\n[Eu\/Fe] ratio, the actual lower limit of [Eu\/Fe] should be lower\nthan the calculated lower limit plotted in Figure \\ref{f1.eps}.\nThe observed low [Eu\/Fe] ratios should imply that the actual initial\n[Eu\/Fe] ratios of these clouds are lower than the mean [Eu\/Fe] ratios.\nFurthermore, because the explosion energy of SNe II depends on their\nprogenitor mass \\citep{kobayashi2006,nomoto2013} and the mass of the\nclouds swept up by the SNe II increases with increasing explosion\nenergy \\citep{machida2005}, the mass of the clouds swept up by the\nSNe II with the progenitor mass $M > 10 M_{\\odot}$ should be larger\nthan that of the clouds swept up by the SNe II with the progenitor\nmass $M \\approx 8-10 M_{\\odot}$.\n\nFor illustrating the formation of the lower boundary of [Eu\/Fe]\nclearly, in the right panel we use the inclined arrows to represent\nthe jumps of [Eu\/Fe] and [Fe\/H]. For the cloud with the initial\nratios [Eu\/Fe]$_{ini} = -0.60$ and [Fe\/H] $= -3.5$, the final\nratios [Eu\/Fe]$_{lw} = -1.25$ and [Fe\/H] $= -2.85$ imply that the\n[Fe\/H] ratio jumps from $-3.5$ to $-2.85$ because of the pollution\nfrom the single SN II-Fe event. In this case, the increasing value\nof [Fe\/H] reaches 0.65 dex, since the initial Fe mass in the cloud\nis lower than the Fe yield of the SN II-Fe event (i.e.,\n$M_{SW,pri}X_{Fe} < Y_{Fe}$). On the other hand, for the cloud with\ninitial ratios [Eu\/Fe]$_{ini} = 0.44$ and [Fe\/H] $= -1.5$, the\nfinal ratios [Eu\/Fe]$_{lw} = 0.42$ and [Fe\/H] $= -1.48$ mean that\nthe increasing value of [Fe\/H] is only 0.02 dex. The two reasons\nlead to the flat of the lower boundary of [Eu\/Fe]: (1) the cloud\ncontains more Eu and (2) the initial Fe mass in the cloud is higher\nthan the Fe yield of the SN II-Fe event (i.e., $M_{SW,pri}X_{Fe} >\nY_{Fe}$). The derived results mean that, for a cloud from which\nmetal-poor stars formed, the formation of the lower limit of [Eu\/Fe]\nis mainly due to the pollution from a single SN II-Fe event.\n\nThe first generation of very massive stars only produces light\nelements and Fe group elements \\citep{komiya2014}. This abundance\npattern is represented by the prompt (P) component \\citep{qian2001}\nwhich should be responsible for the origin of light elements and Fe\nfor [Fe\/H] $\\lesssim -3.5$. For the clouds with [Fe\/H] $\\gtrsim -3.5$,\nthe element Eu originates from the main r-process events (e.g., the\nSNe II with the progenitor mass $M \\approx 8-10 M_{\\odot}$ or NSMs)\nand the element Fe originates from (1) the prompt inventory\n(P-inventory) of Fe at the early universe \\citep{qian2001} and (2)\nthe SN II-Fe events. The effect of the P-inventory became smaller\nwhen the SN II-Fe events began to pollute the ISM \\citep{qian2001,li2013b}.\nOn the other hand, for the most stars with [Fe\/H] $\\lesssim -3.5$,\nthe element Fe should mainly originate from the P-inventory\nand the observed r-process abundances should be ascribed to the\nsurface pollution by the ISM which had contained r-process material\n\\citep{komiya2014,komiya2016}.\n\nFigures \\ref{f2.eps} and \\ref{f3.eps} show the observed [r\/Fe] (r:\nBa, La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Hf, Os, Ir,\nPt, Au and Pb) as a function of [Fe\/H] for the metal-poor stars.\nFrom the figures we can see that the observed trends of the upper\nand lower boundaries of [r\/Fe] are similar to those of [Eu\/Fe]. For\na gas cloud swept up by a single main r-process event, the\nrelationship between the r-process abundance $N_{i,r}$ and the Fe\nabundance $N_{Fe}$ is:\n\\begin{equation}\n\\frac{N_{i,r}}{N_{Fe}}=(\\frac{A_{Fe}}{A_{i}})(\\frac{M_{SW,rm}X_{i}+Y_{i,rm}}{M_{SW,rm}X_{Fe}}),\n\\end{equation}\nwhere $X_{i}$ is the initial mass fraction of the $i$th element.\n$Y_{i,rm}$ is the main r-process yield. $A_{i}$ is the atomic weight.\nThe main r-process yield $Y_{i,rm}$ can be derived as:\n\\begin{equation}\nY_{i,rm}=\\frac{A_{i}N^{*}_{i,rm}}{A_{Eu}N^{*}_{Eu}}Y_{Eu},\n\\end{equation}\nwhere $N^{*}_{i,rm}$ and $N^{*}_{Eu}$ are the component abundances,\nwhich are adopted from \\citet{li2013b}. Using the initial [Eu\/Fe]\nratios of the clouds adopted from \\citet{cescutti2006}\nand the component abundances adopted from \\citet{li2013b}, we can\nderive the initial [r\/Fe] ratios for the clouds, which are\nplotted in Figures \\ref{f2.eps} and \\ref{f3.eps} by the dashed lines.\nThe upper limits of [r\/Fe] of the clouds are also calculated for two\ncases.\n\nCase A: the pollution from the SNe II with the progenitor mass\n$M \\approx 8-10 M_{\\odot}$.\nAdopting the Eu yield $Y_{Eu} = 6.4\\times10^{-7} M_{\\odot}$ and the\npolluted cloud mass $M_{SW,r,m} = 4.5 \\times 10^{4} M_{\\odot}$, from\nequations (3) and (4) we can derive the upper limits of [r\/Fe]\nfor the clouds swept up by the SNe II with the progenitor mass\n$M \\approx 8-10 M_{\\odot}$. The results are plotted in Figures\n\\ref{f2.eps} and \\ref{f3.eps} by the solid lines.\n\nCase B: the pollution from NSMs. Adopting the Eu yield $Y_{Eu} =\n1.5\\times10^{-4} M_{\\odot}$ and the polluted cloud mass\n$M_{SW,rm} = 10^{7} M_{\\odot}$, we derive the upper limits\nof [r\/Fe] for the clouds swept up by NSMs, which are plotted\nin Figures \\ref{f2.eps} and \\ref{f3.eps} by the dotted-dashed lines.\nObviously, the calculated upper boundaries are close to the observed\nupper boundaries of [r\/Fe] of the sample stars. These results could\nprovide more supports for the suggestion that the formation of the\nupper limits of [r\/Fe] in a cloud is mainly due to the pollution\nfrom a single main r-process event.\n\nFor a gas cloud swept up by a single SN II-Fe event, the\nrelationship between the r-process abundance $N_{i,r}$ and the Fe\nabundance $N_{Fe}$ is:\n\\begin{equation}\n\\frac{N_{i,r}}{N_{Fe}}=(\\frac{A_{Fe}}{A_{i}})(\\frac{M_{SW,pri}X_{i}+Y_{i,pri}}{M_{SW,pri}X_{Fe}+Y_{Fe}}),\n\\end{equation}\nwhere $Y_{i,pri}$ is the primary yield of the $i$th\nelement from the SN II-Fe event. The primary yield $Y_{i,pri}$\ncan be derived as:\n\\begin{equation}\nY_{i,pri}=\\frac{A_{i}N^{*}_{i,pri}}{A_{Fe}N^{*}_{Fe}}Y_{Fe},\n\\end{equation}\nwhere $N^{*}_{i,pri}$ and $N^{*}_{Fe}$ are the component abundances,\nwhich are adopted from \\citet{li2013b}. Adopting the derived Fe yield\n$Y_{Fe} = 0.1 M_{\\odot}$ and the polluted cloud mass\n$M_{SW,pri} = 6.5\\times10^{4} M_{\\odot}$, from equations (5) and (6)\nwe can derive the lower limits of [r\/Fe] for the clouds with\ndifferent metallicities. The results are plotted in Figures\n\\ref{f2.eps} and \\ref{f3.eps} by the solid lines. We can see that\nthe lines are close to the observed lower boundaries of [r\/Fe] of\nthe sample stars. The results also mean that the formation of the\nlower limits of [r\/Fe] in a cloud is mainly due to the pollution\nfrom a single SN II-Fe event.\n\n\\section{Conclusions}\n\nThe [r\/Fe] scatter of the heavier n-capture elements in metal-poor\nstars preserves excellent information of the star formation history\nand provides important insights into the various situations of the\nGalactic chemical enrichment. In this respect, the upper and lower\nboundaries of [r\/Fe] could present useful clues for investigating\nthe extreme situations of the chemical enrichment of the clouds. In\nthis paper, we investigate the formation of the upper and lower\nboundaries of [r\/Fe] for the clouds. The main results are listed as\nfollows:\n\n1. Based on the assumptions of (1) the progenitor of the main\nr-process event does not produce Fe and (2) the yields of the main\nr-process event possess the primary nature (i.e., the yields are\nuncorrelated with the initial stellar metallicity), we find that the\nobserved upper boundaries of [r\/Fe] can be explained. For the clouds\nwith the initial metallicities [Fe\/H] $\\lesssim -2.5$, the upper\nboundaries of [r\/Fe] are close to straight lines with slopes\n$\\thicksim -1$, since the mass of the initial r-process elements in\neach of the clouds is lower than the yields of the single main\nr-process event. On the other hand, for the clouds with the initial\nmetallicities [Fe\/H] $> -2.5$, the upper boundaries of [r\/Fe] show\nmild decreasing trends, since the mass of the initial r-process\nelements in each of the clouds is close to or higher than the yields\nof the single main r-process event.\n\n2. Based on the assumptions of (1) the progenitor of the SN II-Fe\nevent does not produce the main r-process elements and (2) the yields of\nthe SN II-Fe event possess the primary nature, we find that the\nobserved lower boundaries of [r\/Fe] can be explained. For the clouds\nwith the initial metallicities [Fe\/H] $\\lesssim -2.5$, the lower\nboundaries of [r\/Fe] show sharp increasing trends, since the initial\nFe mass in each of the clouds is lower than the Fe yield of the\nsingle SN II-Fe event. On the other hand, for the clouds with the\ninitial metallicities [Fe\/H] $> -2.5$, the lower boundaries of [r\/Fe]\nshow mild increasing trends, since the initial Fe mass in each of\nthe clouds is close to or higher than the Fe yield of the single SN\nII-Fe event.\n\n3. The observed upper and lower boundaries of [r\/Fe] present the\nextreme situations of the chemical enrichment in the early Galaxy.\nThe calculated results mean that, for a cloud from which\nmetal-poor stars formed, the formation of the upper limits of [r\/Fe]\nis mainly due to the pollution from a single main r-process event.\nFor a cloud from which metal-poor stars formed, the formation of\nthe lower limits of [r\/Fe] is mainly due to the pollution from a\nsingle SN II-Fe event.\n\nOur results may provide useful clues for investigating the star\nformation history and the early Galactic chemical evolution.\nObviously, more observational and theoretical studies of the main\nr-process event and the SN II-Fe event are desirable.\n\n\n\\acknowledgments\n\nThis work has been supported by the National Natural Science\nFoundation of China under Grants 11673007, 11273011, U1231119,\n10973006, 11403007, 11547041, 11643007 and 11273026, the Natural\nScience Foundation of Hebei Province under Grant A2011205102, the\nProgram for Excellent Innovative Talents in University of Hebei\nProvince under Grant CPRC034, and the Innovation Fund Designated\nfor Graduate Students of Hebei Province under Grant sj2016023.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe big puzzle of modern cosmology is the fact that most of the 95\\% of the {\\it cosmic pie} is composed by dark energy and dark matter. Despite the huge experimental efforts to detect these elusive components at fundamental level, there is no final evidence, at the moment, confirming their particle nature.\nOn the other hand, clustering phenomena at any galactic and extragalactic scale and accelerated expansion of the Hubble flow point out that the paradigm based on the General Relativity and the Standard Model of Particles is not sufficient to explain the observations. According to this state of art, it is mandatory to search for alternative explanations in order to address the cosmological phenomenology unless incontrovertible tests for new particles are found. \n\nSpecifically, observations indicate a spatially flat homogeneous and isotropic universe on large scale. This property is described\nby the Friedmann-Robertson-Walker (FRW) metric. The observed facts require that the Einstein equations \n$ R_{ij} - \\tfrac{1}{2} g_{ij} R= \\kappa T_{ij} $ \ncontain a stress-energy tensor $T_{ij}=T_{ij}^{matt} + T_{ij}^{dark}$ of a perfect fluid that accounts for ordinary matter as well as\n dark matter and dark energy. \nHowever, the alternative road consists in explaining the dark fluid $T_{ij}^{dark}$ as a geometrical effect. In other words, one can attempt at reproducing the observed facts by modifying the left-hand side\nof the equations of gravity, within the appropriate frame of FRW space-times. This is done by replacing\nthe Hilbert-Einstein action, linear in the Ricci scalar $R$, with approaches where more general curvature \n \\cite{CF08,CDL11,Nojiri17} or torsion \\cite{Manos} invariants are considered into the dynamics. The so called $f(R)$ gravity has been the first example in this direction \\cite{Cap}.\n \n\nWith this premise, it has been proved that the stress-energy tensor has the perfect-fluid form\nin any $f(R)$ cosmology in generalized FRW space-times with harmonic Weyl tensor \\cite{CMM2018}. This means that the additional terms, produced by extending geometry (that is from $R$ to functions $f(R)$, non-linear in $R$), sum up to a\nperfect-fluid tensor that may well fit as a source term in the right-hand side of the Einstein equations.\\\\\nIn this paper, the result is generalized to $f (R,{\\cal G})$ theories of gravitation in FRW space-times, where $f (R,{\\cal G})$ is an \narbitrary analytical function of the scalar curvature $R$ and of the Gauss-Bonnet topological invariant \n\\begin{align}\\label{2.2}\n{\\cal G}=R_{jklm}R^{jklm} -4 R_{jk}R^{jk} +R^2\\,.\n\\end{align}\nModified gravity with Gauss-Bonnet scalar in the form $R+f({\\cal G})$ was first introduced in the context of FRW metric, as alternative to dark energy for the late acceleration of the universe \\cite{Nojiri05}. Several other investigations followed \\cite{Cognola06,Santos18,Benetti18, MF, Odintsov17} because the Gauss-Bonnet term results useful to regularize the gravitational theory for quantum fields in curved spaces \\cite{CDL11} and improves the efficiency of inflation giving rise to multiple accelerated expansions \\cite{Paolella} because ${\\cal G}$ behaves as a further {\\it scalaron} besides $R$ \\cite{Starobinsky80}. \n\n\n\nIn $n$ dimensions the gravitational action is \n\\begin{align}\\label{2.1}\nS=\\frac{1}{2\\kappa}\\int d^nx \\sqrt{-g} f(R,{{\\cal G}}) +S^{matt} ,\n\\end{align}\nwhere $S^{matt}$ is the action term of standard matter fields. The first variation in the metric gives the field equations\n\\begin{align}\\label{2.3}\nR_{kl} -\\tfrac{1}{2} g_{kl} R= \\kappa T^{matt}_{kl} + \\Sigma_{kl}\n\\end{align}\nwhere $T^{matt}_{kl} $ results from $S^{matt}$ and the tensor $\\Sigma_{kl}$ arises from the geometry. \nAssuming $f_R = \\partial_R f $ and $f_{{\\cal G}} =\\partial_{\\cal G} f $, the latter is \\cite{Atazadeh14}:\n\\begin{align}\n\\Sigma_{kl} =&( \\nabla_k\\nabla_l -g_{kl}\\nabla^2) f_R + 2R (\\nabla_k\\nabla_l - g_{kl}\\nabla^2) f_{\\cal G} -4(R_k{}^m\\nabla_m\\nabla_l +R_l{}^m\\nabla_m\\nabla_k)f_{\\cal G} \\nonumber\\\\\n& +4 (R_{kl}\\nabla^2 + g_{kl} R^{pq}\\nabla_p\\nabla_q + R_{kpql}\\nabla^p\\nabla^q) f_{\\cal G} -\\tfrac{1}{2}g_{kl}(R f_R+{\\cal G} f_{\\cal G} - f) \\label{Sigma} \\\\\n& + (1-f_R) (R_{kl}-\\tfrac{1}{2} g_{kl}R) \\nonumber\n\\end{align}\nDespite of the complexity of the expression, we prove the following:\n\\begin{theorem}\nIn a Friedmann-Robertson-Walker space-time of dimension $n$, for any analytical $f(R,{\\cal G})$ model of gravity, \nthe tensor $\\Sigma_{kl}$ is a perfect fluid of the form\n\\begin{align} \\frac{1}{\\kappa}\\Sigma_{kl}= (p+\\mu) u_ku_l+p g_{kl} \n\\label{perfect}\n\\end{align}\n\\end{theorem}\n\\noindent\nThe scalar fields $p$ and $\\mu$ can be interpreted as effective pressure and energy density produced by \nthe geometry in the locally comoving frame ($u^0=1$).\nTheir explicit expressions will be given in Eq.\\eqref{expsigma} and following ones. \n\nWe have adopted the following notations: for a scalar $S$ we use $\\dot S = u^m\\nabla_m S $, for a vector $v^k$ we write \n$v^2$ for $v^k v_k$, and $\\nabla^2$ for $\\nabla^k\\nabla_k$. The signature is ($-,+,\\dots,+$).\nBefore giving the proof of the above statement, let us discuss the covariant description of physical quantities in FRW space-times.\n\\section{Covariant description of FRW space-times}\nThe perfect fluid representation of $f(R,{\\cal G})$ gravity can be achieved in the framework of a covariant description of FRW space-times. Based on a theorem by Chen \\cite{Chen14},\\cite{GRWSurv17}\na FRW space-time is\ncharacterized by the Weyl tensor being zero and by the existence of a time-like vector field $u^k$ ($u^k u_k =-1$) such that \\cite{[29]}\n\\begin{align} \\nabla_j u_k =\\varphi (u_ju_k + g_{jk}) \\end{align}\nwhere $\\varphi $ is a scalar field such that $\\nabla_i \\varphi = - u_i \\dot\\varphi $.\nThe scalar $\\varphi $ coincides with\nHubble's parameter $H=\\dot a\/a$ in the comoving frame, where $a(t)$ is the scale factor of the FRW metric \nin its standard warped expression. By evaluating the Riemann tensor\n\\begin{equation}\nR_{jkl}{}^mu_m = [\\nabla_j,\\nabla_k]u_l = (\\dot\\varphi+\\varphi^2)(u_k g_{jl} - u_j g_{kl})\\,, \n\\end{equation}\nand using it in the equation for the Weyl tensor, $0=C_{jkl}{}^mu_m $, one obtains the Ricci tensor \nof a FRW space-time \\cite{MaMoJMP16}. It has the perfect fluid structure being:\n \\begin{align}\nR_{kl} = \\frac{R-n\\xi}{n-1} u_k u_l + \\frac{R-\\xi}{n-1} g_{kl} \\label{Ricci}\\,,\n\\end{align}\nwith the Ricci scalar $R=R^k{}_k$. The eigenvalue is \n\\begin{equation}\n\\xi = (n-1)(\\dot \\varphi + \\varphi^2) =(n-1)\\ddot a\/a\\,.\n\\end{equation} \nThe curvature scalar is then \n\\begin{align}\nR = \\frac{R^*}{a^2} + (n-1)(n-2)\\varphi^2 +2\\xi\n\\end{align}\nwhere $R^*$ is the constant (in space-time) curvature of the space-like surfaces orthogonal to $u^k$.\nIt solves the equation resulting from the covariant derivative of $R_{kj}u^j=\\xi u_k$, that is:\n\\begin{align}\n\\dot R -2\\dot\\xi = -2\\varphi (R - n\\xi)\\,. \n\\end{align} \nAn application of this formalism to a toy-model in cosmology is done in \\cite{toy}.\n\nIn \\cite{CMM2018} we proved the following relations, valid for generalized FRW space-times with harmonic Weyl tensor and,\nin particular, in FRW space-times:\n\\begin{gather}\n\\nabla_k R = -u_k \\dot R, \\qquad \\nabla_k\\nabla_l R = -\\varphi \\dot R \\, g_{kl} - (\\varphi \\dot R-\\ddot R) u_k u_l\\,, \\label{nabR}\\\\\n\\nabla_k \\xi = -u_k \\dot \\xi, \\qquad \\nabla_k\\nabla_l \\xi = -\\varphi \\dot \\xi \\, g_{kl} - (\\varphi \\dot \\xi-\\ddot \\xi) u_k u_l\\,. \\label{nabxi} \n\\end{gather}\n\n\\begin{proposition} In a FRW space-time it is:\n\\begin{equation}\n\\nabla_k {\\cal G} = -u_k \\dot {\\cal G}, \\qquad \\nabla_k\\nabla_l {\\cal G} = -\\varphi \\dot {\\cal G} \\, g_{kl} - (\\varphi \\dot {\\cal G}-\\ddot {\\cal G}) u_k u_l \n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIt is advantageous to express the Gauss-Bonnet scalar ${\\cal G}$ in terms of the Weyl scalar\n\\begin{equation}\nR_{jklm}R^{jklm}=\nC_{jklm}C^{jklm} +\\frac{4}{n-2}R_{jk}R^{jk}-\\frac{2}{(n-1)(n-2)}R^2\\,,\n\\end{equation}\nin the form\n\\begin{align*}\n{\\cal G} = C_{jklm}C^{jklm} - 4 \\frac{n-3}{n-2} R^{kl}R_{kl} + \\frac{n(n-3)}{(n-1)(n-2)} R^2 \n\\end{align*}\nIn a FRW space-time it is $C_{jklm}=0$ then, by Eq.\\eqref{Ricci}, we get:\n \\begin{equation}\nR_{kl} R^{kl}= \\xi^2+\\frac{1}{n-1} (R-\\xi)^2\\,.\n\\end{equation}\n The Gauss-Bonnet invariant reduces to\n \\begin{equation}\n{\\cal G} = \\frac{n-3}{(n-1)(n-2)}[(n-4)R+2n\\xi](R-2\\xi)\\,.\n\\end{equation}\nIt follows that\n\\begin{equation}\n \\nabla_k {\\cal G} = \\tfrac{n-3}{(n-1)(n-2)}[2R(n-4)\\nabla_k R -8n\\xi \\nabla_k \\xi + 8\\xi\\nabla_k R + 8 R \\nabla_k \\xi]\\,.\n \\end{equation}\nBy the properties \\eqref{nabR} and \\eqref{nabxi}, the first assertion is proven being\n\\begin{align}\n\\dot {\\cal G} = \\tfrac{n-3}{(n-1)(n-2)}[(2nR -8R+8\\xi)\\dot R + 8(R-n\\xi)\\dot \\xi ]\\,.\n\\end{align}\nThe other assertion follows by differentiation.\n\\end{proof}\nThese results allow to set the tensor $\\Sigma_{kl}$ in a perfect fluid form. \n\n\\section{Evaluation of $\\Sigma_{kl}$ and the perfect-fluid form}\nFor an analytical function $f(R,{\\cal G})$ the double derivatives are evaluated as\n\\begin{align*}\n\\nabla_k\\nabla_l f_R = \n& f_{RRR} (\\nabla_k R)(\\nabla_l R) + f_{RR{\\cal G}} [(\\nabla_k R) (\\nabla_l {\\cal G}) + (\\nabla_l R)(\\nabla_k {\\cal G})]\\\\\n&+ f_{R{\\cal G}\\G} (\\nabla_k {\\cal G})(\\nabla_l {\\cal G}) + f_{RR} \\nabla_k\\nabla_l R + f_{R{\\cal G}} \\nabla_k\\nabla_l {\\cal G} \\label{RR}\\\\\n\\nabla_k\\nabla_l f_{\\cal G} =& f_{{\\cal G}\\G{\\cal G}} (\\nabla_k {\\cal G})(\\nabla_l {\\cal G}) + f_{R{\\cal G}\\G}[ (\\nabla_k R)( \\nabla_l {\\cal G}) + (\\nabla_l R)(\\nabla_k {\\cal G})]\\\\\n&+ f_{RR{\\cal G}} (\\nabla_k R)(\\nabla_l R) + f_{{\\cal G}\\G} \\nabla_k\\nabla_l {\\cal G} + f_{R{\\cal G}} \\nabla_k\\nabla_l R\\,\n\\end{align*}\nIn a Robertson-Walker space-time they gain the perfect-fluid form being:\n\\begin{align*}\n\\nabla_k\\nabla_l f_R =& -\\varphi (g_{kl}+u_k u_l)(f_{RR} \\dot R + f_{R{\\cal G}} \\dot {\\cal G})\\\\\n&+u_k u_l (f_{RRR} \\dot R \\dot R + 2f_{RR{\\cal G}} \\dot R \\dot {\\cal G} + f_{R{\\cal G}\\G} \\dot {\\cal G}\\dot {\\cal G} +f_{RR}\\ddot R +f_{R{\\cal G}}\\ddot {\\cal G} ) \\\\\n\\nabla_k\\nabla_l f_{\\cal G} =& -\\varphi (g_{kl}+u_ku_l) (f_{{\\cal G}\\G}\\dot {\\cal G} + f_{R{\\cal G}} \\dot R) \\\\\n&+u_k u_l (f_{{\\cal G}\\G{\\cal G}} \\dot {\\cal G} \\dot {\\cal G} +2 f_{R{\\cal G}\\G} \\dot R \\dot {\\cal G} + f_{RR{\\cal G}} \\dot R\\dot R + f_{R{\\cal G}}\\ddot R + f_{{\\cal G}\\G}\\ddot {\\cal G} )\n\\end{align*}\nThe perfect-fluid form is more evident by introducing the simplified expressions:\n\\begin{equation}\n\\nabla_k\\nabla_l f_R = A_R g_{kl}+ B_R u_ku_l, \\qquad \n\\nabla_k\\nabla_l f_{\\cal G} = A_{\\cal G} g_{kl} + B_{\\cal G} u_ku_l \\,.\n\\end{equation} \nIn particular, we have \n\\begin{equation}\n\\nabla^2 f_R=nA_R-B_R, \\quad \\mbox{and} \\quad \\nabla^2 f_{\\cal G}=nA_{\\cal G}-B_{\\cal G}\\,,\n\\end{equation}\nthat can be easily introduced in \\eqref{Sigma}. That is \n\\begin{align*}\n\\Sigma_{kl}=& (A_R g_{kl} + B_R u_ku_l) -g_{kl} (nA_R-B_R)\\\\\n& + 2R (A_{\\cal G} g_{kl} + B_{\\cal G} u_ku_l) -2R g_{kl} (nA_{\\cal G}-B_{\\cal G})\\\\\n&-4[R_k{}^m(A_{\\cal G} g_{ml}+B_{\\cal G} u_m u_l)+R_l{}^m(A_{\\cal G} g_{mk}+B_{\\cal G} u_m u_k)]\\\\\n& +4 R_{kl} (n A_{\\cal G}-B_{\\cal G}) +4g_{kl} R^{pq} (A_{\\cal G} g_{pq}+B_{\\cal G} u_pu_q)+4R_{kpql}(A_{\\cal G} g^{pq}+B_{\\cal G} u^pu^q)\\\\\n& -\\tfrac{1}{2}g_{kl}(R f_R+{\\cal G} f_{\\cal G} - f) + (1-f_R) (R_{kl}-\\tfrac{1}{2} g_{kl}R)\n\\end{align*}\nSimplifying with $R_{jk}u^k=\\xi u_j$ and $R_{kpql}u^pu^q = \\frac{1}{n-1}\\xi(g_{kl}+u_ku_l)$, it is \n\\begin{align*}\n\\Sigma_{kl}=& -g_{kl}(n-1)(A_R+2RA_{\\cal G}) + (g_{kl}+u_ku_l )(B_R+2RB_{\\cal G})-8R_{kl}A_{\\cal G} - 8 \\xi B_{\\cal G} u_ku_l\\\\\n& +4 R_{kl} (nA_{\\cal G}-B_{\\cal G}) +4g_{kl}(RA_{\\cal G}-\\xi B_{\\cal G}) -4 R_{kl}A_{\\cal G} \\\\\n&+\\tfrac{4}{n-1}\\xi B_{\\cal G}(g_{kl}+u_ku_l) -\\tfrac{1}{2}g_{kl}({\\cal G} f_{\\cal G} - f+R) + (1-f_R)R_{kl}\\,.\n\\end{align*}\nFinally, we obtain the expression in a perfect-fluid form as in \\eqref{perfect}, that is:\n\\begin{align}\\label{expsigma}\n\\Sigma_{kl} =& \\frac{g_{kl}}{n-1} [-(n-1)^2A_R + (n-1) B_R +2(n-3)(R-2\\xi)B_{\\cal G}\\nonumber \\\\\n& -2(n-3)(R(n-3)+2\\xi)A_{\\cal G} -\\tfrac{1}{2}(n-1)({\\cal G} f_{\\cal G}-f+R)+(R-\\xi)(1-f_R)] \\nonumber\\\\\n&+\\frac{u_ku_l}{n-1} [(n-1)B_R +2(n-3)(R-2\\xi)B_{\\cal G} + 4(n-3)(R-n\\xi)A_{\\cal G} \\\\\n&+(R-n\\xi)(1-f_R)]\\,.\\nonumber\n\\end{align}\nwhere:\n\\begin{align}\nA_R = &-\\varphi (f_{RR} \\dot R + f_{R{\\cal G}} \\dot {\\cal G})\\,,\\\\\nB_R =& f_{RRR} (\\dot R)^2 + 2f_{RR{\\cal G}} \\dot R \\dot {\\cal G} + f_{R{\\cal G}\\G} (\\dot {\\cal G} )^2 -f_{RR}(\\varphi \\dot R -\\ddot R) - f_{R{\\cal G}}(\\varphi \\dot{\\cal G} -\\ddot {\\cal G})\\,,\\\\\nA_{\\cal G}=&-\\varphi (f_{{\\cal G}\\G} \\dot {\\cal G} + f_{R{\\cal G}} \\dot R)\\,,\\\\\nB_{\\cal G} =& f_{{\\cal G}\\G{\\cal G}} (\\dot {\\cal G} )^2 +2 f_{R{\\cal G}\\G} \\dot R \\dot {\\cal G} + f_{RR{\\cal G}} (\\dot R)^2 - f_{R{\\cal G}}(\\varphi \\dot R- \\ddot R) -f_{{\\cal G}\\G}(\\varphi \\dot{\\cal G} -\\ddot {\\cal G})\\,,\n\\end{align}\nwith $\\dot A_R = \\dot\\varphi (A_R\/\\varphi ) - \\varphi B_R$ and $\\dot A_{\\cal G} = \\dot\\varphi (A_{\\cal G}\/\\varphi ) - \\varphi B_{\\cal G}$. \nThe interpretation of the coefficients in Eq.\\eqref{expsigma} is straightforward. The term in $g_{kl}$ is the effective pressure\n\\begin{align}\\label{expsigma}\np =& \\left(\\frac{1}{n-1}\\right) \\left[-(n-1)^2A_R + (n-1) B_R +2(n-3)(R-2\\xi)B_{\\cal G}\\right. \\\\\n&\\left. -2(n-3)(R(n-3)+2\\xi)A_{\\cal G} -\\tfrac{1}{2}(n-1)({\\cal G} f_{\\cal G}-f+R)+(R-\\xi)(1-f_R)\\right] \\,, \\nonumber \n\\end{align}\nand the term in the velocities $u_k u_l$ is the sum of the effective pressure and effective energy density\n\\begin{align}\\label{expsigma}\np+\\mu =&\\left(\\frac{1}{n-1}\\right) \\left[(n-1)B_R +2(n-3)(R-2\\xi)B_{\\cal G} + 4(n-3)(R-n\\xi)A_{\\cal G}\\right. \\nonumber\\\\\n&\\left.+(R-n\\xi)(1-f_R)\\right]\\,.\n\\end{align}\n\n\\section{Discussion and Conclusions}\n\nIn this paper, we extended the results reported in \\cite{CMM2018} where extra-contributions to the Einstein field equations, coming from $f(R)$ gravity, were recast as an effective perfect fluid of geometrical origin. Here we continued this ``geometrization\" program by including contributions coming from the Gauss-Bonnet topological invariant ${\\cal G}$ into the dynamics. This approach improves the perspective of interpreting the dark side of the universe within the standard of geometric effects \\cite{Cardo}. As for the previous case, we showed that the extra source produces a perfect-fluid term in the right hand side of the field equations. The addition of the Gauss-Bonnet invariant exhausts all \npossibilities of fourth-order dynamics coming from curvature invariants. In fact, an action involving curvature invariants means to take into account combinations\/functions of $R$, $R_{ik}$ and $R^k_{ilm}$. The inclusion of ${\\cal G}$ fixes a relation among them and so $f(R,{\\cal G})$ is a general fourth-order action of gravity where all possible curvature invariants are taken into account. Clearly, the program can be enlarged by considering terms like $\\Box R$, non-local terms like $\\Box^{-1} R$, torsion invariants and so on. The perspective is to achieve a classification of perfect fluids coming from geometry. This ``geometric view\" could address the puzzle of dark side and explain why no particles beyond the Standard Model have yet been detected. However, this statement has to be confirmed, at least, by a stringent cosmographic analysis where possible components to the cosmic bulk should be observationally discriminated \\cite{OrlandoDunsby,ester1,ester2,revcosmo,aviles,gruber}. In a forthcoming paper, this problem will be considered.\n \n\\section*{Acknowledgments}\nS. C. acknowledges the support of INFN ({\\it iniziative specifiche} MOONLIGHT2 and QGSKY).\nThis paper is based upon work from COST action CA15117 (CANTATA), supported by COST (European \nCooperation in Science and Technology).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn this paper we continue the investigation we began in \\cite{DS-F-S-WK}, about multiscale analysis of mathematical models for geophysical flows.\nOur focus here is on the effect of gravity in regimes of \\emph{low stratification}, but which go beyond a choice of the scaling that, in light\nof previous results, we call ``critical''.\n\nIn order to explain better all this, let us introduce some physics about the problem we are interested in, and give an overview of related studies.\nWe present in Section \\ref{s:result} the precise system we will work on, and the statements of our main results.\n\n\\subsection{Some physical considerations}\n\nBy definition (see \\textsl{e.g.} \\cite{C-R}, \\cite{Ped}), geophysical flows are flows whose dynamics is characterised by large time and space scales.\nTypical examples are currents in the atmosphere and the ocean, but of course there are many other cases where such fluids occur out of the Earth, like flows on stars or other celestial bodies.\n\nAt those scales, the effects of the rotation of the ambient space (which in the previous examples is the Earth) are no more negligible, and the fluid motion undergoes the action of a strong Coriolis force.\nA simplistic assumption, which is however often adopted in physical and mathematical studies, consists in restricting the attention to flows at mid-latitudes, \\textsl{i.e.} flows which take place far enough from the poles and the equatorial zone. \nThus, if we denote by $\\vr\\,\\geq\\,0$ the density of the fluid and by $\\vu\\,\\in\\,\\mathbb{R}^3$ its velocity field, the Coriolis force may be represented in the following form:\n\\begin{equation} \\label{def:Coriolis}\n\\mathfrak C(\\vr,\\vu)\\,:=\\,\\frac{1}{\\rm Ro}\\,\\vec e_3\\times\\vr\\,\\vu\\,,\n\\end{equation}\nwhere $\\vec e_3=(0,0,1)$,\nthe symbol $\\times$ denotes the classical external product of vectors in $\\mathbb{R}^3$ and $\\rm Ro>0$ is the so-called \\emph{Rossby number},\na physical adimensional parameter linked to the speed of rotation of the Earth.\nIn particular, the previous definition implies that the rotation is approximated to take place around the vertical axis, and its strength does not depend\non the latitude. We point out that, despite all these simplifications, the obtained model is already able to give a quite accurate description of several physically relevant phenomena occurring in the dynamics of geophysical fluids (see \\textsl{e.g.} \\cite{Ped}, \\cite{C-D-G-G}).\n\nIn geophysical fluid dynamics, effects of the fast rotation are predominant; this translates into the fact that the value of $\\rm Ro$\nis very small.\nAs a matter of fact, the Rossby number $\\rm Ro$ is defined as the ratio between the nonlinear acceleration to the Coriolis parameter term, namely\n$$ {\\rm Ro}:=\\frac{U_{\\rm ref}}{f_{\\rm ref}\\,L_{\\rm ref}}\\,, $$\nwhere $U_{\\rm ref},\\, L_{\\rm ref}$ and $f_{\\rm ref}$ are respectively the horizontal velocity scale, the horizontal length scale and the reference Coriolis frequency (see \\textsl{e.g.} \\cite{K-C-D} for more details).\nFor instance, for a typical atmospheric value of $U_{\\rm ref} \\sim 10$ m\/s, $f_{\\rm ref}\\sim 10^{-4} \\text{ s}^{-1}$ and $L_{\\rm ref}\\sim 1000$ km, the Rossby number\nturns out to be $0.1$; its value is even smaller for many flows in the oceans.\nAs established by the \\emph{Taylor-Proudman theorem} in geophysics, the fast rotation imposes a certain rigidity\/stability, as it undresses the motion of any vertical variation, and forces it to take place on planes orthogonal to the rotation axis.\nThus, the dynamics becomes purely two-dimensional and horizontal, and the fluid tends to move along vertical columns.\n\nHowever, such an ideal configuration is hinder by another fundamental force acting at geophysical scales, the gravity, which works to restore vertical stratification of the density. The gravitational force may be represented by the term\n\\[\n \\mathcal G(\\vr)\\,:=\\,-\\,\\frac{1}{\\rm Fr^2}\\,\\vr\\,\\vec e_3\\,,\n\\]\nwhere $\\rm Fr>0$ is the \\emph{Froude number}, another physical adimensional parameter, which measures the importance of the stratification effects in the dynamics.\nIn geophysics (see again \\cite{K-C-D} for details), $\\rm Fr$ represents the square root of the ratio between inertia and gravity, namely\n$$ {\\rm Fr}:=\\frac{U_{\\rm ref}}{\\sqrt{g\\,L_{\\rm ref}}}\\,, $$\nwhere $g$ is the acceleration of gravity.\n\nAs it happens for the Rossby number,\nat large scales also the Froude parameter is typically very small. Thus, the competition between the stabilisation effect of the Coriolis force and the vertical stratification due to gravity, is translated in the model into the competition between the orders of magnitude of the two parameters $\\rm Ro$ and $\\rm Fr$.\n\nActually, it turns out that the gravity $\\mathcal G$ acts in combination with pressure forces. Restricting from now on our attention to the case of compressible fluids, like currents in the atmosphere for instance, and neglecting for a while heat transfer processes, the pressure term arising in the mathematical model takes the form\n\\[\n\\mathfrak P(\\vr)\\,:=\\,\\frac{1}{\\rm Ma^2}\\,\\nabla p(\\vr)\\,,\n\\]\nwhere $p$ is a known smooth function of the density (and, in the general case, of the temperature of the fluid) and $\\rm Ma>0$ is the so-called \\emph{Mach number}, a third fundamental adimensional parameter which sets the size of isentropic departures from incompressible flow: the more $\\rm Ma$ is small, the more compressibility effects are low. Also the value of $\\rm Ma$ is very small for geophysical flows, since it is defined as\n$$ {\\rm Ma}:=\\frac{U_{\\rm ref}}{c}\\, $$\nwith $c$ being the sound speed (for instance, in the oceans the typical sound speed is $c\\sim 1520$ m\/s).\n\n\n\\medbreak\nAs it is customary in physical studies, because of the complexity of the model, one would like to derive reduced models for geophysical flows,\nwhich however are able to retain most of the properties of the original system.\nThe problem is that the three terms $\\mathfrak C$, $\\mathcal G$ and $\\mathfrak P$ enter into play in the model with a very large prefactor in front of them, owing to the\nsmallness of the values of $\\rm Ro$, $\\rm Fr$ and $\\rm Ma$ respectively. \nFor assessing their relative importance and their influence in the dynamics, one fixes a choice of their orders of magnitude. Actually (see \\textsl{e.g.} the discussion in Section 1.4 of \\cite{C-R}), there is some arbitrariness in doing so, depending on the specific properties of the physical\nsystem and on the processes one wants to put the accent on.\n\nIn general, geophysical fluid dynamics is a multiscale process, meaning that Earth's rotation, gravity and pressure forces act, and interact, at different\nscales in the system. In other words, $\\rm Ro$, $\\rm Fr$ and $\\rm Ma$ have different orders of magnitude, and only for some specific choices,\nall (or some) of them are in balance.\n\n\n\\subsection{Multiscale analysis: an overview of previous results}\n\nAt the mathematical level, in the last 30 years there has been a huge amount of works devoted to the rigorous justification, in various functional frameworks,\nof the reduced models considered in geophysics. Studies have been carried out in various contexts: for instance, focusing only on the effects of the low Mach number, or on its interplay with a low Froude number regime.\n\nReviewing the whole literature about this subject goes far beyond the scopes of this introduction, therefore we make the choice to report only on works which\ndeal with the presence of the Coriolis force \\eqref{def:Coriolis}.\nWe also decide to leave aside from the discussion the case of incompressible models, because (owing to the rigidity imposed by the divergence-free constraint\non the velocity field of the fluid) less pertinent for multiscale analysis. We refer to book \\cite{C-D-G-G} and the references therein for a panorama of\nknown results for incompressible homogeneous fluids, even though more recent developments have been made (see \\textsl{e.g.} \\cite{Scrobo} for a case where stratification is considered).\nNotice that there are also a few recent works \\cite{Fan-G}, \\cite{C-F}, \\cite{Sbaiz}, dealing with incompressible non-homogeneous fluids, but results in\nthat direction are only partial and the general picture still remains poorly understood at present.\n\nThe framework of compressible fluid models, instead, provides a much richer setting for the multiscale analysis of geophysical flows.\nIn what we are going to say, we make the choice of focusing on works which deal with viscous flows and which perform the asymptotic study for general\nill-prepared initial data. However, the literature about the subject is more ample than that.\n\nFirst results in that direction were presented in \\cite{F-G-N}, \\cite{F-G-GV-N} for the barotropic Navier-Stokes system. There, the authors investigated the\ncombined effect of a strong Coriolis force (low Rossby number limit) and of weak compressibility of the fluid (low Mach number limit),\nunder the scaling\n\\begin{equation} \\label{eq:scale}\n{\\rm Ma}\\,=\\,\\varepsilon^m\\qquad\\mbox{ and }\\qquad {\\rm Ro}\\,=\\,\\varepsilon\\,,\\qquad\\qquad \\mbox{ with }\\quad m\\geq1\\,,\n\\end{equation}\nwhere $\\varepsilon\\in\\,]0,1]$ is a small parameter, which one wants to let go to $0$ in order to derive an asymptotic model.\nNotice that in \\cite{F-G-GV-N} the effects due to the centrifugal force were considered as well, but this imposed the severe restriction $m>10$.\nIn the case $m=1$ in \\eqref{eq:scale}, the system presents an isotropic scaling, since the Rossby and Mach numbers act at the same order of magnitude and they keep in balance in the limit process. This balance takes the name of quasi-geostrophic balance, and the limit system\nis identified as the so-called \\emph{quasi-geostrophic equation} for the stream function of the target velocity field.\nWhen $m>1$, instead, the pressure and Coriolis forces act at different scales, the former one having a predominant effect on the dynamics of the fluid. At the mathematical level, the anisotropy of scaling generates some complications in the analysis; in \\cite{F-G-GV-N} this issue was handled by the use of dispersive\nestimates, which allowed to show convergence to a $2$-D incompressible Navier-Stokes system.\n\nWe refer to \\cite{F_MA} for a similar study in the context of capillary models. There, the choice $m=1$ was made, but the anisotropy was given by the scaling\nfixed for the internal forces term (the so-called Korteweg stress tensor). In addition, we refer to \\cite{F_2019} for the case of large Mach numbers,\nnamely for the case when $0\\leq m<1$ in \\eqref{eq:scale}. Since, in that instance, the pressure gradient is not strong enough to compensate the Coriolis force,\nin order to find some interesting dynamics in the limit one has to introduce a penalisation of the bulk viscosity coefficient.\n\nIn \\cite{F-N_AMPA}, \\cite{F-N_CPDE} the effects of gravity were added, under the scaling\n\\begin{equation} \\label{eq:scale-G}\n{\\rm Fr}\\,=\\,\\varepsilon^n\\,,\\qquad\\qquad\\mbox{ with }\\quad 1\\,\\leq\\,n\\,<\\,\\frac{m}{2}\\,.\n\\end{equation}\nIn particular, in those works one had $m>2$. As before, a planar incompressible Navier-Stokes system was identified as the limiting system, but, as already\nmentioned, the anisotropy of scaling created several difficulties in the analysis. We refer to \\cite{K-M-N} and \\cite{K-N} for related studies in the\ncontext of the full Navier-Stokes-Fourier system, under the same choices of the scaling (notice that, in \\cite{K-N}, the case $m=1$ was considered, but the gravitational force was not penalised at all).\nThe asymptotic results of \\cite{F-N_AMPA}, \\cite{F-N_CPDE}, \\cite{K-M-N} and \\cite{K-N} are all based on a fine combination of the relative entropy\/relative energy method with dispersive estimates derived from oscillatory integrals, and a strong argument which allows to handle the ill-preparation of the data (typically, the use of relative energy estimates requires to consider well-prepared initial data).\nIn all those works, a vanishing viscosity regime was also considered.\n\nIn our recent work \\cite{DS-F-S-WK}, devoted to the full Navier-Stokes-Fourier system in presence of stratification, we were able to improve the choice of the scaling \\eqref{eq:scale-G} and take the endpoint case $n\\,=\\,m\/2$, with $m\\geq 1$ as in \\eqref{eq:scale}. In passing, we mention that also effects of\nthe centrifugal force were considered in \\cite{DS-F-S-WK}, but this imposed the additional constraint $m\\geq 2$ on the order of the Mach number\n(which, besides, refined the restriction in \\cite{F-G-GV-N}).\nThe improvement on the orders of the scaling was possible, essentially due to a different technique employed for proving convergence, based on \\emph{compensated compactness} arguments.\nWe refer to \\cite{G-SR_2006} for the first implementation of that method in the context of fast rotating fluids, to \\cite{F-G-GV-N}, \\cite{F_JMFM},\n\\cite{F_2019} for other applications in the case of non-homogeneous flows. In particular, the convergence is not quantitative at all, but only qualitative. This technique is purely based on the algebraic structure of the system, which allows to find smallness (and vanishing to the limit) of suitable non-linear quantities, and fundamental compactness properties for other quantities (linked to the vorticity of the fluid and to the variations of the density);\nsuch compactness properties were already put in evidence in \\cite{Fan-G} (see also \\cite{C-F}) in the context of non-homogeneous incompressible fluids in fast\nrotation. All these features were enough to pass to the limit in the primitive system, and derive the limiting dynamics: a $2$-D incompressible Navier-Stokes system when $m>1$, a quasi-geostrophic equation for the stream function of the limit velocity when $m=1$.\n\nAn important point of the study performed in \\cite{DS-F-S-WK} is that the scaling $n=m\/2$ allowed to deduce some stratification effect in the limit. More precisely, although the limit dynamics was horizontal and two-dimensional, as dictated by the Taylor-Proudman theorem, stratification appeared in the functions representing\ndepartures of the density and temperature from the respective equilibria. On the contrary, in previous works like \\cite{F-N_AMPA}, \\cite{F-N_CPDE}, \\cite{K-M-N}, \\cite{K-N}, based on the scaling \\eqref{eq:scale-G}, stratification effects were completely absent. In this sense, we call\nthe endpoint case $n=m\/2$ ``critical''.\n\nTo conclude this part, we mention that all the results quoted so far concern various regimes of \\emph{low stratification}, meaning that,\naccording to the scaling in \\eqref{eq:scale}, \\eqref{eq:scale-G}, one has\n\\[\n \\frac{\\rm Ma}{\\rm Fr}\\,\\longrightarrow\\,0\\qquad\\qquad\\mbox{ when }\\qquad \\varepsilon\\,\\rightarrow\\,0^+\\,.\n\\]\nThe \\emph{strong stratification} regime, namely when the ratio ${\\rm Ma}\/{\\rm Fr}$ is of order $O(1)$, is particularly delicate for fast rotating fluids.\nThis is in stark contrast with the results available about the derivation of the anelastic approximation, where rotation is neglected:\nwe refer \\textsl{e.g.} to \\cite{Masm}, \\cite{BGL}, \\cite{F-K-N-Z} and, more recently, \\cite{F-Z} (see also \\cite{F-N} and references therein for a\nmore detailed account of previous works). The reason for that has to be ascribed exactly to the competition between vertical stratification (due to gravity) and horizontal stability (which the Coriolis force tends to impose): in the strong stratification regime, vertical oscillations of the solution\n(seem to) persist in the limit, and the available techniques do not allow at present to deal with this problem in its full generality.\nNonetheless, partial results have been obtained in the case of well-prepared initial data, by means of a relative entropy method: we refer to \\cite{F-L-N}\nfor the first result, where the mean motion is derived, and to \\cite{B-F-P} for an analysis of Ekman boundary layers in that framework.\n\n\n\\subsection{A short overview of the contents of the paper}\n\nIn the present work, we continue our investigation from \\cite{DS-F-S-WK}, devoted to the multiscale analysis of systems for geophysical fluids and the derivation\nof reduced models.\n\nFor clarity of exposition, we neglect here heat transfer processes in the fluid, and focus on the classical barotropic Navier-Stokes system\nas the primitive system; the more general case of the Navier-Stokes-Fourier system can be handled at the price of some additional technicalities (as done in \\cite{DS-F-S-WK}). Also, we simplify the model by neglecting effects due to the centrifugal force. On the one hand, this choice is not dramatic from the physical viewpoint (see the discussion in \\cite{C-R}, for instance); on the other hand, we could include the presence of the centrifugal force, after imposing\nsome additional restrictions on the order of magnitude of the Mach number. We refer to Section \\ref{s:result} below for the presentation of the precise equations\nwe are going to consider in this paper.\n\nWe work in the context of global in time \\emph{finite energy} weak solutions to the barotropic Navier-Stokes system with Coriolis force, which provides\na good setting for studying singular limits for that system. We consider the general case of \\emph{ill-prepared} initial data.\n\nOur goal here is to go beyond the ``critical'' choice ${\\rm Fr}\\,=\\,\\sqrt{\\rm Ma}$ performed in \\cite{DS-F-S-WK}, and investigate\nother regimes where the stratification has an even more important effect.\nMore precisely, we fix the following choice for the parameters $m$ and $n$ appearing in \\eqref{eq:scale} and \\eqref{eq:scale-G}: we assume that\n\\begin{equation} \\label{eq:scale-our}\n\\mbox{ either }\\qquad m\\,>\\,1\\quad\\mbox{ and }\\quad m\\,<\\,2\\,n\\,\\leq\\,m+1\\,,\\qquad\\qquad\\mbox{ or }\\qquad\nm\\,=\\,1\\quad\\mbox{ and }\\quad \\frac{1}{2}\\,<\\,n\\,<\\,1\\,.\n\\end{equation}\nThe restriction $n<1$ when $m=1$ is imposed in order to avoid a strong stratification regime: as already mentioned before, it is not clear\nat present how to deal with this case for general ill-prepared initial data, as all the available techniques seem to break down in that case.\nOn the other hand, the restriction $2\\,n\\leq m+1$ (for $m>1$) looks to be of technical nature. However, it comes out naturally\nin at least two points of our analysis, and it is not clear to us if, and how, it is possible to bypass it and consider the remaining range of values\n$(m+1)\/2\\,<\\,n\\,<\\,m$. \nLet us point out that, in our considerations, the relation $n1$ and $m= 1$). We refer again to Section \\ref{s:result} for the precise statements.\nIn this paper, the main point we put the accent on is how using in a fine way not only the structure of the system, but also the precise structure of each term in order to pass to the limit. To be more precise, the fact of considering\nvalues of $n$ going above the threshold $2n=m$ is made possible thanks to special algebraic cancellations involving the gravity term in the system of wave equations.\nSuch cancellations owe very much to the peculiar form of the gravitational force, which depends on the vertical variable only, and they do not appear, in general, if one wants to consider the action of different forces on the system. As one may easily guess, the case $2n=m+1$ is more involved: indeed, this choice\nof the scaling implies the presence of an additional bilinear term of order $O(1)$ in the computations; in turn, this term might not vanish in the limit, differently to what happens in the case $2n1$ and $m=1$, respectively, \\textsl{via} a compensated compactness argument.\nWe conclude the paper with Appendix \\ref{app:LP}, where we present some tools from Littlewood-Paley decomposition,\nwhich have been needed in our analysis.\n\n\\paragraph*{Some notation and conventions.} \n\nLet $B\\subset\\mathbb{R}^n$. The symbol ${\\mathchoice {\\rm 1\\mskip-4mu l_B$ denotes the characteristic function of $B$.\nThe notation $C_c^\\infty (B)$ stands for the space of $C^\\infty$ functions on $\\mathbb{R}^n$ and having compact support in $B$. The dual space $\\mathcal D^{\\prime}(B)$ is the space of\ndistributions on $B$.\n\nGiven $p\\in[1,+\\infty]$, by $L^p(B)$ we mean the classical space of Lebesgue measurable functions $g$, where $|g|^p$ is integrable over $B$ (with the usual modifications for the case $p=+\\infty$).\nSometimes, given $T>0$ and $(p,q)\\in[1,+\\infty]^2$, we use the symbol $L_T^p(L^q)$ to denote the space $L^p\\big(0,T;L^q(B)\\big)$.\nGiven $k \\geq 0$, we denote by $W^{k,p}(B)$ the Sobolev space of functions which belongs to $L^p(B)$ together with all their derivatives up to order $k$. When $p=2$, we set $W^{k,2}(B)=H^k(B)$.\nFor the sake of simplicity, we will often omit from the notation the set $B$, that we will explicitly point out if needed.\n\nIn the whole paper, the symbols $c$ and $C$ will denote generic multiplicative constants, which may change from line to line, and which do not depend on the small parameter $\\varepsilon$.\nSometimes, we will explicitly point out the quantities on which these constants depend, by putting them inside brackets.\n\nLet $\\big(f_\\varepsilon\\big)_{0<\\varepsilon\\leq1}$ be a sequence of functions in a normed space $Y$. If this sequence is bounded in $Y$, we use the notation $\\big(f_\\varepsilon\\big)_{\\varepsilon} \\subset Y$.\n \n\\medbreak\nNext, let us introduce some notation specific to fluids in fast rotation.\n\nIf $B$ is a domain in $\\mathbb{R}^3$, we decompose $x\\in B$\ninto $x=(x^h,x^3)$, with $x^h\\in\\mathbb{R}^2$ denoting its horizontal component. Analogously,\nfor a vector-field $v=(v^1,v^2,v^3)\\in\\mathbb{R}^3$, we set $v^h=(v^1,v^2)$ and we define the differential operators\n$\\nabla_h$ and ${\\rm div}\\,_{\\!h}$ as the usual operators, but acting just with respect to $x^h$.\nIn addition, we define the operator $\\nabla^\\perp_h\\,:=\\,\\bigl(-\\partial_2\\,,\\,\\partial_1\\bigr)$.\nFinally, the symbol $\\mathbb{H}$ denotes the Helmholtz projector onto the space of solenoidal vector fields in $B$, \nwhile $\\mathbb{H}_h$ denotes the Helmholtz projection on $\\mathbb{R}^2$.\nObserve that, in the sense of Fourier multipliers, one has $\\mathbb{H}_h\\vec f\\,=\\,-\\nabla_h^\\perp(-\\Delta_h)^{-1}{\\rm curl}_h\\vec f$.\n\nMoreover, since we will deal with a periodic problem in the $x^{3}$-variable, we also introduce the following decomposition: for a vector-field $X$, we write\n\\begin{equation} \\label{eq:decoscil}\nX(x)=\\langle X\\rangle (x^{h})+\\widetilde{X}(x)\\quad\\qquad\n \\text{ with }\\quad \\langle X\\rangle(x^{h})\\,:=\\,\\frac{1}{\\left|\\mathbb{T}^1\\right|}\\int_{\\mathbb{T}^1}X(x^{h},x^{3})\\, dx^{3}\\,,\n\\end{equation}\nwhere $\\mathbb{T}^1\\,:=\\,[-1,1]\/\\sim$ is the one-dimensional flat torus (here $\\sim$ denotes the equivalence relation which identifies $-1$ and $1$)\nand $\\left|\\mathbb{T}^1\\right|$ denotes its Lebesgue measure.\nNotice that $\\widetilde{X}$ has zero vertical average, and therefore we can write $\\widetilde{X}(x)=\\partial_{3}\\widetilde{Z}(x)$ with $\\widetilde{Z}$ having zero vertical average as well.\n\n\n\n\\subsection*{Acknowledgements}\n{\\small \nThe work of the second and third authors has been partially supported by the project CRISIS (ANR-20-CE40-0020-01), operated by the French National Research Agency (ANR). The last author is supported by (Polish) National Center of Science grant 2020\/38\/E\/ST1\/00469.\n\nThe first and the third authors are members of the Italian Institute for Advanced Mathematics (INdAM) group. \n}\n\n\n\\section{Setting of the problem and main results} \\label{s:result}\n\nIn this section, we introduce the primitive system and formulate our working hypotheses (see Section \\ref{ss:FormProb}), then we state our main results\n(in Section \\ref{ss:results}).\n\n\n \\subsection{The primitive system} \\label{ss:FormProb}\n\nAs already said in the introduction, in this paper we assumed that the motion of the fluid is described by a rescaled version of the\nbarotropic Navier-Stokes system with Coriolis and gravitational forces.\n\nThus, given a small parameter $\\varepsilon\\in\\,]0,1]$, the system reads as follows:\n\\begin{align}\n&\t\\partial_t \\vre + {\\rm div}\\, (\\vre\\ue)=0 \\label{ceq}\\tag{NSC$_{\\ep}^1$} \\\\\n&\t\\partial_t (\\vre\\ue)+ {\\rm div}\\,(\\vre\\ue\\otimes\\ue) + \\frac{1}{\\ep}\\,\\vec{e}_3 \\times \\vre\\ue + \\frac{1}{\\ep^{2m}} \\nabla_x p(\\vre) \n\t={\\rm div}\\, \\mathbb{S}(\\nabla_x\\ue) + \\frac{\\vre}{\\ep^{2n}} \\nabla_x G\\, ,\n\t\\label{meq}\\tag{NSC$_{\\ep}^2$}\n\t\\end{align}\nwhere we recall that $m$ and $n$ are taken according to \\eqref{eq:scale-our}.\nThe unknowns in the previous equations are the density $\\vre=\\vre(t,x)\\geq0$ of the fluid and its velocity field $\\ue=\\ue(t,x)\\in\\mathbb{R}^3$, where $t\\in\\mathbb{R}_+$ and $x\\in \\Omega:=\\mathbb{R}^2 \\times\\; ]0,1[$.\nThe viscous stress tensor in \\eqref{meq} is given by Newton's rheological law\n\t\\begin{equation}\\label{S}\n\t\\mathbb{S}(\\nabla_x \\ue) = \\mu\\left( \\nabla_x\\ue + \\nabla_x^T \\ue - \\frac{2}{3}{\\rm div}\\, \\ue \\tens{Id} \\right)\n\t+ \\eta\\, {\\rm div}\\,\\ue \\tens{Id}\\,,\n\t\\end{equation}\nwhere $\\mu>0$ is the shear viscosity and $\\eta\\geq 0$ represents the bulk viscosity. The term $\\vec{e}_3\\times\\varrho_\\varepsilon\\vec u_\\varepsilon$ takes into account the (strong) Coriolis force acting on the fluid.\nAs for the gravitational force, it is physically relevant to assume that \n\t\\begin{equation}\\label{assG}\n\t G(x)= -x^3\\,.\n\t\\end{equation}\nThe precise expression of $G$ will be useful in some computations below,\nalthough some generalisations are certainly possible.\n\n\t\nThe system is supplemented with \\emph{complete slip boundary conditions}, namely\n\t\\begin{align}\n\n\t\\big(\\ue \\cdot \\n\\big) _{|\\partial \\Omega} = 0\n\t\\quad &\\mbox{ and } \\quad\n\t\\bigl([ \\mathbb{S} (\\nabla_x \\ue) \\n ] \\times \\n\\bigr)_{|\\partial\\Omega} = 0\\,, \\label{bc1-2}\n\n\t\\end{align}\nwhere $\\vec{n}$ denotes the outer normal to the boundary $\\partial\\Omega\\,=\\,\\{x_3=0\\}\\cup\\{x_3=1\\}$.\nNotice that this is a true simplification, because it avoid complications due to the presence of Ekman boundary layers, when passing to the limit\n$\\varepsilon\\ra0^+$.\n\n\n\\begin{remark} \\label{r:period-bc}\nAs is well-known (see \\textsl{e.g.} \\cite{Ebin}), equations \\eqref{ceq}--\\eqref{meq}, supplemented by the complete slip boundary boundary conditions\nfrom \\eqref{bc1-2},\ncan be recasted as a periodic problem with respect to the vertical variable, in the new domain\n$$\n\\Omega\\,=\\,\\mathbb{R}^2\\,\\times\\,\\mathbb{T}^1\\,,\\qquad\\qquad\\mbox{ with }\\qquad\\mathbb{T}^1\\,:=\\,[-1,1]\/\\sim\\,,\n$$\nwhere $\\sim$ denotes the equivalence relation which identifies $-1$ and $1$. Indeed, the equations are invariant if we extend\n$\\rho$ and $u^h$ as even functions with respect to $x^3$, and $u^3$ as an odd function.\n\nIn what follows, we will always assume that such modifications have been performed on the initial data, and\nthat the respective solutions keep the same symmetry properties.\n\\end{remark}\n\n\nNow we need to impose structural restrictions on the pressure function $p$. We assume that\n\t\\begin{equation}\\label{pp1}\n\tp\\in C^1 [0,\\infty)\\cap C^2(0,\\infty),\\qquad p(0)=0,\\qquad p'(\\varrho )>0\\quad \\mbox{ for all }\\,\\varrho\\geq 0\\, .\n\t\\end{equation}\nAdditionally to \\eqref{pp1}, we require that \n\t\\begin{equation}\\label{pp2}\n\\mbox{ exists }\\;\\gamma\\,>\\,\\frac{3}{2}\\quad\\mbox{ such that }\\qquad\n\\lim\\limits_{\\varrho \\to +\\infty} \\frac{p^\\prime(\\varrho)}{\\varrho^{\\gamma -1}} = p_\\infty >0\\, .\n\t\\end{equation}\nWithout loss of generality, we can suppose that $p$ has been renormalised so that $p^\\prime (1)=1$. \n\n\n\n\n\\subsubsection{Equilibrium states} \\label{sss:equilibrium}\n\nNext, we focus our attention on the so-called \\emph{equilibrium states}. For each value of $\\varepsilon\\in\\,]0,1]$ fixed, the equilibria of system \\eqref{ceq}--\\eqref{meq} consist of static densities $\\vret$ satisfying\n\t\\begin{equation}\\label{prF}\n\\nabla_x p(\\vret) = \\ep^{2(m-n)} \\vret \\nabla_x G \\qquad \\mbox{ in }\\; \\Omega\\,.\n\t\\end{equation}\t\n\n\n\nEquation \\eqref{prF} identifies $\\widetilde{\\varrho}_\\varepsilon$ up to an additive constant: taking the target density to be $1$, we get\n\\begin{equation} \\label{eq:target-rho}\n H^\\prime(\\vret)=\\, \\ep^{2(m-n)} G + H^\\prime (1)\\,,\\qquad\\qquad \\mbox{ where }\\qquad \nH(\\varrho) = \\varrho \\int_1^{\\varrho} \\frac{ p(z)}{z^2} {\\rm d}z\\,.\n\\end{equation}\nNotice that relation \\eqref{eq:target-rho} implies that \n\\begin{equation*}\nH^{\\prime \\prime}(\\varrho)=\\frac{p^\\prime (\\varrho)}{\\varrho} \\quad \\text{ and }\\quad H^{\\prime \\prime}(1)=1\\, .\n\\end{equation*}\n\nTherefore, we infer that, whenever $m\\geq1$ and $m>n$ as in the present paper, for any $x\\in\\Omega$ one has $\\widetilde{\\varrho}_\\varepsilon(x)\\longrightarrow 1$ in the limit $\\varepsilon\\ra0^+$.\nMore precisely, the next statement collects all the necessary properties of the static states. It corresponds to Lemma 2.3 and Proposition 2.5 of\n\\cite{DS-F-S-WK}. \n\\begin{proposition} \\label{p:target-rho_bound}\nLet the gravitational force $G$ be given by \\eqref{assG}.\nLet $\\bigl(\\widetilde{\\varrho}_\\varepsilon\\bigr)_{0<\\varepsilon\\leq1}$ be a family of static solutions to equation \\eqref{prF}\nin $\\Omega = \\mathbb{R}^2\\times\\, ]0,1[$.\n\nThen, there exist an $\\varepsilon_0>0$ and a $0<\\rho_*<1$ such that $\\widetilde{\\varrho}_\\varepsilon\\geq\\rho_*$ for all $\\varepsilon\\in\\,]0,\\varepsilon_0]$\nand all $x\\in\\Omega$.\nIn addition, for any $\\varepsilon\\in\\,]0,\\varepsilon_0]$, one has:\n\\begin{equation*}\n\\left|\\widetilde{\\varrho}_\\varepsilon(x)\\,-\\,1\\right|\\,\\leq\\,C\\,\\varepsilon^{2(m-n)}\\, ,\n\\end{equation*}\nfor a constant $C>0$ which is uniform in $x\\in\\Omega$ and in $\\varepsilon\\in\\,]0,1]$.\n\\end{proposition}\n\nWithout loss of any generality, we can assume that $\\varepsilon_0=1$ in Proposition \\ref{p:target-rho_bound}.\n\n\\medbreak\nIn light of this analysis, it is natural to try to solve system \\eqref{ceq}--\\eqref{meq} in $\\Omega$, supplemented with the \\emph{far field conditions}\n\\begin{equation} \\label{ff}\n\\varrho_{\\varepsilon}\\rightarrow \\vret \\qquad \\mbox{ and } \\qquad \\ue \\rightarrow 0 \\qquad\\qquad \\text{ as }\\quad |x|\\rightarrow +\\infty \\, .\n\\end{equation}\n\n\n\\subsubsection{Initial data and finite energy weak solutions} \\label{sss:data-weak}\n\nIn view of the boundary conditions \\eqref{ff} ``at infinity'', we assume that the initial data are close (in a suitable sense) to the equilibrium states $\\vret$ that we have just identified.\nNamely, we consider initial densities of the following form:\n\t\\begin{equation}\\label{in_vr}\n\t\\vrez = \\vret + \\ep^m \\vrez^{(1)} \\, \n\t\\end{equation}\nFor later use, let us introduce also the following decomposition of the initial densities:\n\\begin{equation} \\label{eq:in-dens_dec}\n\\varrho_{0,\\varepsilon}\\,=\\,1\\,+\\,\\varepsilon^{2(m-n)}\\,R_{0,\\varepsilon}\\qquad\\qquad\\mbox{ with }\\qquad\nR_{0,\\varepsilon}\\,=\\,\\widetilde r_\\varepsilon\\,+\\,\\varepsilon^{2n-m}\\, \\varrho_{0,\\varepsilon}^{(1)}\\,,\\qquad \\widetilde r_\\varepsilon\\,:=\\,\\frac{\\widetilde\\varrho_\\varepsilon-1}{\\varepsilon^{2(m-n)}}\\,.\n\\end{equation}\nNotice that the $\\widetilde r_\\varepsilon$'s are in fact data of the system, since they only depend on $p$ and $G$.\n\n\nWe suppose the density perturbations $\\vrez^{(1)}$ to be measurable functions and satisfy the control\n\t\\begin{align}\n\\sup_{\\varepsilon\\in\\,]0,1]}\\left\\| \\vrez^{(1)} \\right\\|_{(L^2\\cap L^\\infty)(\\Omega)}\\,\\leq \\,c\\,,\\label{hyp:ill_data}\n\t\\end{align}\ntogether with the ``mean-free condition''\n$\n\\int_{\\Omega} \\vrez^{(1)} \\dx = 0\\,.\n$\nAs for the initial velocity fields, we assume the following uniform bound:\n\\begin{equation} \\label{hyp:ill-vel}\n \t\\sup_{\\varepsilon\\in\\,]0,1]}\\left\\| \\sqrt{\\widetilde\\varrho_\\varepsilon} \\vec{u}_{0,\\ep} \\right\\|_{L^2(\\Omega)}\\, \\leq\\, c\\,.\n\\end{equation}\n\n\n\n\\begin{remark} \\label{r:ill_data}\nIn view of Proposition \\ref{p:target-rho_bound}, the condition in \\eqref{hyp:ill-vel} immediately implies that\n$$\n\\sup_{\\varepsilon\\in\\,]0,1]}\\,\\left\\| \\vec{u}_{0,\\ep} \\right\\|_{L^2(\\Omega)}\\,\\leq\\,c\\,.\n$$\n\\end{remark}\n\n\n\nThanks to the previous uniform estimates, up to extraction, we can identify the limit points\n\\begin{align} \n\\varrho^{(1)}_0\\,:=\\,\\lim_{\\varepsilon\\ra0}\\varrho^{(1)}_{0,\\varepsilon}\\qquad\\text{ weakly-$\\ast$ in }&\\qquad L^\\infty (\\Omega) \\cap L^2(\\Omega)\\label{conv:in_data_vrho}\\\\\n\\vec{u}_0\\,:=\\,\\lim_{\\varepsilon\\ra0}\\vec{u}_{0,\\varepsilon}\\qquad \\text{ weakly in }&\\qquad L^2(\\Omega)\\label{conv:in_data_vel}\\,.\n\\end{align}\n\n\\medbreak\n\n\nAt this point, let us specify better what we mean for \\emph{finite energy weak solution} (see \\cite{F-N} for details). \n\\begin{definition} \\label{d:weak}\nLet $\\Omega = \\mathbb{R}^2 \\times\\, ]0,1[\\,$. Fix $T>0$ and $\\varepsilon>0$. Let $(\\varrho_{0,\\varepsilon}, \\vec u_{0,\\varepsilon})$ be \nan initial datum satisfying \\eqref{in_vr}--\\eqref{hyp:ill-vel}. We say that the couple $(\\varrho_\\varepsilon, \\vec u_\\varepsilon)$ is a \\emph{finite energy weak solution} of the system \n\\eqref{ceq}--\\eqref{meq} in $\\,]0,T[\\,\\times \\Omega$,\nsupplemented with the boundary conditions \\eqref{bc1-2} and far field conditions \\eqref{ff}, related to the initial datum $(\\varrho_{0,\\varepsilon}, \\vec u_{0,\\varepsilon})$, if the following conditions hold:\n\\begin{enumerate}[(i)]\n\\item the functions $\\varrho_\\varepsilon$ and $\\ue$ belong to the class\n\\begin{equation*}\n\\varrho_\\varepsilon\\geq 0\\,,\\; \\varrho_\\varepsilon - \\widetilde{\\varrho}_\\varepsilon\\,\\in L^\\infty\\big(0,T; L^2+L^\\gamma (\\Omega)\\big)\\,,\\;\n\\ue \\in L^2\\big(0,T;H^1(\\Omega)\\big),\\; \\big(\\ue \\cdot \\n\\big) _{|\\partial \\Omega} = 0\\, ;\n\\end{equation*}\n\\item the equations have to be satisfied in a distributional sense:\n\t\\begin{equation}\\label{weak-con}\n\t-\\int_0^T\\int_{\\Omega} \\left( \\vre \\partial_t \\varphi + \\vre\\ue \\cdot \\nabla_x \\varphi \\right) \\dxdt = \n\t\\int_{\\Omega} \\vrez \\varphi(0,\\cdot) \\dx\n\t\\end{equation}\nfor any $\\varphi\\in C^\\infty_c([0,T[\\,\\times \\overline\\Omega)$ and\n\t\\begin{align}\n\t&\\int_0^T\\!\\!\\!\\int_{\\Omega} \n\t\\left( - \\vre \\ue \\cdot \\partial_t \\vec\\psi - \\vre [\\ue\\otimes\\ue] : \\nabla_x \\vec\\psi \n\t+ \\frac{1}{\\ep} \\, \\vec{e}_3 \\times (\\vre \\ue ) \\cdot \\vec\\psi - \\frac{1}{\\ep^{2m}} p(\\vre) {\\rm div}\\, \\vec\\psi \\right) \\dxdt \\label{weak-mom} \\\\\n\t& =\\int_0^T\\!\\!\\!\\int_{\\Omega} \n\t\\left(- \\mathbb{S}(\\nabla_x\\vec u_\\varepsilon) : \\nabla_x \\vec\\psi + \\frac{1}{\\ep^{2n}} \\vre \\nabla_x G\\cdot \\vec\\psi \\right) \\dxdt \n\t+ \\int_{\\Omega}\\vrez \\uez \\cdot \\vec\\psi (0,\\cdot) \\dx \\nonumber\n\t\\end{align}\nfor any test function $\\vec\\psi\\in C^\\infty_c([0,T[\\,\\times \\overline\\Omega; \\mathbb{R}^3)$ such that $\\big(\\vec\\psi \\cdot \\n \\big)_{|\\partial {\\Omega}} = 0$;\n\\item the energy inequality holds for almost every $t\\in (0,T)$:\n\\begin{align}\n&\\hspace{-0.7cm} \\int_{\\Omega}\\frac{1}{2}\\vre|\\ue|^2(t) \\dx\\,+\\,\\frac{1}{\\ep^{2m}}\\int_{\\Omega}\\mathcal E\\left(\\varrho_\\varepsilon,\\widetilde\\varrho_\\varepsilon\\right)(t) \\dx\n+ \\int_0^t\\int_{\\Omega} \\mathbb S(\\nabla_x \\ue):\\nabla_x \\ue \\, \\dx {\\rm d}\\tau \\label{est:dissip} \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\\,\\leq\\,\n\\int_{\\Omega}\\frac{1}{2}\\vrez|\\uez|^2 \\dx\\,+\\,\n\\frac{1}{\\ep^{2m}}\\int_{\\Omega}\\mathcal E\\left(\\varrho_{0,\\varepsilon},\\widetilde\\varrho_\\varepsilon\\right) \\dx\\, ,\n\\nonumber\n\\end{align}\nwhere the function\n\\begin{equation} \\label{def:rel-entropy}\n\\mathcal E\\left(\\rho,\\widetilde\\varrho_\\varepsilon\\right)\\,:=\\,H(\\rho) - (\\rho - \\vret)\\, H^\\prime(\\vret)\n- H(\\vret)\n\\end{equation}\n is the \\emph{relative internal energy} of the fluid, with $H$ given by \\eqref{eq:target-rho}.\n\\end{enumerate}\nThe solutions is \\emph{global} if the previous conditions hold for all $T>0$.\n\\end{definition}\n\t\nUnder the assumptions fixed above, for any \\emph{fixed} value of the parameter $\\varepsilon\\in\\,]0,1]$,\nthe existence of a global in time finite energy weak solution $(\\varrho_\\varepsilon,\\vec u_\\varepsilon)$ to system \\eqref{ceq}--\\eqref{meq}, related to the initial datum\n$(\\varrho_{0,\\varepsilon},\\vec u_{0,\\varepsilon})$, in the sense of the previous definition, can be proved as in the classical case, see \\textsl{e.g.} \\cite{Lions_2}, \\cite{Feireisl}. \nNotice that the mapping $t \\mapsto (\\vre\\ue)(t,\\cdot)$ is weakly continuous, and one has $(\\vre)_{|t=0} = \\vrez$ together with $(\\vre\\ue)_{|t=0}= \\vrez\\uez$. \n\nWe remark also that, in view of \\eqref{ceq}, the total mass is conserved in time, in the following sense: for almost every $t\\in[0,+\\infty[\\,$,\none has\n\\begin{equation*} \\label{eq:mass_conserv}\n\\int_{\\Omega}\\bigl(\\vre(t)\\,-\\,\\vret\\bigr)\\,\\dx\\,=\\,0\\,.\n\\end{equation*}\n\nTo conclude, we point out that, in our framework of finite energy weak solutions to the primitive system,\ninequality \\eqref{est:dissip} will be the only tool to derive uniform estimates for the family of weak solutions we are going to consider. \n\n \n\n\n\\subsection{Main results}\\label{ss:results}\n\nWe can now state our main results. We point out that, due to the scaling \\eqref{eq:scale-our}, the relation $m>n$ is always true,\nso we will always be in a low stratification regime.\n\n\nThe first statement concerns the case when the effects linked to the pressure term are predominant with respect to the fast rotation, \\textsl{i.e.} $m>1$. \n\n \n\\begin{theorem}\\label{th:m>1}\nLet $\\Omega= \\mathbb{R}^2 \\times\\,]0,1[\\,$ and $G\\in W^{1,\\infty}(\\Omega)$ be as in \\eqref{assG}. Take $m>1$ and $m+1\\geq 2n >m$.\nFor any fixed value of $\\varepsilon \\in \\; ]0,1]$, let initial data $\\left(\\varrho_{0,\\varepsilon},\\vec u_{0,\\varepsilon}\\right)$ verify the hypotheses fixed in Paragraph \\ref{sss:data-weak}, and let\n$\\left( \\vre, \\ue\\right)$ be a corresponding weak solution to system \\eqref{ceq}--\\eqref{meq}, supplemented with the structural hypotheses \\eqref{S} on $\\mathbb{S}(\\nabla_x \\ue)$ and with boundary conditions \\eqref{bc1-2} and far field conditions \\eqref{ff}.\nLet $\\vec u_0$ be defined as in \\eqref{conv:in_data_vel}.\n\nThen, for any $T>0$ one has the convergence properties\n\t\\begin{align*}\n\t\\varrho_\\ep \\rightarrow 1 \\qquad\\qquad &\\mbox{ strongly in } \\qquad L^{\\infty}\\big(0,T; L_{\\rm loc}^{\\min\\{2,\\gamma\\}}(\\Omega )\\big) \\\\\n\n\t\\vec{u}_\\ep \\weak \\vec{U}\n\t\\qquad\\qquad &\\mbox{ weakly in }\\qquad L^2\\big(0,T;H^{1}(\\Omega)\\big)\\,,\n\t\\end{align*}\t\nwhere $\\vec{U} = (\\vec U^h,0)$, with $\\vec U^h=\\vec U^h(t,x^h)$ such that ${\\rm div}_h\\vec U^h=0$. In addition, the vector field $\\vec{U}^h $ is a weak solution\nto the following homogeneous incompressible Navier-Stokes system in $\\mathbb{R}_+ \\times \\mathbb{R}^2$,\n\\begin{align}\n& \\partial_t \\vec U^{h}+{\\rm div}_h\\left(\\vec{U}^{h}\\otimes\\vec{U}^{h}\\right)+\\nabla_h\\Gamma-\\mu \\Delta_{h}\\vec{U}^{h}=0\\, , \\label{eq_lim_m:momentum}\n\\end{align}\nfor a suitable pressure function $\\Gamma\\in\\mathcal D'(\\mathbb{R}_+\\times\\mathbb{R}^2)$ and related to the initial condition\n$\n\\vec{U}_{|t=0}=\\mathbb{H}_h\\left(\\langle\\vec{u}^h_{0}\\rangle\\right)\\, .\n$\n\\end{theorem}\n\n\n\n\nWhen $m=1$, the Mach and Rossby numbers have the same order of magnitude, and they keep in balance in the whole asymptotic process,\nrealising in this way the so-called \\emph{quasi-geostrophic balance} in the limit. \nThe next statement is devoted to this case.\n\\begin{theorem} \\label{th:m=1}\nLet $\\Omega = \\mathbb{R}^2 \\times\\,]0,1[\\,$ and let $G\\in W^{1,\\infty}(\\Omega)$ be as in \\eqref{assG}. Take $m=1$ and $1\/20$ one has the following convergence properties:\n\t\\begin{align*}\n\t\\varrho_\\ep \\rightarrow 1 \\qquad\\qquad &\\mbox{ strongly in } \\qquad L^{\\infty}\\big(0,T; L_{\\rm loc}^{\\min\\{2,\\gamma\\}}(\\Omega )\\big) \\\\\n\t\\varrho^{(1)}_\\varepsilon:=\\frac{\\varrho_\\ep - \\widetilde{\\varrho_\\varepsilon}}{\\ep} \\weakstar \\varrho^{(1)} \\qquad\\qquad &\\mbox{ weakly-$*$ in }\\qquad L^{\\infty}\\big(0,T; L^{2}+L^{\\gamma}(\\Omega )\\big) \\\\\n\t\\vec{u}_\\ep \\weak \\vec{U}\n\t\\qquad\\qquad &\\mbox{ weakly in }\\qquad L^2\\big(0,T;H^{1}(\\Omega)\\big)\\,,\n\t\\end{align*}\t\nwhere, as above, $\\vec{U} = (\\vec U^h,0)$, with $\\vec U^h=\\vec U^h(t,x^h)$ such that ${\\rm div}_h\\vec U^h=0$. \nMoreover, one has the relation $\\vec U_h=\\nabla_h^\\perp \\varrho^{(1)}$, and $\\varrho^{(1)}$ satisfies (in the weak sense) the quasi-geostrophic equation\n\\begin{align}\n& \\partial_{t}\\left(\\varrho^{(1)}-\\Delta_{h}\\varrho^{(1)}\\right) -\\nabla_{h}^{\\perp}\\varrho^{(1)}\\cdot\n\\nabla_{h}\\left( \\Delta_{h}\\varrho^{(1)}\\right) +\\mu \n\\Delta_{h}^{2}\\varrho^{(1)}\\,=\\,0\\,, \\label{eq_lim:QG} \n\\end{align}\nsupplemented with the initial condition\n$$\n\\left(\\varrho^{(1)}-\\Delta_{h}\\varrho^{(1)}\\right)_{|t=0}= \\langle \\varrho_0^{(1)}\\rangle-{\\rm curl}_h\\langle\\vec u^h_{0}\\rangle\\,.\n$$\n\\end{theorem}\n\n\n\n\\section{Consequences of the energy inequality} \\label{s:energy}\n\nIn Definition \\ref{d:weak}, we have postulated that the family of weak solutions $\\big(\\varrho_\\varepsilon,\\vu_\\varepsilon\\big)_\\varepsilon$ considered in Theorems\n\\ref{th:m>1} and \\ref{th:m=1} satisfies the energy inequality \\eqref{est:dissip}. \nIn this section we take advantage of that fact to infer uniform bounds for $\\big(\\varrho_\\varepsilon,\\vu_\\varepsilon\\big)_\\varepsilon$, see Section \\ref{ss:unif-est}.\nThanks to those bounds, we can extract weak-limit points of the sequence of solutions and deduce some properties those limit points\nhave to satisfy, see Section \\ref{ss:ctl1}.\n\n\\subsection{Uniform bounds and weak limits}\\label{ss:unif-est}\n\nThis section is devoted to establish uniform bounds on the sequence $\\bigl(\\varrho_\\varepsilon,\\vec u_\\varepsilon\\bigr)_\\varepsilon$.\nThis can be done as in the classical case (see \\textsl{e.g.} \\cite{F-N} for details), since\nthe Coriolis term does not contribute to the total energy balance of the system.\nHowever, for the reader's convenience, let us present some details.\n\nTo begin with, let us introduce a partition of the space domain $\\Omega$ into the so-called ``essential'' and ``residual'' sets.\nFor this, for $t>0$ and for all $\\varepsilon\\in\\,]0,1]$, we define the sets\n$$\n\\Omega_\\ess^\\varepsilon(t)\\,:=\\,\\left\\{x\\in\\Omega\\;\\big|\\quad \\varrho_\\varepsilon(t,x)\\in\\left[1\/2\\,\\rho_*\\,,\\,2\\right]\\right\\}\\,,\\qquad\\Omega^\\varepsilon_\\res(t)\\,:=\\,\\Omega\\setminus\\Omega^\\varepsilon_\\ess(t)\\,,\n$$\nwhere the positive constant $\\rho_*>0$ has been defined in Proposition \\ref{p:target-rho_bound}.\nThen, given a function $h$, we write\n$$\nh\\,=\\,\\left[h\\right]_\\ess\\,+\\,\\left[h\\right]_\\res\\,,\\qquad\\qquad\\mbox{ where }\\qquad \\left[h\\right]_\\ess\\,:=\\,h\\,\\mathds{1}_{\\Omega_\\ess^\\varepsilon(t)}\\,.\n$$\nHere above, $\\mathds{1}_A$ denotes the characteristic function of a set $A\\subset\\Omega$.\n\nNext, we observe that\n\\[\n\\Big[\\mathcal E\\big(\\rho(t,x),\\widetilde\\varrho_\\varepsilon(x)\\big)\\Big]_\\ess\\,\\sim\\,\\left[\\rho-\\widetilde\\varrho_\\varepsilon(x)\\right]_\\ess^2\n\\qquad\\quad \\mbox{ and }\\qquad\\quad\n\\Big[\\mathcal E\\big(\\rho(t,x),\\widetilde\\varrho_\\varepsilon(x)\\big)\\Big]_\\res\\,\\geq\\,C\\left(1\\,+\\,\\big[\\rho(t,x)\\big]_\\res^\\gamma\\right)\\,,\n\\]\nwhere $\\vret$ is the static density state identified in Section \\ref{sss:equilibrium} and $\\mathcal E$ is given by \n\\eqref{def:rel-entropy}.\n\nHere above, the multiplicative constants are all strictly positive and may depend on $\\rho_*$ and we agree to write $A\\sim B$ whenever there exists a ``universal'' constant $c>0$ such that $(1\/c)\\, B\\leq A\\leq c\\, B$.\n\nThanks to the previous observations, we easily see that, under the assumptions fixed in Section \\ref{s:result} on the initial data,\nthe right-hand side of \\eqref{est:dissip} is \\emph{uniformly bounded} for all $\\varepsilon\\in\\,]0,1]$. Specifically, we have\n$$\n\\int_{\\Omega} \\frac{1}{2}\\vrez|\\uez|^2\\,\\dx + \\frac{1}{\\ep^{2m}}\\int_{\\Omega}\\mathcal E\\left(\\varrho_{0,\\varepsilon},\n\\, \\widetilde\\varrho_\\varepsilon\\right)\\,\\dx\\,\\leq\\,C\\,.\n$$\n\nOwing to the previous inequalities and the finite energy condition \\eqref{est:dissip} on the family of weak solutions,\nit is quite standard to derive, for any time $T>0$ fixed and any $\\varepsilon\\in\\,]0,1]$, the following estimates:\n\\begin{align}\n\t\\sup_{t\\in[0,T]} \\| \\sqrt{\\vre}\\ue\\|_{L^2(\\Omega;\\, \\mathbb{R}^3)}\\, &\\leq\\,c \\label{est:momentum} \\\\\t\n\t\\sup_{t\\in[0,T]} \\left\\| \\left[ \\dfrac{\\vre - \\vret}{\\ep^m}\\right]_\\ess (t) \\right\\|_{L^2(\\Omega)}\\,&\\leq\\, c \\label{est:rho_ess} \\\\\n\t\\sup_{t\\in[0,T]} \\int_{\\Omega}\t{\\mathchoice {\\rm 1\\mskip-4mu l_{\\mathcal{M}^\\varepsilon_\\res[t]} \\,dx\\,&\\leq \\, c\\,\\ep^{2m} \\label{est:M_res-measure}\\\\\n\t\\sup_{t\\in [0,T]} \\int_{\\Omega} [ \\vre]^{\\gamma}_\\res (t)\\,\\dx \\,\n\t\\,&\\leq\\,c\\,\\ep^{2m} \\label{est:rho_res} \\\\\n\n\t\\int_0^T \\left\\| \\nabla_x \\ue + \\nabla_x^T \\ue - \\frac{2}{3} {\\rm div}\\, \\ue \\tens{Id} \\right\\|^2_{L^2(\\Omega ;\\, \\mathbb{R}^{3\\times3})}\\, \\dt\\,\n\t&\\leq\\, c\\, . \\label{est:Du} \n\t\\end{align}\nWe refer to \\cite{F-N} (see also \\cite{F-G-N}, \\cite{F-G-GV-N}, \\cite{F_2019} and \\cite{DS-F-S-WK}) for the details of the computations.\n\n\nOwing to \\eqref{est:Du} and a generalisation of the Korn inequality (see \\textsl{e.g.} Chapter 10 of \\cite{F-N}), we gather that\n$\\big(\\nabla\\vu_\\varepsilon\\big)_\\varepsilon\\,\\subset\\,L^2_T(L^2)$. On the other hand, by arguing as in \\cite{F-G-N}, we can use\n\\eqref{est:momentum}, \\eqref{est:M_res-measure} and \\eqref{est:rho_res} to deduce that also\n$\\big(\\vu_\\varepsilon\\big)_\\varepsilon\\,\\subset\\,L^2_T(L^2)$. Putting those bounds together, we finally infer that\n\\begin{equation}\\label{unif-bound-for-vel}\n\\int_0^T \\left\\|\\ue \\right\\|^2_{H^{1}(\\Omega ;\\, \\mathbb{R}^{3})}\\, \\dt\\,\\leq \\, c\\, .\n\\end{equation} \nIn particular, there exist $\\vU\\,\\in\\,L^2_{\\rm loc}\\big(\\mathbb{R}_+;H^1(\\Omega;\\mathbb{R}^3)\\big)$ such that, up to a suitable extraction (not relabelled here),\nwe have\n\\begin{equation} \\label{conv:u}\n\\vu_\\varepsilon\\,\\rightharpoonup\\,\\vU\\qquad\\qquad \\mbox{ in }\\quad L^2_{\\rm loc}\\big(\\mathbb{R}_+;H^1(\\Omega;\\mathbb{R}^3)\\big)\\,.\n\\end{equation}\n\nLet us move further and consider the density functions. The previous estimates on the density tell us that we must find a finer decomposition for the densities. As a matter of fact, for any time $T>0$ fixed, we have\n\t\\begin{equation}\\label{rr1}\n\\| \\vre - 1 \\|_{L^\\infty_T(L^2 + L^{\\gamma} + L^\\infty)}\\,\\leq\\,c\\, \\ep^{2(m-n)}\\,.\n\t\\end{equation}\nIn order to see \\eqref{rr1} we write \n\\begin{equation}\\label{rel:density_1}\n|\\varrho_\\varepsilon-1|\\,\\leq\\,|\\varrho_\\varepsilon-\\widetilde{\\varrho}_\\varepsilon|+|\\widetilde{\\varrho}_\\varepsilon-1|\\,.\n\\end{equation}\t\nFrom \\eqref{est:rho_ess}, we infer that $\\big[\\vr_\\varepsilon\\,-\\,\\widetilde{\\vr}_\\varepsilon\\big]_\\ess$ is of order $O(\\varepsilon^m)$ in $L^\\infty_T(L^2)$.\nFor the residual part of the same term, we can use \\eqref{est:rho_res} to discover that it is of order $O(\\varepsilon^{2m\/\\gamma})$.\nObserve that, if $1<\\gamma<2$, the higher order is $O(\\varepsilon^m)$, whereas, in the case $\\gamma\\geq2$, by use of \\eqref{est:rho_res} and \\eqref{est:M_res-measure} again, it is easy to get\n\\begin{equation} \\label{est:res_g>2}\n\\left\\|\\left[\\vr_\\varepsilon\\,-\\,\\widetilde{\\vr}_\\varepsilon\\right]_\\res\\right\\|_{L^\\infty_T(L^2)}^2\\,\\leq\\,C\\,\\varepsilon^{2m}\\,.\n\\end{equation}\nFinally, we apply Proposition \\ref{p:target-rho_bound} to control the last term in the right-hand side of \\eqref{rel:density_1}.\nIn the end, estimate \\eqref{rr1} is proved.\n\nThis having been established, and keeping in mind the notation introduced in \\eqref{in_vr} and \\eqref{eq:in-dens_dec}, we can introduce the\ndensity oscillation functions\n\\[\nR_\\varepsilon\\,:=\\, \\frac{\\varrho_\\ep -1}{\\ep^{2(m-n)}}\\, =\\,\\widetilde{r}_\\varepsilon\\,+\\,\\varepsilon^{2n-m}\\,\\varrho_\\varepsilon^{(1)}\\,,\n\\]\nwhere we have defined\n\\begin{equation} \\label{def_deltarho}\n\\varrho_\\varepsilon^{(1)}(t,x)\\,:=\\,\\frac{\\vre-\\widetilde{\\varrho}_\\varepsilon}{\\ep^m}\\qquad\\mbox{ and }\\qquad\n\\widetilde{r}_\\varepsilon(x)\\,:=\\,\\frac{\\widetilde{\\varrho}_\\varepsilon-1}{\\ep^{2(m-n)}}\\,.\n\\end{equation}\n\nThanks again to \\eqref{est:rho_ess}, \\eqref{est:rho_res} and Proposition \\ref{p:target-rho_bound}, we see that the above quantities verify the following uniform bounds, for any time $T>0$ fixed:\n\\begin{equation}\\label{uni_varrho1}\n\\sup_{\\varepsilon\\in\\,]0,1]}\\left\\|\\varrho_\\varepsilon^{(1)}\\right\\|_{L^\\infty_T(L^2+L^{\\gamma}({\\Omega}))}\\,\\leq\\, c \\qquad\\qquad\\mbox{ and }\\qquad\\qquad\n\\sup_{\\varepsilon\\in\\,]0,1]}\\left\\| \\widetilde{r}_\\varepsilon \\right\\|_{L^{\\infty}(\\Omega)}\\,\\leq\\, c \\,.\n\\end{equation}\nIn view of the previous properties, there exist $\\varrho^{(1)}\\in L^\\infty_{\\rm loc}(\\mathbb{R}_+;L^2+L^{\\gamma})$ and\n$\\widetilde{r}\\in L^\\infty$ such that (up to the extraction of a new suitable subsequence), for any $T>0$ we have\n\\begin{equation} \\label{conv:rr}\n\\varrho_\\varepsilon^{(1)}\\,\\weakstar\\,\\varrho^{(1)}\\quad\\mbox{ weakly-$*$ in } L^\\infty(0,T;L^2+L^{\\gamma}(\\Omega))\\qquad \n\\mbox{ and }\\qquad \\widetilde{r}_\\varepsilon\\,\\weakstar\\,\\widetilde{r} \\quad\\mbox{ weakly-$*$ in } L^\\infty(\\Omega).\n\\end{equation}\nIn particular, we get\n\t\\begin{equation*}\n\tR_\\varepsilon\\, \\weakstar\\,\\widetilde{r} \\quad \\mbox{ weakly-$*$ in } L^\\infty\\bigl(0,T; L^{\\min\\{\\gamma,2\\}}_{\\rm loc}(\\Omega)\\bigr)\\, .\n\t\\end{equation*}\n\n\\begin{remark} \\label{r:g>2}\nObserve that, owing to \\eqref{est:res_g>2}, when $\\gamma\\geq2$ we get\n\\[\n\\sup_{\\varepsilon\\in\\,]0,1]}\\left\\|\\varrho_\\varepsilon^{(1)}\\right\\|_{L^\\infty_T(L^2)}\\,\\leq\\, c\\, .\n\\]\nTherefore, in that case we actually have that $\\varrho^{(1)}\\,\\in\\,L^\\infty_T(L^2)$ and that $\\varrho_\\varepsilon^{(1)}\\,\\stackrel{*}{\\rightharpoonup}\\,\\varrho^{(1)}$ in $L^\\infty_T(L^2)$.\n\nAnalogously, when $\\gamma\\geq2$ we also get\n\\[\n\\| \\vre - 1 \\|_{L^\\infty_T(L^2 + L^\\infty)}\\,\\leq\\,c\\, \\ep^{2(m-n)}\\, .\n\\]\n\n\\end{remark}\n\n\n\\subsection{Constraints on the limit}\\label{ss:ctl1}\n\nIn this section, we establish some properties that the limit points of the family $\\bigl(\\varrho_\\varepsilon,\\vec u_\\varepsilon \\bigr)_\\varepsilon$, which have been\nidentified here above, have to satisfy.\n\nWe first need a preliminary result about the decomposition of the pressure function, which will be useful in the following computations.\n\\begin{lemma}\\label{lem:manipulation_pressure}\nLet $(m,n)\\in\\mathbb{R}^2$ verify the condition $m+1\\,\\geq\\,2n\\,>\\,m\\, \\geq 1$. Let $p$ be the pressure term satisfying the structural hypotheses \\eqref{pp1}--\\eqref{pp2}. Then, for any $\\varepsilon\\in\\,]0,1]$, one has\n \\begin{equation}\\label{p_decomp_lemma}\n\\begin{split}\n\\frac{1}{\\varepsilon^{2m}}\\,\\nabla_x\\Big(p(\\varrho_\\varepsilon)\\,-\\,p(\\widetilde{\\varrho}_\\varepsilon)\\Big)\\,=\\,\n\\frac{1}{\\varepsilon^m}\\nabla_x \\Big(p^\\prime (1)\\varrho_\\varepsilon^{(1)}\\Big)\\,+\\,\n\\frac{1}{\\varepsilon^{2n-m}}\\,\\nabla_x \\Pi_\\varepsilon\\, ,\n\\end{split}\n \\end{equation}\n where the functions $\\vr_\\varepsilon^{(1)}$ have been introduced in \\eqref{def_deltarho} and, for all $T>0$, the family $\\big(\\Pi_\\varepsilon\\big)_\\varepsilon$ verifies the uniform bound\n\\begin{equation}\\label{unif-bound-Pi}\n\\left\\|\\Pi_\\varepsilon\\right\\|_{L^\\infty_T(L^1+L^2+L^\\gamma)}\\,\\leq\\,C\\,.\n\\end{equation}\nWhen $\\gamma\\geq2$, one can dispense of the space $L^\\gamma$ in the above control of $\\big(\\Pi_\\varepsilon\\big)_\\varepsilon$.\n\\end{lemma}\n\n\\begin{proof}\nWe start by writing simple algebraic computations:\n\\begin{equation}\\label{rel_p}\n\\begin{split}\n\\frac{1}{\\varepsilon^{2m}}\\,\\nabla_x\\Big(p(\\varrho_\\varepsilon)\\,-\\,p(\\widetilde{\\varrho}_\\varepsilon)\\Big)\\,&=\\,\\frac{1}{\\varepsilon^{2m}}\\,\\nabla_x\\Big(p(\\varrho_\\varepsilon)\\,-\\,p(\\widetilde{\\varrho}_\\varepsilon)-p^\\prime(\\widetilde{\\varrho}_\\varepsilon)(\\varrho_\\varepsilon -\\widetilde{\\varrho}_\\varepsilon ) \\Big)\\\\\n&\\qquad\\qquad+\\,\\frac{1}{\\varepsilon^{m}}\\,\\nabla_x\\Big(\\big(p^\\prime(\\widetilde{\\varrho}_\\varepsilon)\\,-\\,p'(1)\\big)\\,\\vr_\\varepsilon^{(1)}\\Big)\\,+\\,\n\\frac{1}{\\varepsilon^m}\\nabla_x \\Big(p^\\prime (1)\\varrho_\\varepsilon^{(1)}\\Big)\\,.\n\\end{split}\n\\end{equation}\n\nWe start by analysing the first term on the right-hand side of \\eqref{rel_p}. For the essential part, we can employ a Taylor expansion\nto write\n$$\n\\left[p(\\varrho_\\varepsilon)-p(\\widetilde{\\varrho}_\\varepsilon)-p^\\prime(\\widetilde{\\varrho}_\\varepsilon)(\\varrho_\\varepsilon-\\widetilde{\\varrho}_\\varepsilon)\\right]_\\ess=\\left[p^{\\prime \\prime}(z_\\varepsilon )(\\varrho_\\varepsilon - \\widetilde{\\varrho}_\\varepsilon)^2\\right]_\\ess\\, ,\n$$\nwhere $z_\\varepsilon$ is a suitable point between $\\varrho_\\varepsilon$ and $\\widetilde{\\varrho}_\\varepsilon$. Thanks to the uniform bound \\eqref{est:rho_ess}, we have that this term is of order $O(\\varepsilon^{2m})$ in $L^\\infty_T(L^1)$, for any $T>0$ fixed. For the residual part, we can use \\eqref{est:M_res-measure} and \\eqref{est:rho_res}, together with the boundedness of the profiles $\\widetilde\\vr_\\varepsilon$ (keep in mind Proposition \\ref{p:target-rho_bound}), to deduce that\n\\[\n\\left\\|\\left[p(\\varrho_\\varepsilon)-p(\\widetilde{\\varrho}_\\varepsilon)-p^\\prime(\\widetilde{\\varrho}_\\varepsilon)(\\varrho_\\varepsilon-\\widetilde{\\varrho}_\\varepsilon)\\right]_\\res\\right\\|_{L^\\infty_T(L^1)}\\,\\leq C\\,\\varepsilon^{2m}\\,.\n\\]\nWe refer to \\textsl{e.g.} Lemma 4.1 of \\cite{F_2019} for details.\n\nIn a similar way, a Taylor expansion for the second term on the right-hand side of \\eqref{rel_p} gives\n$$\n\\big( p^\\prime(\\widetilde{\\varrho}_\\varepsilon)-p^\\prime(1)\\big)\\varrho^{(1)}_\\varepsilon\\,=\\,p''(\\eta_\\varepsilon )( \\widetilde{\\varrho}_\\varepsilon -1)\\varrho^{(1)}_\\varepsilon\n$$\nwhere $\\eta_\\varepsilon$ is a suitable point between $\\widetilde{\\varrho}_\\varepsilon$ and 1. Owing to Proposition \\ref{p:target-rho_bound} again and to bound \\eqref{uni_varrho1},\nwe infer that this term is of order $O(\\varepsilon^{2(m-n)})$ in $L^\\infty_T(L^2+L^\\gamma)$, for any time $T>0$ fixed. Then, defining \n$$ \\Pi_\\varepsilon:=\\frac{1}{\\varepsilon^{2(m-n)}}\\left[\\frac{p(\\varrho_\\varepsilon )-p(\\widetilde{\\varrho}_\\varepsilon)}{\\varepsilon^m}-p^\\prime (1)\\varrho_\\varepsilon^{(1)}\\right] $$\nwe have the control \\eqref{unif-bound-Pi}.\n\nThe final statement concerning the case $\\gamma\\geq2$ easily follows from Remark \\ref{r:g>2}.\nThis completes the proof of the lemma. \n\\qed\n\\end{proof}\n\n\n\\begin{remark} \\label{r:pressure}\nNotice that the last term appearing in \\eqref{p_decomp_lemma} is singular in $\\varepsilon$. This is in stark contrast with the situation considered in previous works, see\n\\textsl{e.g.} \\cite{F-G-N}, \\cite{F-G-GV-N}, \\cite{F-N_CPDE} and \\cite{F_2019}. However, its gradient structure will play a fundamental role in the computations below.\n\\end{remark}\n\nThis having been pointed out, we can now analyse the constraints on the weak-limit points $\\big(\\vr^{(1)},\\vec U\\big)$, identified in relations \\eqref{conv:u} and \\eqref{conv:rr} above. \n\n\\subsubsection{The case of large values of the Mach number: $m>1$} \\label{ss:constr_2}\n\nWe start by considering the case of anisotropic scaling, namely $m>1$ and $m+1\\geq 2n>m$. Notice that, in particular, one has $m>n$.\n\n\n\\begin{proposition} \\label{p:limitpoint}\nLet $m>1$ and $m+1\\geq 2n>m$ in \\eqref{ceq}--\\eqref{meq}.\nLet $\\left( \\vre, \\ue \\right)_{\\varepsilon}$ be a family of weak solutions, related to initial data $\\left(\\varrho_{0,\\varepsilon},\\vec u_{0,\\varepsilon}\\right)_\\varepsilon$\nverifying the hypotheses of Section \\ref{sss:data-weak}. Let $(\\varrho^{(1)}, \\vec{U} )$ be a limit point of the sequence\n$\\left(\\varrho_\\varepsilon^{(1)}, \\ue\\right)_{\\varepsilon}$, as identified in Section \\ref{ss:unif-est}. Then\n\\begin{align}\n&\\vec{U}\\,=\\,\\,\\Big(\\vec{U}^h\\,,\\,0\\Big)\\,,\\qquad\\qquad \\mbox{ with }\\qquad \\vec{U}^h\\,=\\,\\vec{U}^h(t,x^h)\\quad \\mbox{ and }\\quad {\\rm div}\\,_{\\!h}\\,\\vec{U}^h\\,=\\,0\\,, \\label{eq:anis-lim_1} \\\\[1ex]\n&\\nabla_x \\varrho^{(1)}\\,=\\, 0\n\\qquad\\qquad\\mbox{ in }\\;\\,\\mathcal D^\\prime(\\mathbb{R}_+\\times \\Omega)\\,. \\label{eq:anis-lim_2} \n\\end{align}\n\\end{proposition}\n\n\n\n\\begin{proof} First of all, let us consider the weak formulation of the mass equation \\eqref{ceq}. Take a test function\n$\\varphi\\in C_c^\\infty\\bigl(\\mathbb{R}_+\\times\\Omega\\bigr)$ and denote $[0,T]\\times K\\,:=\\,{\\rm supp} \\, \\varphi$. Then by \\eqref{weak-con} we have\n$$\n-\\int^T_0\\int_K\\bigl(\\varrho_\\varepsilon-1\\bigr)\\,\\partial_t\\varphi \\dxdt\\,-\\,\\int^T_0\\int_K\\varrho_\\varepsilon\\,\\vec{u}_\\varepsilon\\,\\cdot\\,\\nabla_{x}\\varphi \\dxdt\\,=\\,\n\\int_K\\bigl(\\varrho_{0,\\varepsilon}-1\\bigr)\\,\\varphi(0,\\,\\cdot\\,)\\dx\\,.\n$$\nWe can easily pass to the limit in this equation, thanks to the strong convergence $\\varrho_\\varepsilon\\longrightarrow1$, provided by \\eqref{rr1}, and the weak convergence of\n$\\vec{u}_\\varepsilon$ in $L_T^2\\bigl(L^6_{\\rm loc}\\bigr)$, provided by \\eqref{conv:u} and Sobolev embeddings. Notice that one always has $1\/\\gamma\\,+\\,1\/6\\,\\leq\\,1$. In this way,\nwe find\n \n\\begin{equation}\\label{001_U}\n-\\,\\int^T_0\\int_K\\vec{U}\\,\\cdot\\,\\nabla_{x}\\varphi \\dxdt\\,=\\,0\n\\end{equation}\nfor $\\varphi$ taken as above. Since the choice of $\\varphi$ is arbitrary,\nwe obtain that\n\\begin{equation} \\label{eq:div-free}\n{\\rm div}\\, \\U = 0 \\qquad\\qquad\\mbox{ a.e. in }\\; \\,\\mathbb{R}_+\\times \\Omega\\,.\n\\end{equation}\n\nNext, we test the momentum equation \\eqref{meq} on $\\varepsilon^m\\,\\vec\\phi$, for a smooth compactly supported $\\vec\\phi$.\nUsing the uniform bounds established in Section \\ref{ss:unif-est}, it is easy to see that the term presenting the time derivative, the viscosity term and the convective\nterm all converges to $0$, in the limit $\\varepsilon\\ra0^+$. Since $m>1$, also the Coriolis term vanishes when $\\varepsilon\\ra0^+$. It remains us to consider\nthe pressure and gravity terms in the weak formulation \\eqref{weak-mom} of the momentum equation:\nusing relation \\eqref{prF}, we see that we can couple them to write\n\\begin{align}\n\\frac{1}{\\varepsilon^{2m}}\\,\\nabla_x p(\\varrho_\\varepsilon)-\\, \\frac{1}{\\varepsilon^{2n}}\\,\\varrho_\\varepsilon\\nabla_x G\\,=\\,\\frac{1}{\\varepsilon^{2m}}\\nabla_x\\Big(p(\\varrho_\\varepsilon)\\,-\\,p(\\widetilde{\\varrho}_\\varepsilon)\\Big)-\\,\\varepsilon^{m-2n}\\varrho_\\varepsilon^{(1)}\\nabla_x G\\,. \\label{eq:mom_rest_1}\n\\end{align}\nBy \\eqref{uni_varrho1} and the fact that $m>n$, we readily see that the last term in the right-hand side of \\eqref{eq:mom_rest_1} converges to $0$,\nwhen tested against any smooth compactly supported $\\varepsilon^m\\,\\vec\\phi$.\nAt this point, we use Lemma \\ref{lem:manipulation_pressure} to treat the first term on the right-hand side of \\eqref{eq:mom_rest_1}. \nSo, taking $\\vec\\phi \\in C^\\infty_c([0,T[\\, \\times \\Omega)$ (for some $T>0$), we\ntest the momentum equation against $\\varepsilon^m\\,\\vec\\phi$: using \\eqref{conv:rr}, in the limit $\\varepsilon\\ra0^+$ we find that \n$$ \\int_0^T \\int_\\Omega p'(1) \\vr^{(1)} {\\rm div}\\, \\vec\\phi \\dxdt = 0\\,.$$\nRecalling that $p^\\prime (1)=1$, the previous relation implies \\eqref{eq:anis-lim_2} for $\\vr^{(1)}$.\nIn particular, that relation implies that $\\varrho^{(1)}(t,x)\\,=\\,c(t)$ for almost all $(t,x)\\in\\mathbb{R}_+\\times\\Omega$, for a suitable function $c=c(t)$ depending only on time.\n\nNow, in order to see effects due to the fast rotation in the limit, we need to ``filter out'' the contribution coming from the low Mach number.\nTo this end, we test \\eqref{meq} on $\\varepsilon\\,\\vec\\phi$, where this time we take $\\vec\\phi\\,=\\,{\\rm curl}\\,\\vec\\psi$, for some smooth compactly supported $\\vec\\psi\\,\\in C^\\infty_c\\bigl([0,T[\\,\\times\\Omega\\bigr)$, with $T>0$.\nOnce again, by uniform bounds we infer that the $\\partial_t$ term, the convective term and the viscosity term all converge to $0$ when $\\varepsilon\\ra0^+$.\nAs for the pressure and the gravitational force, we argue as in \\eqref{eq:mom_rest_1}. Since the structure of $\\vec\\phi$ kills any gradient term, we are left with the convergence\nof the integral\n$$\n\\int^T_0\\int_\\Omega\\varepsilon^{m-2n+1}\\varrho_\\varepsilon^{(1)}\\nabla_x G\\cdot\\vec\\phi\\,\\dx\\,dt\\,\\longrightarrow\\,\n\\delta_0(m-2n+1)\\int^T_0\\int_\\Omega\\varrho^{(1)}\\nabla_x G\\cdot\\vec\\phi\\,\\dx\\,dt\\,,\n$$\nwhere $\\delta_0(\\zeta)\\,=\\,1$ if $\\zeta=0$, $\\delta_0(\\zeta)\\,=\\,0$ otherwise.\nFinally, arguing as done for the mass equation, we see that the Coriolis term converges to the integral $\\int^T_0\\int_\\Omega\\vec{e}_3\\times\\vec{U}\\cdot\\vec\\phi$.\n\nConsider the case $m+1>2n$ for a while.\nPassing to the limit for $\\varepsilon\\ra0^+$, we find that $\\mathbb{H}\\left(\\vec{e}_3\\times\\vec{U}\\right)\\,=\\,0$, which implies that\n$\\vec{e}_3\\times\\vec{U}\\,=\\,\\nabla_x\\Phi$, for some potential function $\\Phi$. From this relation, one easily deduces that $\\Phi=\\Phi(t,x^h)$, \\textsl{i.e.} $\\Phi$ does not depend\non $x^3$, and that the same property is inherited by $\\vec{U}^h\\,=\\,\\bigl(U^1,U^2\\bigr)$, \\textsl{i.e.} one has $\\vec{U}^h\\,=\\,\\vec{U}^h(t,x^h)$. Furthermore, since $\\vec U^h\\,=\\,-\\,\\nabla^\\perp\\Phi$, we get that ${\\rm div}\\,_{\\!h}\\,\\vec{U}^h\\,=\\,0$.\nAt this point, we\ncombine this fact with \\eqref{eq:div-free} to infer that $\\partial_3 U^3\\,=\\,0$; but, thanks to the boundary condition\n\\eqref{bc1-2}, we must have $\\bigl(\\vec{U}\\cdot\\vec{n}\\bigr)_{|\\partial\\Omega}\\,=\\,0$, which implies that $U^3$ has to vanish at the boundary of $\\Omega$.\nThus, we finally deduce that $U^3\\,\\equiv\\,0$, whence \\eqref{eq:anis-lim_1} follows.\n\nNow, let us focus on the case when $m+1=2n$. The previous computations show that, when $\\varepsilon\\ra0^+$, we get\n\\begin{equation}\\label{eq:streamfunction_1} \n\\vec{e}_{3}\\times \\vec{U}+\\varrho^{(1)}\\nabla_x G\\,=\\,\\nabla_x\\Phi \\qquad\\qquad\\mbox{ in }\\; \\mathcal D^\\prime(\\mathbb{R}_+\\times \\Omega)\\,,\n\\end{equation}\nfor a new suitable function $\\Phi$. However, owing to \\eqref{eq:anis-lim_2}, we see that $\\varrho^{(1)}\\nabla_x G\\,=\\,\\nabla_x\\big(\\vr^{(1)}\\,G\\big)$; hence, the previous relations\ncan be recasted as $\\vec{e}_3\\times\\vec U\\,=\\,\\nabla_x\\widetilde\\Phi$, for a new scalar function $\\widetilde\\Phi$. Therefore, the same analysis as above applies,\nallowing us to gather \\eqref{eq:anis-lim_1} also in the case $m+1=2n$.\n\\qed\n\\end{proof}\n\n\n\\subsubsection{The case $m=1$} \\label{ss:constr_1}\n\nNow we focus on the case $m=1$. In this case, the fast rotation and weak compressibility\neffects are of the same order: this allows to reach the so-called \\emph{quasi-geostrophic balance} in the limit.\n\n\n\\begin{proposition} \\label{p:limit_iso}\nTake $m=1$ and $1\/21$. Now, since the most singular terms are of order $\\varepsilon^{-1}$ (keep in mind Lemma \\ref{lem:manipulation_pressure}), we test\nthe weak formulation \\eqref{weak-mom} of the momentum equation against $\\varepsilon\\,\\vec\\phi$, where $\\vec \\phi$ is a smooth compactly supported function. Similarly to what done above, the uniform bounds of Section \\ref{ss:unif-est} allow us to infer that the only quantity which does not vanish in the limit is the sum of the terms involving the Coriolis force, the pressure and the gravitational force: more precisely, using also Lemma \\ref{lem:manipulation_pressure} and \\eqref{prF}, we have\n$$\n\\vec{e}_{3}\\times \\varrho_{\\varepsilon}\\ue\\,+\\frac{\\nabla_x \\Big( p(\\varrho_\\varepsilon)-p(\\widetilde{\\varrho}_\\varepsilon)\\Big)}{\\varepsilon}\\,-\\,\n\\varepsilon^{2(1-n)}\\varrho_\\varepsilon^{(1)}\\nabla_x G\\,=\\,\\mathcal O(\\varepsilon)\n$$\nin the sense of $\\mathcal D'(\\mathbb{R}_+\\times\\Omega)$.\nFollowing the same computations performed in the proof of Proposition \\ref{p:limitpoint}, in the limit $\\varepsilon\\ra0^+$ it is easy to get that\n$$\n\\vec{e}_{3}\\times \\vec{U}+\\nabla_x\\left(p^\\prime (1) \\varrho^{(1)}\\right)\\,=\\,0\\qquad\\qquad\\mbox{ in }\\; \\mathcal D'\\big(\\mathbb{R}_+\\times \\Omega\\big)\\,.\n$$\nAfter recalling that $p^\\prime (1)=1$, this equality can be equivalently written as\n$$\n\\vec{e}_{3}\\times \\vec{U}+\\nabla_x \\varrho^{(1)}\\,=\\,0 \\qquad\\qquad\\mbox{ a.e. in }\\; \\mathbb{R}_+ \\times \\Omega\\,.\n$$\nNotice that $\\vec U$ is in fact in $L^2_{\\rm loc}(\\mathbb{R}_+;L^2)$, therefore so is $\\nabla_x \\vr^{(1)}$; hence the previous relation is in fact satisfied almost everywhere\nin $\\mathbb{R}_+\\times\\Omega$.\n\nAt this point, we can repeat the same argument used in the proof of Proposition \\ref{p:limitpoint} to deduce \\eqref{eq:for q}.\nThe proposition is thus proved.\n\\qed\n\\end{proof}\n\n\n\\section{Convergence in the case $m>1$}\\label{s:proof}\n\nIn this section, we complete the proof of Theorem \\ref{th:m>1}. Namely, we show convergence in the weak formulation of the primitive system, in the case when $m>1$ and $m+1\\geq 2n>m$. \n\nIn Proposition \\ref{p:limitpoint}, we have already seen how passing to the limit in the mass equation.\nHowever, problems arise when tackling the convergence in the momentum equation. Indeed, the analysis carried out so far\nis not enough to identify the weak limit of the convective term $\\varrho_\\varepsilon\\,\\vec u_\\varepsilon\\otimes\\vec u_\\varepsilon$, which is highly non-linear.\nFor proving that this term converges to the expected limit $\\vec U\\otimes\\vec U$, the key point is to control the strong oscillations in time of the solutions,\ngenerated by the singular terms in the momentum equation. For this, we will use a compensated compactness argument and exploit the algebraic structure of the wave system\nunderlying the primitive equations \\eqref{ceq}--\\eqref{meq}.\n\nIn Section \\ref{ss:acoustic}, we start by giving a quite accurate description of those fast oscillations.\nThen, using that description, we are able, in Section \\ref{ss:convergence}, to establish two fundamental properties: on the one hand, strong convergence of a suitable quantity related to the velocity fields; on the other hand, that the other terms which do not involve that quantity tend to vanish when $\\varepsilon\\ra0^+$.\nIn turn, this allows us to complete, in Section \\ref{ss:limit}, the proof of the convergence.\n\n\\subsection{Analysis of the acoustic waves} \\label{ss:acoustic}\n\nThe goal of the present subsection is to describe the fast time oscillations of the solutions. First of all, we recast our equations into a wave system. Then, we establish uniform bounds for the quantities appearing in the wave system. Finally, we apply a regularisation\nin space procedure for all the quantities, which is preparatory in view of the computations of Section \\ref{ss:convergence}.\n\n\n\\subsubsection{Formulation of the acoustic equation} \\label{sss:wave-eq}\n\nWe introduce the quantity\n$$\n\\vec{V}_\\varepsilon\\,:=\\,\\varrho_\\varepsilon\\vec{u}_\\varepsilon\\,.\n$$\nThen, straightforward computations show that we can recast the continuity equation in the form\n\\begin{equation} \\label{eq:wave_mass}\n\\varepsilon^m\\,\\partial_t\\varrho^{(1)}_\\varepsilon\\,+\\,{\\rm div}\\,\\vec{V}_\\varepsilon\\,=\\,0\\,,\n\\end{equation}\nwhere $\\varrho^{(1)}_\\varepsilon$ is defined in \\eqref{def_deltarho}.\nNext, thanks to Lemma \\ref{lem:manipulation_pressure} and the static relation \\eqref{prF}, we can derive the following form of the momentum equation:\n\\begin{align}\n\\varepsilon^m\\,\\partial_t\\vec{V}_\\varepsilon\\,+\\,\\varepsilon^{m-1}\\,\\vec{e}_3\\times \\vec V_\\varepsilon\\,+p^\\prime(1)\\,\\nabla_x \\varrho_{\\varepsilon}^{(1)}\\,&=\\,\n\\varepsilon^{2(m-n)}\\left(\\varrho_\\varepsilon^{(1)}\\nabla_x G\\,-\\,\\nabla_x\\Pi_\\varepsilon\n\\right) \\label{eq:wave_momentum} \\\\\n&\\qquad\\qquad\n+\\,\\varepsilon^m\\,\\Big({\\rm div}\\,\\mathbb{S}\\!\\left(\\nabla_x\\vec{u}_\\varepsilon\\right)\\,-\\,{\\rm div}\\,\\!\\left(\\varrho_\\varepsilon\\vec{u}_\\varepsilon\\otimes\\vec{u}_\\varepsilon\\right)\n\\Big)\\,. \\nonumber\n\\end{align}\n\nThen, if we define\n\\begin{equation}\\label{def_f-g}\n\\vec f_\\varepsilon :={\\rm div}\\,\\big(\\mathbb{S}\\!\\left(\\nabla_x\\vec{u}_\\varepsilon\\right)\\,-\\,\\varrho_\\varepsilon\\vec{u}_\\varepsilon\\otimes\\vec{u}_\\varepsilon\\big)\\qquad \\mbox{ and }\\qquad\n\\vec g_\\varepsilon :=\\varrho_\\varepsilon^{(1)}\\nabla_x G\\,-\\,\\nabla_x\\Pi_\\varepsilon\\,,\n\\end{equation}\nrecalling that we have normalised the pressure function so that $p^\\prime (1)=1$, we can recast the primitive system \\eqref{ceq}--\\eqref{meq} in the following form:\n\\begin{equation} \\label{eq:wave_syst}\n\\left\\{\\begin{array}{l}\n \\varepsilon^m\\,\\partial_t \\varrho^{(1)}_\\varepsilon\\,+\\,{\\rm div}\\,\\vec{V}_\\varepsilon\\,=\\,0 \\\\[1ex]\n \\varepsilon^m\\,\\partial_t\\vec{V}_\\varepsilon\\,+\\,\\nabla_x \\varrho_\\varepsilon^{(1)}\\,+\\,\\varepsilon^{m-1}\\,\\vec{e}_3\\times \\vec V_\\varepsilon\\,=\\,\\varepsilon^m\\,\\vec f_\\varepsilon +\\varepsilon^{2(m-n)}\\vec g_\\varepsilon\\,.\n \\end{array}\n\\right.\n\\end{equation}\n\nWe remark that system \\eqref{eq:wave_syst} has to be read in the weak sense: for any $\\varphi\\in C_c^\\infty\\bigl([0,T[\\,\\times \\overline\\Omega\\bigr)$, one has\n$$\n-\\,\\varepsilon^m\\,\\int^T_0\\int_{\\Omega} \\varrho^{(1)}_\\varepsilon\\,\\partial_t\\varphi\\,-\\,\\int^T_0\\int_{\\Omega} \\vec{V}_\\varepsilon\\cdot\\nabla_x\\varphi\\,=\\,\n\\varepsilon^{m}\\int_{\\Omega} \\varrho^{(1)}_{0,\\varepsilon}\\,\\varphi(0)\\,\\,,\n$$\nand also, for any $\\vec{\\psi}\\in C_c^\\infty\\bigl([0,T[\\,\\times \\overline\\Omega;\\mathbb{R}^3\\bigr)$ such that $(\\vec \\psi \\cdot \\vec n)_{|\\partial \\Omega}=0$, one has\n\\begin{align*}\n&\\hspace{-0.5cm}\n-\\,\\varepsilon^m\\,\\int^T_0\\int_{\\Omega}\\vec{V}_\\varepsilon\\cdot\\partial_t\\vec{\\psi}\\,-\\,\\int^T_0\\int_{\\Omega} \\varrho^{(1)}_\\varepsilon\\,{\\rm div}\\,\\vec{\\psi}\\,+\\,\\varepsilon^{m-1}\\int^T_0\\int_{\\Omega} \\vec{e}_3\\times\\vec V_\\varepsilon\\cdot\\vec\\psi \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\n=\\,\\varepsilon^{m}\\int_{\\Omega}\\varrho_{0,\\varepsilon}\\,\\vec{u}_{0,\\varepsilon}\\cdot\\vec{\\psi}(0)\\,+\\,\\varepsilon^m\\,\\int^T_0\\int_{\\Omega} \\vec f_\\varepsilon \\cdot\\vec{\\psi}+\\,\\varepsilon^{2(m-n)}\\,\\int^T_0\\int_{\\Omega} \\vec g_\\varepsilon \\cdot\\vec{\\psi}\\,.\n\\end{align*}\n\n\n\nHere we use estimates of Section \\ref{ss:unif-est} in order to show uniform bounds for the solutions and the data in the wave equation \\eqref{eq:wave_syst}.\nWe start by dealing with the ``unknown'' $\\vec V_\\varepsilon$. Splitting the term into essential and residual parts, one can obtain for all $T>0$: \n\\begin{equation}\\label{eq:V_bounds}\n\\|\\vec V_\\varepsilon\\|_{L^\\infty_T(L^2+L^{2\\gamma\/(\\gamma +1)})}\\leq c\\, .\n\\end{equation}\nIn the next lemma, we establish bounds for the source terms in the system of acoustic waves \\eqref{eq:wave_syst}.\n\n\\begin{lemma} \\label{l:source_bounds}\nWrite $\\vec f_\\varepsilon\\,=\\,{\\rm div}\\, \\widetilde{\\vec f}_\\varepsilon$ and $\\vec g_\\varepsilon\\,=\\,\\vec g^1_\\varepsilon\\,-\\,\\nabla_x\\Pi_\\varepsilon$, where we have defined\n$\\vec g^1_\\varepsilon\\,:=\\,\\vr_\\varepsilon^{(1)}\\,\\nabla_xG$ and the functions $\\Pi_\\varepsilon$ have been introduced in Lemma \\ref{lem:manipulation_pressure}.\n\nFor any $T>0$ fixed, one has the uniform embedding properties\n\\[\n\\big(\\widetilde{\\vec f}_\\varepsilon\\big)_\\varepsilon\\,\\subset\\,L^2_T(L^2+L^1)\\qquad\\mbox{ and }\\qquad \\big(\\vec g^1_\\varepsilon\\big)_\\varepsilon\\,\\subset\\,L^2_T(L^2+L^\\gamma)\\,.\n\\]\nIn the case $\\gamma\\geq2$, we may dispense with the space $L^\\gamma$ in the control of $\\big(\\vec g^1_\\varepsilon\\big)_\\varepsilon$.\n\nIn particular, the sequences $\\bigl(\\vec f_\\varepsilon\\bigr)_\\varepsilon$ and\n$\\bigl(\\vec g_\\varepsilon\\bigr)_\\varepsilon$, defined in system \\eqref{eq:wave_syst}, are uniformly bounded in the space $L^{2}\\big([0,T];H^{-s}(\\Omega)\\big)$, for all $s>5\/2$.\n\\end{lemma}\n\n\\begin{proof}\nFrom \\eqref{est:momentum}, \\eqref{est:Du} and \\eqref{unif-bound-for-vel}, we immediately infer the uniform bound for the family $\\big(\\widetilde{\\vec f}_\\varepsilon\\big)_\\varepsilon$ in $L^2_T(L^1+L^2)$,\nfrom which we deduce also the uniform boundedness of $\\big(\\vec f_\\varepsilon\\big)_\\varepsilon$ in $L^2_T(H^{-s})$, for any $s>5\/2$.\n\nNext, for bounding $\\big(\\vec g^1_\\varepsilon\\big)_\\varepsilon$ we simply use \\eqref{uni_varrho1}, together with Remark \\ref{r:g>2} when $\\gamma\\geq2$.\nKeeping in mind the bounds established in Lemma \\ref{lem:manipulation_pressure}, the uniform estimate for $\\big(\\vec g_\\varepsilon\\big)_\\varepsilon$ follows.\n\\qed\n\\end{proof}\n\n\\subsubsection{Regularization and description of the oscillations}\\label{sss:w-reg}\n\n\n\nAs already mentioned in Remark \\ref{r:period-bc}, in order to apply the Littlewood-Paley theory, it is convenient to reformulate problem \\eqref{ceq}--\\eqref{meq} in the new domain (which we keep calling $\\Omega$, with a little abuse of notation) \n$$ \\Omega:=\\mathbb{R}^2 \\times \\mathbb{T}^1\\, , \\quad \\text{with}\\quad \\mathbb{T}^1:=[-1,1]\/\\sim\\, .$$ \nIn addition, to avoid the appearing of (irrelevant) multiplicative constants in the computations, we suppose that the torus $\\mathbb{T}^1$ has been renormalised so that its Lebesgue measure is equal to 1.\n\nNow, for any $M\\in\\mathbb{N}$ we consider the low-frequency cut-off operator ${S}_{M}$ of a Littlewood-Paley decomposition, as introduced in equation \\eqref{eq:S_j} of the Appendix. Then, we define \n\\begin{equation}\\label{def_reg_vrho-V}\n\\varrho^{(1)}_{\\varepsilon ,M}={S}_{M}\\varrho^{(1)}_{\\varepsilon}\\qquad\\qquad \\text{ and }\\qquad\\qquad \\vec{V}_{\\varepsilon ,M}={S}_{M}\\vec{V}_{\\varepsilon}\\, .\n\\end{equation} \n\nThe previous regularised quantities satisfy the following properties.\n\n\\begin{proposition} \\label{p:prop approx}\nFor any $T>0$, we have the following convergence properties, in the limit $M\\rightarrow +\\infty $:\n\\begin{equation}\\label{eq:approx var}\n\\begin{split}\n&\\sup_{0<\\varepsilon\\leq1}\\, \\left\\|\\varrho^{(1)}_{\\varepsilon }-\\vrm\\right\\|_{L^{\\infty}([0,T];H^{-s})}\\longrightarrow 0\\qquad\n\\forall\\,s>\\max\\left\\{0,3\\left(\\frac{1}{\\gamma}\\,-\\,\\frac12\\right)\\right\\}\\\\\n&\\sup_{0<\\varepsilon\\leq1}\\, \\left\\|\\vec{V}_{\\varepsilon }-\\vec{V}_{\\varepsilon ,M}\\right\\|_{L^{\\infty}([0,T];H^{-s})}\\longrightarrow 0\\qquad\n\\forall\\,s>\\frac{3}{2\\,\\gamma}\\,.\n\\end{split}\n\\end{equation}\nMoreover, for any $M>0$, the couple $(\\vrm,\\vec V_{\\varepsilon ,M})$ satisfies the approximate wave equations\n\\begin{equation}\\label{eq:approx wave}\n\\left\\{\\begin{array}{l}\n \\varepsilon^m\\,\\partial_t \\vrm \\,+\\,\\,{\\rm div}\\,\\vec{V}_{\\varepsilon ,M}\\,=\\,0 \\\\[1ex]\n \\varepsilon^m\\,\\partial_t\\vec{V}_{\\varepsilon ,M}\\,+\\varepsilon^{m-1}\\,e_{3}\\times \\vec{V}_{\\varepsilon ,M}+\\,\\nabla_x \\vrm\\,=\\,\\varepsilon^m\\,\\vec f_{\\varepsilon ,M}\\,+\\varepsilon^{2(m-n)} \\vec g_{\\varepsilon,M} \n \\end{array}\n\\right.\n\\end{equation}\nwhere $(\\vec f_{\\varepsilon ,M})_{\\varepsilon}$ and $(\\vec g_{\\varepsilon ,M})_{\\varepsilon}$ are families of smooth (in the space variables) functions satisfying, for any $s\\geq0$, the uniform bounds\n\\begin{equation}\\label{eq:approx force}\n\\sup_{0<\\varepsilon\\leq1}\\, \\left\\|\\vec f_{\\varepsilon ,M}\\right\\|_{L^{2}([0,T];H^{s})}\\,+\\,\\sup_{0<\\varepsilon\\leq1}\\,\\left\\|\\vec g_{\\varepsilon ,M}\\right\\|_{L^{\\infty}([0,T];H^{s})}\\,\\leq\\, C(s,M)\\,,\n\\end{equation}\nwhere the constant $C(s,M)$ depends on the fixed values of $s\\geq 0$ and $M>0$, but not on $\\varepsilon>0$.\n\\end{proposition}\n\n\\begin{proof}\nThanks to characterization \\eqref{eq:LP-Sob} of $H^{s}$, properties \\eqref{eq:approx var} are straightforward consequences of the uniform bounds establish in Section \\ref{ss:unif-est}. For instance, let us consider the functions $\\vr^{(1)}_\\varepsilon$: when $\\gamma\\geq2$, owing to Remark \\ref{r:g>2} one has\n$\\big(\\vr^{(1)}_\\varepsilon\\big)_\\varepsilon\\,\\subset\\,L^\\infty_T(L^2)$, and then we use estimate \\eqref{est:sobolev} from the Appendix.\nWhen $1<\\gamma<2$, instead, we first apply the dual Sobolev embedding to infer that $\\big(\\vr^{(1)}_\\varepsilon\\big)_\\varepsilon\\,\\subset\\,L^\\infty_T(H^{-\\sigma})$,\nwith $\\sigma\\,=\\,\\sigma(\\gamma)\\,=\\,3\\big(1\/\\gamma-1\/2\\big)$, and then we use \\eqref{est:sobolev} again. The bounds for the momentum\n$\\big(\\vec V_\\varepsilon\\big)_\\varepsilon$ can be deduced by a similar argument, after observing that $2\\gamma\/(\\gamma+1)<2$ always.\n\nNext, applying the operator ${S}_{M}$ to \\eqref{eq:wave_syst} immediately gives us system \\eqref{eq:approx wave}, where we have set \n\\begin{equation*}\n\\vec f_{\\varepsilon ,M}:={S}_{M}\\vec f_\\varepsilon \\qquad \\text{ and }\\qquad \\vec g_{\\varepsilon ,M}:={S}_{M}\\vec g_\\varepsilon\\,.\n\\end{equation*}\nThanks to Lemma \\ref{l:source_bounds} and \\eqref{eq:LP-Sob}, it is easy to verify inequality \\eqref{eq:approx force}. \n\\qed\n\\end{proof}\n\n\\medbreak\nWe will need also the following important decomposition for the momentum vector fields $\\vec V_{\\varepsilon,M}$ and their ${\\rm curl}\\,$.\n\\begin{proposition} \\label{p:prop dec}\nFor any $M>0$ and any $\\varepsilon\\in\\,]0,1]$, the following decompositions hold true:\n\\begin{equation*}\n\\vec{V}_{\\varepsilon ,M}\\,=\\,\n\\varepsilon^{2(m-n)}\\vec{t}_{\\varepsilon ,M}^{1}+\\vec{t}_{\\varepsilon ,M}^{2}\\qquad\\mbox{ and }\\qquad\n{\\rm curl}\\,_{x}\\vec{V}_{\\varepsilon ,M}=\\varepsilon^{2(m-n)}\\vec{T}_{\\varepsilon ,M}^{1}+\\vec{T}_{\\varepsilon ,M}^{2}\\,,\n\\end{equation*}\nwhere, for any $T>0$ and $s\\geq 0$, one has \n\\begin{align*}\n&\\left\\|\\vec{t}_{\\varepsilon ,M}^{1}\\right\\|_{L^{2}([0,T];H^{s})}+\\left\\|\\vec{T}_{\\varepsilon ,M}^{1}\\right\\|_{L^{2}([0,T];H^{s})}\\leq C(s,M) \\\\\n&\\left\\|\\vec{t}_{\\varepsilon ,M}^{2}\\right\\|_{L^{2}([0,T];H^{1})}+\\left\\|\\vec{T}_{\\varepsilon ,M}^{2}\\right\\|_{L^{2}\\left([0,T];L^2\\right)}\\leq C\\,,\n\\end{align*}\nfor suitable positive constants $C(s,M)$ and $C$, which are uniform with respect to $\\varepsilon\\in\\,]0,1]$.\n\\end{proposition}\n\n\\begin{proof}\nWe decompose $\\vec{V}_{\\varepsilon ,M}\\,=\\,\\varepsilon^{2(m-n)}\\vec t_{\\varepsilon,M}^{1}\\,+\\,\\vec t_{\\varepsilon,M}^{2}$, where we define\n\\begin{equation} \\label{eq:t-T}\n\\vec{t}_{\\varepsilon,M}^{1}\\,:=\\,{S}_{M}\\left(\\frac{\\varrho_\\varepsilon -1}{\\varepsilon^{2(m-n)}}\\, \\vec{u}_{\\varepsilon}\\right) \\qquad\\mbox{ and }\\qquad\n\\vec{t}_{\\varepsilon,M}^{2}\\,:=\\,{S}_{M}\\left(\\vec{u}_{\\varepsilon}\\right)\\,.\n\\end{equation}\nThe decomposition of ${\\rm curl}\\,_x\\vec V_{\\varepsilon,M}$ follows after setting $\\vec T_{\\varepsilon,M}^j\\,:=\\,{\\rm curl}\\,_x\\vec t_{\\varepsilon,M}^j$, for $j=1,2$.\n\nWe have to prove uniform bounds for all those terms, by using the estimates established in Section \\ref{ss:unif-est} above.\nFirst of all, we have that $\\big(\\vu_\\varepsilon\\big)_\\varepsilon\\,\\subset\\,L^2_T(H^1)$, for any $T>0$ fixed. Then, we immediately gather the sought bounds for the vector fields $\\vec t_{\\varepsilon,M}^2$ and $\\vec T_{\\varepsilon,M}^2$.\n\nFor the families of $\\vec t_{\\varepsilon,M}^1$ and $\\vec T_{\\varepsilon,M}^1$, instead, we have to use the bounds provided by \\eqref{rr1} and (when $\\gamma\\geq2$)\nRemark \\ref{r:g>2}. In turn, we see that for any $T>0$:\n\\[\n\\left(\\frac{\\varrho_\\varepsilon -1}{\\varepsilon^{2(m-n)}}\\, \\vec{u}_{\\varepsilon}\\right)\\,\\subset\\,L^2_T(L^1+L^2+L^{6\\gamma\/(\\gamma+6)})\\,\\hookrightarrow\\,\nL^2_T(H^{-\\sigma})\\,,\n\\]\nfor some $\\sigma>0$ large enough. Therefore, the claimed bounds follow thanks to the regularising effect of the operators $S_M$. The proof of the proposition\nis thus completed.\n\\qed\n\\end{proof}\n\n\n\\subsection{Convergence of the convective term} \\label{ss:convergence}\nIn this subsection we show the convergence of the convective term. \nThe first step is to reduce its analysis to the case of smooth vector fields $\\vec{V}_{\\varepsilon ,M}$.\n\n\\begin{lemma} \\label{lem:convterm}\nLet $T>0$. For any $\\vec{\\psi}\\in C_c^\\infty\\bigl([0,T[\\,\\times\\Omega;\\mathbb{R}^3\\bigr)$, we have \n\\begin{equation*}\n\\lim_{M\\rightarrow +\\infty} \\limsup_{\\varepsilon \\rightarrow 0^+}\\left|\\int_{0}^{T}\\int_{\\Omega} \\varrho_\\varepsilon\\,\\vec{u}_\\varepsilon\\otimes \\vec{u}_\\varepsilon: \\nabla_{x}\\vec{\\psi}\\, dx \\, dt-\n\\int_{0}^{T}\\int_{\\Omega} \\vec{V}_{\\varepsilon ,M}\\otimes \\vec{V}_{\\varepsilon,M}: \\nabla_{x}\\vec{\\psi}\\, dx \\, dt\\right|=0\\, .\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nThe proof is very similar to the one of Lemma 4.5 from \\cite{DS-F-S-WK}, for this reason we just outline it.\n\nOne starts by using the decomposition $\\vr_\\varepsilon\\,=\\,1\\,+\\,\\varepsilon^{2(m-n)}\\,R_\\varepsilon$ to reduce (owing to the uniform bounds of Section \\ref{ss:unif-est}) the convective term to the ``homogeneous counterpart'': for any test function $\\vec\\psi\\in C^\\infty_c\\big(\\mathbb{R}_+\\times\\Omega;\\mathbb{R}^3\\big)$, one has\n\\[\n\\lim_{\\varepsilon \\rightarrow 0^+}\\left|\\int_{0}^{T}\\int_{\\Omega} \\varrho_\\varepsilon\\,\\vec{u}_\\varepsilon\\otimes \\vec{u}_\\varepsilon: \\nabla_{x}\\vec{\\psi}\\, dx \\, dt-\n\\int_{0}^{T}\\int_{\\Omega}\\vec{u}_\\varepsilon\\otimes\\vec{u}_\\varepsilon:\\nabla_{x}\\vec{\\psi}\\,\\dx\\,\\dt\\right|\\,=\\,0\\,.\n\\]\nNotice that, here, one has to use that $\\gamma\\geq 3\/2$.\n\nAfter that, we write\n$\\vu_\\varepsilon\\,=\\,S_M(\\vu_\\varepsilon)\\,+\\,({\\rm Id}\\,-S_M)\\vu_\\varepsilon\\,=\\,\\vec t^2_{\\varepsilon,M}\\,+\\,({\\rm Id}\\,-S_M)\\vu_\\varepsilon$.\nUsing Proposition \\ref{p:prop dec} and the fact that $\\left\\|({\\rm Id}\\,-{S}_{M})\\,\\vec{u}_\\varepsilon\\right\\|_{L_{T}^{2}(L^{2})}\\,\\leq C\\,2^{-M}\\, \\|\\nabla_x\\vec u_\\varepsilon\\|_{L^2_T(L^2)}\\leq C\\,2^{-M}$, which holds in view of estimate \\eqref{est:sobolev} from the Appendix and the uniform bound \\eqref{unif-bound-for-vel},\none can conclude.\n\\qed\n\\end{proof}\n\n\\medbreak\n\nFrom now on, for notational convenience, we generically denote by $\\mathcal{R}_{\\varepsilon ,M}$ any remainder term, that is any term satisfying the property\n\\begin{equation} \\label{eq:reminder}\n\\lim_{M\\rightarrow +\\infty}\\limsup_{\\varepsilon \\rightarrow 0^+}\\left|\\int_{0}^{T}\\int_{\\Omega}\\mathcal{R}_{\\varepsilon ,M}\\cdot \\vec{\\psi}\\, dx \\, dt\\right|=0\n\\end{equation}\nfor all test functions $\\vec{\\psi}\\in C_c^\\infty\\bigl([0,T[\\,\\times\\Omega;\\mathbb{R}^3\\bigr)$ lying in the kernel of the singular perturbation operator,\nnamely (in view of Proposition \\ref{p:limitpoint}) such that\n\\begin{equation} \\label{eq:test-f}\n\\vec\\psi\\in C_c^\\infty\\big([0,T[\\,\\times\\Omega;\\mathbb{R}^3\\big)\\qquad\\qquad \\mbox{ such that }\\qquad {\\rm div}\\,\\vec\\psi=0\\quad\\mbox{ and }\\quad \\partial_3\\vec\\psi=0\\,.\n\\end{equation}\nNotice that, in order to pass to the limit in the weak formulation of the momentum equation and derive the limit system, it is enough to use test functions $\\vec\\psi$ as above.\n\nThus, for $\\vec\\psi$ as in \\eqref{eq:test-f}, \nwe have to pass to the limit in the term \n\\begin{align*}\n-\\int_{0}^{T}\\int_{\\Omega} \\vec{V}_{\\varepsilon ,M}\\otimes \\vec{V}_{\\varepsilon ,M}: \\nabla_{x}\\vec{\\psi}\\,&=\\,\\int_{0}^{T}\\int_{\\Omega} {\\rm div}\\,\\left(\\vec{V}_{\\varepsilon ,M}\\otimes\n\\vec{V}_{\\varepsilon ,M}\\right) \\cdot \\vec{\\psi}\\,.\n\\end{align*}\nNotice that the integration by parts above is well-justified, since all the quantities inside the integrals are smooth with respect to the space variable. Owing to the structure of the test function, and\nresorting to the notation introduced in \\eqref{eq:decoscil}, we remark that we can write\n$$\n\\int_{0}^{T}\\int_{\\Omega} {\\rm div}\\,\\left(\\vec{V}_{\\varepsilon ,M}\\otimes \\vec{V}_{\\varepsilon ,M}\\right) \\cdot \\vec{\\psi}\\,=\\,\n\\int_{0}^{T}\\int_{\\mathbb{R}^2} \\left(\\mathcal{T}_{\\varepsilon ,M}^{1}+\\mathcal{T}_{\\varepsilon, M}^{2}\\right)\\cdot\\vec{\\psi}^h\\,,\n$$\nwhere we have defined the terms\n\\begin{equation} \\label{def:T1-2}\n\\mathcal T^1_{\\varepsilon,M}\\,:=\\, {\\rm div}_h\\left(\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\otimes \\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\right)\\qquad \\mbox{ and }\\qquad\n\\mathcal T^2_{\\varepsilon,M}\\,:=\\, {\\rm div}_h\\left(\\langle \\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\otimes \\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\rangle \\right)\\,.\n\\end{equation}\n\nIn the next two paragraphs, we will deal with each one of those terms separately. We borrow most of the arguments from \\cite{DS-F-S-WK}\n(see also \\cite{F-G-GV-N}, \\cite{F_2019} for a similar approach). However, the special structure of the gravity force will play a key role here,\nin order (loosely speaking) to compensate the stronger singularity due to our scaling $2n>m$.\n\n\n\\subsubsection{Convergence of the vertical averages}\\label{sss:term1}\nWe start by dealing with $\\mathcal T^1_{\\varepsilon,M}$. It is standard to write\n\\begin{align}\n\\mathcal{T}_{\\varepsilon ,M}^{1}\\,&=\\,{\\rm div}_h\\left(\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\otimes \\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\right)=\n{\\rm div}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\, \\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle+\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle \\cdot \\nabla_{h}\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle \\label{eq:T1} \\\\\n&={\\rm div}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\, \\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle+\\frac{1}{2}\\, \\nabla_{h}\\left(\\left|\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\right|^{2}\\right)+\n{\\rm curl}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle^{\\perp}\\,. \\nonumber\n\\end{align}\nNotice that the second term is a perfect gradient, so it vanishes when tested against divergence-free test functions. Hence, we can treat it as\na remainder, in the sense of \\eqref{eq:reminder}.\n\n\nFor the first term in the second line of \\eqref{eq:T1}, instead, we take advantage of system \\eqref{eq:approx wave}: averaging the first equation with respect to $x^{3}$ and multiplying it by $\\langle \\vec{V}^h_{\\varepsilon ,M}\\rangle$, we arrive at\n$$\n{\\rm div}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,=\\,-\\varepsilon^m\\partial_t\\langle \\vrm\\rangle \\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,=\\,\n\\varepsilon^m\\langle \\vrm\\rangle \\partial_t \\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle +\\mathcal{R}_{\\varepsilon ,M}\\,,\n$$\nin the sense of distributions.\nWe remark that the term presenting the total derivative in time is in fact a remainder, thanks to the factor $\\varepsilon^m$ in front of it.\nNow, we use the horizontal part of \\eqref{eq:approx wave}, where we first take the vertical average and then multiply by $\\langle \\vrm\\rangle$:\nsince $m>1$, we gather\n\\begin{align*}\n&\\varepsilon^m\\langle \\vrm \\rangle \\partial_t \\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle \\\\\n&\\qquad\\quad=\n- \\langle \\vrm\\rangle \\nabla_{h}\\langle \\vrm \\rangle-\n\\varepsilon^{m-1}\\langle \\vrm \\rangle\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle^{\\perp}\n+\\varepsilon^{m}\\langle \\vrm \\rangle \\langle \\vec f_{\\varepsilon ,M}^{h}\\rangle+\\varepsilon^{2(m-n)}\\langle \\vrm \\rangle \\langle \\vec g_{\\varepsilon ,M}^{h}\\rangle\\\\\n&\\qquad\\quad=-\\varepsilon^{m-1}\\langle \\vrm \\rangle\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle^{\\perp}+\\mathcal{R}_{\\varepsilon ,M}\\, ,\n\\end{align*}\nwhere we have repeatedly exploited the properties proved in Proposition \\ref{p:prop approx} and we have included in the remainder term also the perfect gradient.\nInserting this relation into \\eqref{eq:T1} yields\n\\begin{equation*}\n\\mathcal{T}_{\\varepsilon ,M}^{1}= \\left({\\rm curl}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,-\\,\\varepsilon^{m-1}\\langle \\vrm \\rangle\\right) \n\\langle\\vec{V}_{\\varepsilon,M}^{h}\\rangle^{\\perp}+\\mathcal{R}_{\\varepsilon,M}\\,.\n\\end{equation*}\nObserve that the first term appearing in the right-hand side of the previous relation is bilinear. Thus, in order to pass to the limit in it, one needs some strong convergence property. As a matter of fact, in the next computations we will work on the regularised wave system \\eqref{eq:approx wave} to show that the quantity\n\\[\n\\gamma_{\\varepsilon, M}:={\\rm curl}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,-\\,\\varepsilon^{m-1}\\langle \\vrm \\rangle\n\\]\nis \\emph{compact} in some suitable space. In particular, as $m>1$, also ${\\rm curl}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle$ is compact.\n\nIn order to see this, we write the vertical average of the first equation in \\eqref{eq:approx wave} as\n\\begin{equation*}\n\\varepsilon^{2m-1}\\,\\partial_t \\langle \\vrm \\rangle\\,+\\,\\varepsilon^{m-1}{\\rm div}\\,_{h} \\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,=0\\,.\n\\end{equation*}\nNext, we take the vertical average of the horizontal components of the second equation in \\eqref{eq:approx wave} and then apply ${\\rm curl}_h$: we obtain\n\\begin{equation*}\n\\varepsilon^m\\,\\partial_t{\\rm curl}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,+\\varepsilon^{m-1}\\,{\\rm div}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\, =\\,\\varepsilon^m {\\rm curl}_h\\langle\\vec f_{\\varepsilon ,M}^{h}\\rangle+\\varepsilon^{2(m-n)} {\\rm curl}_h\\langle\\vec g_{\\varepsilon ,M}^{h}\\rangle\\, .\n\\end{equation*}\nAt this point, we recall the definition \\eqref{def_f-g} of $\\vec g_\\varepsilon$, and we see that ${\\rm curl}_h\\langle\\vec g_{\\varepsilon ,M}^{h}\\rangle\\,\\equiv\\,0$.\nThis property is absolutely fundamental, since it allows to erase the last term in the previous relation, which otherwise would have represented an obstacle to get compactness\nof the $\\gamma_{\\varepsilon,M}$'s. Indeed, thanks to this observation, we can sum up the last two equations to get\n\\begin{equation} \\label{eq:gamma}\n\\partial_{t}\\gamma_{\\varepsilon,M}\\,=\\,{\\rm curl}_h\\langle \\vec f_{\\varepsilon ,M}^{h}\\rangle\\, .\n\\end{equation}\nUsing estimate \\eqref{eq:approx force} in Proposition \\ref{p:prop approx}, we discover that, for any $M>0$ fixed, the family \n$\\left(\\partial_{t}\\,\\gamma_{\\varepsilon,M}\\right)_{\\varepsilon}$ is uniformly bounded (with respect to $\\varepsilon$) in \\textsl{e.g.} the space $L_{T}^{2}(L^{2})$. \nOn the other hand, we have that, again for any $M>0$ fixed,\nthe sequence $(\\gamma_{\\varepsilon,M})_{\\varepsilon}$ is uniformly bounded (with respect to $\\varepsilon$) \\textsl{e.g.} in the space $L_{T}^{2}(H^{1})$.\nSince the embedding $H_{\\rm loc}^{1}\\hookrightarrow L_{\\rm loc}^{2}$ is compact, the Aubin-Lions Lemma implies that, for any $M>0$ fixed, the family $(\\gamma_{\\varepsilon,M})_{\\varepsilon}$ is compact\nin $L_{T}^{2}(L_{\\rm loc}^{2})$. Then, up to extraction of a suitable subsequence (not relabelled here), that family converges strongly to a tempered distribution $\\gamma_M$ in the same space. \n\nNow, we have that $\\gamma_{\\varepsilon ,M}$ converges strongly to $\\gamma_M$ in $L_{T}^{2}(L_{\\rm loc}^{2})$ and $\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle$ converges weakly to\n$\\langle \\vec{V}_{M}^{h}\\rangle$ in $L_{T}^{2}(L_{\\rm loc}^{2})$ (owing to Proposition \\ref{p:prop dec}, for instance). Then, we deduce that\n\\begin{equation*}\n\\gamma_{\\varepsilon,M}\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle^{\\perp}\\longrightarrow \\gamma_M \\langle \\vec{V}_{M}^{h}\\rangle^{\\perp}\\qquad \\text{ in }\\qquad \\mathcal{D}^{\\prime}\\big(\\mathbb{R}_+\\times\\mathbb{R}^2\\big)\\,.\n\\end{equation*}\nObserve that, by definition of $\\gamma_{\\varepsilon,M}$, we must have $\\gamma_M={\\rm curl}_h\\langle \\vec{V}_{M}^{h}\\rangle$. On the other hand, owing to Proposition \\ref{p:prop dec} and \\eqref{eq:t-T}, we know that $\\langle \\vec{V}_{M}^{h}\\rangle= \\langle{S}_{M}\\vec{U}^{h}\\rangle$.\nTherefore, in the end we have proved that, for $m>1$ and $m+1\\geq 2n >m$,\none has the convergence (at any $M\\in\\mathbb{N}$ fixed, when $\\varepsilon\\ra0^+$)\n\\begin{equation} \\label{eq:limit_T1}\n\\int_{0}^{T}\\int_{\\mathbb{R}^2}\\mathcal{T}_{\\varepsilon ,M}^{1}\\cdot\\vec{\\psi}^h\\,dx^h\\,dt\\,\\longrightarrow\\,\n\\int^T_0\\int_{\\mathbb{R}^2}{\\rm curl}_h\\langle{S}_{M}\\vec{U}^{h}\\rangle\\; \\langle{S}_{M}(\\vec{U}^{h})^{\\perp}\\rangle\\cdot\\vec\\psi^h\\,dx^h\\,dt\\, ,\n\\end{equation}\nfor any $T>0$ and for any test-function $\\vec \\psi$ as in \\eqref{eq:test-f}.\n\n\\subsubsection{Vanishing of the oscillations}\\label{sss:term2}\n\nWe now focus on the term $\\mathcal{T}_{\\varepsilon ,M}^{2}$, defined in \\eqref{def:T1-2}. Recall that $m>1$. In what follows, we consider separately the two cases $m+1>2n$ and $m+1=2n$. As a matter of fact, in the case when $m+1=2n$, a bilinear term involving $\\vec g_{\\varepsilon,M}$ has no power of $\\varepsilon$ in front of it, so it is not clear that it converges to $0$ and, in fact, it might persist in the limit, giving rise to an additional term in the target system. To overcome this issue and show that this actually does not happen, we deeply exploit the structure of the wave system to recover a quantitative smallness for that term (namely, in terms of positive powers of $\\varepsilon$).\n\n\\subsubsection*{The case $m+1>2n$}\\label{sss:term2_osc}\n\nStarting from the definition of $\\mathcal T_{\\varepsilon,M}^2$, the same computations as above yield\n\\begin{align}\n\\mathcal{T}_{\\varepsilon ,M}^{2}\\,\n&=\\,\\langle {\\rm div}_h (\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h})\\;\\;\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\rangle+\\frac{1}{2}\\, \\langle \\nabla_{h}| \\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}|^{2} \\rangle+\n\\langle {\\rm curl}_h\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\,\\left( \\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\rangle\\, . \\label{eq:T2}\n\\end{align}\n\nLet us now introduce the quantities\n$$\n\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\,:=\\,( \\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h})^{\\perp}-\\partial_{3}^{-1}\\nabla_{h}^{\\perp}\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{3}\\qquad\\mbox{ and }\\qquad\n\\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\,:=\\,{\\rm curl}_h \\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\,.\n$$\nThen we can write\n\\begin{equation*}\n\\left( {\\rm curl}\\, \\widetilde{\\vec{V}}_{\\varepsilon ,M}\\right)^{h}\\,=\\,\\partial_3 \\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\qquad \\text{ and }\\qquad\n\\left( {\\rm curl}\\, \\widetilde{\\vec{V}}_{\\varepsilon ,M}\\right)^{3}\\,=\\,\\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\,.\n\\end{equation*}\nIn addition, from the momentum equation in \\eqref{eq:approx wave}, where we take the mean-free part and then the ${\\rm curl}\\,$, we deduce the equations\n\\begin{equation} \\label{eq:eq momentum term2}\n\\begin{cases}\n\\varepsilon^{m}\\partial_t\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}-\\varepsilon^{m-1}\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}=\\varepsilon^m\\left(\\partial_{3}^{-1}{\\rm curl}\\,\\widetilde{\\vec f}_{\\varepsilon ,M} \\right)^{h}+\\varepsilon^{2(m-n)}\\left(\\partial_{3}^{-1}{\\rm curl}\\,\\widetilde{\\vec g}_{\\varepsilon ,M} \\right)^{h}\\\\[1ex]\n\\varepsilon^{m}\\partial_t\\widetilde{\\omega}_{\\varepsilon ,M}^{3}+\\varepsilon^{m-1}{\\rm div}_h\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}=\\varepsilon^m\\,{\\rm curl}_h\\widetilde{\\vec f}_{\\varepsilon ,M}^{h}\\, .\n\\end{cases}\n\\end{equation}\nMaking use of the relations above, recalling the definitions in \\eqref{def_f-g}, and thanks to Propositions \\ref{p:prop approx} and \\ref{p:prop dec}, we can write\n\\begin{equation}\\label{rel_oscillations}\n\\begin{split}\n{\\rm curl}_h\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\;\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\right)^{\\perp}&=\\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\,\\\\\n&=\\varepsilon \\partial_t\\!\\left( \\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\widetilde{\\omega}_{\\varepsilon ,M}^{3}-\n\\varepsilon\\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\left(\\left(\\partial_{3}^{-1}{\\rm curl}\\,\\widetilde{\\vec f}_{\\varepsilon ,M}\\right)^{h}\\right)^\\perp\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n-\\varepsilon^{m+1-2n}\\, \\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\left(\\left(\\partial_{3}^{-1}{\\rm curl}\\,\\widetilde{\\vec g}_{\\varepsilon ,M}\\right)^{h}\\right)^\\perp \\\\\n&=-\\varepsilon \\left( \\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\partial_t\\widetilde{\\omega}_{\\varepsilon ,M}^{3}+\\mathcal{R}_{\\varepsilon ,M}=\n\\left( \\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\,{\\rm div}_h\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\mathcal{R}_{\\varepsilon ,M}\\,,\n\\end{split}\n\\end{equation}\nwhere, again, the equalities hold in the sense of distributions.\nWe point out that, thanks to the scaling $m+1>2n$, we could include in the remainder also the last term appearing in the second equality, which was of order $O(\\varepsilon^{m+1-2n})$.\n\nHence, putting the gradient term into $\\mathcal R_{\\varepsilon,M}$, from \\eqref{eq:T2} we arrive at \n\\begin{equation}\\label{002_T}\n\\begin{split}\n\\mathcal{T}_{\\varepsilon ,M}^{2}\\,&=\\,\\langle {\\rm div}_h\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\,\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right) \\rangle+\\mathcal{R}_{\\varepsilon ,M} \\\\\n&=\\,\\langle {\\rm div}\\, \\widetilde{\\vec{V}}_{\\varepsilon ,M}\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right) \\rangle -\n\\langle \\partial_3 \\widetilde{\\vec{V}}_{\\varepsilon ,M}^{3}\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right) \\rangle+\\mathcal{R}_{\\varepsilon ,M}\\, .\n\\end{split}\n\\end{equation}\n\nAt this point, the computations mainly follow the same lines of \\cite{F-G-GV-N} (see also \\cite{F_2019}, \\cite{DS-F-S-WK}).\nFirst of all, we notice that (in the last line) the second term on the right-hand side is another remainder. Indeed, using the definition of the function $\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}$ and the fact\nthat the test function $\\vec\\psi$ does not depend on $x^3$, one has\n\\begin{equation*}\n\\begin{split}\n\\partial_3 \\widetilde{\\vec{V}}_{\\varepsilon ,M}^{3}\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right)&=\\partial_3 \\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{3}\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right)\\right) - \\widetilde{\\vec{V}}_{\\varepsilon ,M}^{3}\\, \\partial_3\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right)\\\\\n&=\\mathcal{R}_{\\varepsilon ,M}-\\frac{1}{2}\\nabla_{h}\\left|\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{3}\\right|^{2}=\\mathcal{R}_{\\varepsilon ,M}\\, .\n\\end{split}\n\\end{equation*}\nNext, in order to deal with the first term on the right-hand side of \\eqref{002_T}, we use the first equation in \\eqref{eq:approx wave} to obtain\n(in the sense of distributions)\n\\begin{equation*}\n\\begin{split}\n{\\rm div}\\, \\widetilde{\\vec{V}}_{\\varepsilon ,M}\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right)&=-\\varepsilon^{m} \\partial_t \\widetilde{\\varrho}^{(1)}_{\\varepsilon ,M}\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right)+\\mathcal{R}_{\\varepsilon ,M}\\\\\n&=\\varepsilon^{m} \\widetilde{\\varrho}_{\\varepsilon,M}^{(1)}\\, \\partial_t\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right)+\\mathcal{R}_{\\varepsilon ,M}\\,.\n\\end{split}\n\\end{equation*}\nNow, equations \\eqref{eq:approx wave} and \\eqref{eq:eq momentum term2} immediately yield that\n\\begin{equation*}\n\\varepsilon^{m}\\widetilde{\\varrho}^{(1)}_{\\varepsilon ,M}\\, \\partial_t\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}+\\left(\\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\right)=\n\\mathcal{R}_{\\varepsilon ,M}-\\widetilde{\\varrho}_{\\varepsilon ,M}^{(1)}\\, \\nabla_{h}\\widetilde{\\varrho}^{(1)}_{\\varepsilon ,M}=\n\\mathcal{R}_{\\varepsilon ,M}-\\frac{1}{2}\\nabla_{h}\\left|\\widetilde{\\varrho}_{\\varepsilon ,M}^{(1)}\\right|^{2}=\\mathcal{R}_{\\varepsilon ,M}\\,.\n\\end{equation*}\n\nThis relation finally implies that $\\mathcal{T}_{\\varepsilon ,M}^{2}\\,=\\,\\mathcal R_{\\varepsilon,M}$ is a remainder, in the sense of relation \\eqref{eq:reminder}:\nfor any $T>0$ and any test-function $\\vec \\psi$ as in \\eqref{eq:test-f},\none has the convergence\n(at any $M\\in\\mathbb{N}$ fixed, when $\\varepsilon\\ra0^+$)\n\\begin{equation} \\label{eq:limit_T2}\n\\int_{0}^{T}\\int_{\\mathbb{R}^2}\\mathcal{T}_{\\varepsilon ,M}^{2}\\cdot\\vec{\\psi}^h\\,dx^h\\,dt\\,\\longrightarrow\\,0\\,.\n\\end{equation}\n\n\\subsubsection*{The case $m+1=2n$}\\label{sss:term2_osc_bis}\nIn the case $m+1=2n$, most of the previous computations may be reproduced exactly in the same way.\nThe only (fundamental) change concerns relation \\eqref{rel_oscillations}: since now $m+1-2n=0$, that equation now reads\n\\begin{equation}\\label{rel_oscillations_bis}\n\\begin{split}\n{\\rm curl}_h\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\;\\left(\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\,=\\,\\left( \\widetilde{\\Phi}_{\\varepsilon ,M}^{h}\\right)^{\\perp}\\,{\\rm div}_h\\widetilde{\\vec{V}}_{\\varepsilon ,M}^{h}\\,- \\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\left(\\left(\\partial_{3}^{-1}{\\rm curl}\\,\\widetilde{\\vec g}_{\\varepsilon ,M}\\right)^{h}\\right)^\\perp+\\mathcal{R}_{\\varepsilon ,M}\\,,\n\\end{split}\n\\end{equation}\nand, repeating the same computations performed for $\\mathcal T^2_{\\varepsilon, M}$ in the previous paragraph, we have\n\\begin{equation*}\\label{T^2-bis}\n\\mathcal T^2_{\\varepsilon, M}= \\mathcal R_{\\varepsilon, M}-\\langle\\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\left(\\left(\\partial_{3}^{-1}{\\rm curl}\\,\\widetilde{\\vec g}_{\\varepsilon ,M}\\right)^{h}\\right)^\\perp \\rangle\\, .\n\\end{equation*}\nHence, the main difference with respect to the previous case is that, now, we have to take care of the term $\\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\left(\\left(\\partial_{3}^{-1}{\\rm curl}\\,\\widetilde{\\vec g}_{\\varepsilon ,M}\\right)^{h}\\right)^\\perp $, which is non-linear and of order $O(1)$, so it may potentially\ngive rise to oscillations which persist in the limit.\n\nIn order to show that this does not happen, we make use of definition \\eqref{def_f-g} of $\\vec g_{\\varepsilon}$ to compute\n\\begin{align*}\n\\left({\\rm curl}\\,\\widetilde{\\vec g}_{\\varepsilon ,M}\\right)^{h,\\perp}\\,&=\\,\\left({\\rm curl}\\, \\left(\\widetilde{\\varrho}_{\\varepsilon, M}^{(1)}\\nabla_x G-\\nabla_x \\widetilde{\\Pi}_{\\varepsilon,M}\\right)\\right)^{h,\\perp} \\\\\n&=\\,\n\\begin{pmatrix}\n-\\partial_2 \\widetilde{\\varrho}^{(1)}_{\\varepsilon, M} \\\\ \n\\partial_1 \\widetilde{\\varrho}^{(1)}_{\\varepsilon ,M} \\\\ \n0\n\\end{pmatrix}^{h,\\perp}\\,=\\,-\\,\\nabla_h \\widetilde{\\varrho}_{\\varepsilon,M}^{(1)}\\, .\n\\end{align*}\n\n\nFrom this relation, in turn we get\n\\begin{equation}\\label{T^2}\n\\mathcal T^2_{\\varepsilon, M}\\,=\\,\\mathcal R_{\\varepsilon, M}\\,+\\,\\langle \\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\, \\partial_3^{-1}\\nabla_h \\widetilde{\\varrho}_{\\varepsilon,M}^{(1)} \\rangle \\, .\n\\end{equation} \nNow, we have to employ the potential part of the momentum equation in \\eqref{eq:approx wave}, which has not been used so far. Taking the oscillating\ncomponent of the solutions, we obtain \n\\begin{equation*}\n\\nabla_h \\widetilde{\\varrho}_{\\varepsilon,M}^{(1)}\\,=-\\, \\varepsilon^m\\,\\partial_t\\widetilde{\\vec{V}}^h_{\\varepsilon ,M}\\,-\\varepsilon^{m-1} (\\widetilde{\\vec{V}}^h_{\\varepsilon ,M})^\\perp+\\varepsilon^m\\,\\widetilde{\\vec f}^h_{\\varepsilon ,M}\\,+\\varepsilon^{2(m-n)} \\widetilde{\\vec g}^h_{\\varepsilon,M}= -\\, \\varepsilon^m\\,\\partial_t\\widetilde{\\vec{V}}^h_{\\varepsilon ,M}\\,+ \\mathcal R_{\\varepsilon,M}\\,.\n\\end{equation*}\nInserting this relation into \\eqref{T^2} and using \\eqref{eq:eq momentum term2}, we finally gather\n\\begin{equation*}\n\\mathcal T^2_{\\varepsilon, M}=-\\varepsilon^m \\langle \\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\, \\partial_t\\partial_3^{-1}\\widetilde{\\vec{V}}^h_{\\varepsilon ,M} \\rangle +\\mathcal R_{\\varepsilon,M}=\n\\varepsilon^m \\langle \\partial_t \\widetilde{\\omega}_{\\varepsilon ,M}^{3}\\, \\partial_3^{-1}\\widetilde{\\vec{V}}^h_{\\varepsilon ,M} \\rangle +\\mathcal R_{\\varepsilon,M}=\\mathcal R_{\\varepsilon,M}\\, ,\n\\end{equation*}\nbecause we have taken $m>1$.\n\nThis relation finally implies that, also in the case when $m+1=2n$, $\\mathcal{T}_{\\varepsilon ,M}^{2}$ is a remainder: for any $T>0$ and any test-function $\\vec \\psi$ as in \\eqref{eq:test-f}, one has the convergence \\eqref{eq:limit_T2}.\n\n\n\n\n\\subsection{The limit system} \\label{ss:limit} \nThanks to the computations of the previous subsection, we can now pass to the limit in equation \\eqref{weak-mom}. Recall that $m>1$ and $m+1\\geq 2n >m$ here.\n\nTo begin with, we take a divergence-free test function $\\vec\\psi$ as in \\eqref{eq:test-f}, specifically\n\\begin{equation} \\label{eq:test-2}\n\\vec{\\psi}=\\big(\\nabla_{h}^{\\perp}\\phi,0\\big)\\,,\\qquad\\qquad\\mbox{ with }\\qquad \\phi\\in C_c^\\infty\\big([0,T[\\,\\times\\mathbb{R}^2\\big)\\,,\\quad \\phi=\\phi(t,x^h)\\,.\n\\end{equation}\nWe point out that, since all the integrals will be made on $\\mathbb{R}^2$ (in view of the choice of the test functions in \\eqref{eq:test-2} above), we can safely work on the domain $\\Omega=\\mathbb{R}^2\\times \\, ]0,1[\\,$.\nIn addition, for $\\vec\\psi$ as in \\eqref{eq:test-2}, all the gradient terms vanish identically, as well as all the contributions\ndue to the vertical component of the equation. In particular, we do not see any contribution of the pressure and gravity terms: equation \\eqref{weak-mom} becomes\n\\begin{align}\n\\int_0^T\\!\\!\\!\\int_{\\Omega} \n& \\left( -\\vre \\ue^h \\cdot \\partial_t \\vec\\psi^h -\\vre \\ue^h\\otimes\\ue^h : \\nabla_h \\vec\\psi^h\n+ \\frac{1}{\\ep}\\vre\\big(\\ue^{h}\\big)^\\perp\\cdot\\vec\\psi^h\\right)\\, dx \\, dt \\label{eq:weak_to_conv}\\\\\n&\\qquad\\qquad\\qquad\\qquad =-\\int_0^T\\!\\!\\!\\int_{\\Omega} \n\\mathbb{S}(\\nabla_x\\vec\\ue): \\nabla_x \\vec\\psi\\,dx\\,dt+\n\\int_{\\Omega}\\vrez \\uez \\cdot \\vec\\psi(0,\\cdot)\\,dx\\,. \\nonumber\n\\end{align}\n\nMaking use of the uniform bounds of Section \\ref{ss:unif-est}, we can pass to the limit in the $\\partial_t$ term and in the viscosity term.\nMoreover, our assumptions imply that $\\varrho_{0,\\varepsilon}\\vec{u}_{0,\\varepsilon}\\rightharpoonup \\vec{u}_0$ in e.g. $L_{\\rm loc}^2$. \nNext, the Coriolis term can be dealt with in a standard way: using the structure of $\\vec\\psi$ and the mass equation \\eqref{weak-con}, we can write\n\\begin{align*}\n\\int_0^T\\!\\!\\!\\int_{\\Omega}\\frac{1}{\\ep}\\vre\\big(\\ue^{h}\\big)^\\perp\\cdot\\vec\\psi^h\\,&=\\,\\int_0^T\\!\\!\\!\\int_{\\mathbb{R}^2}\\frac{1}{\\ep}\\langle\\vre \\ue^{h}\\rangle \\cdot \\nabla_{h}\\phi\\,=\\,\n-\\varepsilon^{m-1}\\int_0^T\\!\\!\\!\\int_{\\mathbb{R}^2}\\langle \\varrho^{(1)}_\\varepsilon\\rangle\\, \\partial_t\\phi\\,-\\,\\varepsilon^{m-1}\\int_{\\mathbb{R}^2}\\langle \\varrho^{(1)}_{0,\\varepsilon}\\rangle\\, \\phi(0,\\cdot )\\,,\n\\end{align*}\nwhich of course converges to $0$ when $\\varepsilon\\ra0^+$.\n\nIt remains us to tackle the convective term $\\varrho_\\varepsilon \\ue^h \\otimes \\ue^h$.\nFor it, we take advantage of Lemma \\ref{lem:convterm} and relations \\eqref{eq:limit_T1} and \\eqref{eq:limit_T2}, but\nwe still have to take care of the convergence for $M\\rightarrow+\\infty$ in \\eqref{eq:limit_T1}.\nWe start by performing equalities \\eqref{eq:T1} backwards in the term on the right-hand side of \\eqref{eq:limit_T1}: thus, we have to pass to the limit for $M\\rightarrow+\\infty$\nin\n\\[\n\\int^T_0\\int_{\\mathbb{R}^2}\\vec U_M^h\\otimes\\vec U_M^h : \\nabla_h \\vec\\psi^h\\,\\dx^h\\,\\dt\\,.\n\\]\nNow, we remark that, since $\\vec U^h\\in L^2_T(H^1)$ by \\eqref{conv:u}, from \\eqref{eq:LP-Sob} we gather the strong convergence\n$S_M \\vec U^h\\longrightarrow \\vec{U}^{h}$ in $L_{T}^{2}(H^{s})$ for any $s<1$, in the limit for $M\\rightarrow +\\infty$.\nThen, passing to the limit for $M\\rightarrow+\\infty$ in the previous relation is an easy task: we finally get that, for $\\varepsilon\\ra0^+$, one has\n\\begin{equation*}\n\\int_0^T\\int_{\\Omega} \\vre \\ue^h\\otimes\\ue^h : \\nabla_h \\vec\\psi^h\\, \\longrightarrow\\, \\int_0^T\\int_{\\mathbb{R}^2}\\vec{U}^h\\otimes\\vec{U}^h : \\nabla_h \\vec\\psi^h\\,.\n\\end{equation*}\n\nIn the end, we have shown that, letting $\\varepsilon \\rightarrow 0^+$ in \\eqref{eq:weak_to_conv}, one obtains\n\\begin{align*}\n&\\int_0^T\\!\\!\\!\\int_{\\mathbb{R}^2} \\left(\\vec{U}^{h}\\cdot \\partial_{t}\\vec\\psi^h+\\vec{U}^{h}\\otimes \\vec{U}^{h}:\\nabla_{h}\\vec\\psi^h\\right)\\, dx^h\\, dt=\n\\int_0^T\\!\\!\\!\\int_{\\mathbb{R}^2} \\mu \\nabla_{h}\\vec{U}^{h}:\\nabla_{h}\\vec\\psi^h \\, dx^h\\, dt-\n\\int_{\\mathbb{R}^2}\\langle\\vec{u}_{0}^{h}\\rangle\\cdot \\vec\\psi^h(0,\\cdot)\\,dx^h\\, ,\n\\end{align*}\nfor any test function $\\vec\\psi$ as in \\eqref{eq:test-f}.\nThis implies \\eqref{eq_lim_m:momentum}, concluding the proof of Theorem \\ref{th:m>1}.\n\n\n\\section{Proof of the convergence for $m=1$} \\label{s:proof-1}\n\nIn the present section, we complete the proof of the convergence in the case $m=1$ and $1\/21$, and it comes into play in the compensated compactness argument. Therefore, despite most of the computations may be repeated identical as in the previous section, let us present\nthe main points of the argument.\n\n\\subsection{Handling the convective term when $m=1$} \\label{ss:convergence_1}\n\nLet us take care of the convergence of the convective term in the case when $m=1$. \n\nFirst of all, it is easy to see that Lemma \\ref{lem:convterm} still holds true. Therefore, \ngiven a test function $\\vec\\psi\\in C_c^\\infty\\big([0,T[\\,\\times\\Omega;\\mathbb{R}^3\\big)$ such that ${\\rm div}\\,\\vec\\psi=0$ and $\\partial_3\\vec\\psi=0$,\nwe have to pass to the limit in the term\n\\begin{align*}\n-\\int_{0}^{T}\\int_{\\Omega} \\vec{V}_{\\varepsilon ,M}\\otimes \\vec{V}_{\\varepsilon ,M}: \\nabla_{x}\\vec{\\psi}\\,&=\\,\n\\int_{0}^{T}\\int_{\\Omega}{\\rm div}\\,\\left(\\vec{V}_{\\varepsilon ,M}\\otimes \\vec{V}_{\\varepsilon ,M}\\right) \\cdot \\vec{\\psi}\\,=\\,\n\\int_{0}^{T}\\int_{\\mathbb{R}^2} \\left(\\mathcal{T}_{\\varepsilon ,M}^{1}+\\mathcal{T}_{\\varepsilon, M}^{2}\\right)\\cdot\\vec{\\psi}^h\\,,\n\\end{align*}\nwhere we agree again that the torus $\\mathbb{T}^1$ has been normalised so that its Lebesgue measure is equal to $1$ and we have adopted the same notation as in \\eqref{def:T1-2}.\n\nAt this point, we notice that the analysis of $\\mathcal{T}_{\\varepsilon ,M}^{2}$ can be performed as in Section \\ref{sss:term2}, because we have\n$m+1>2n$, \\textsl{i.e.} $n<1$. \\textsl{Mutatis mutandis}, we find relation \\eqref{eq:limit_T2} also in the case $m=1$.\n\nLet us now deal with the term $\\mathcal{T}_{\\varepsilon ,M}^{1}$. Arguing as in Section \\ref{sss:term1}, we may write it as\n\\begin{equation*}\n\\mathcal{T}_{\\varepsilon ,M}^{1}\\,=\\,\\left({\\rm curl}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle-\\langle \\varrho^{(1)}_{\\varepsilon ,M}\\rangle \\right)\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle^{\\perp}+\\mathcal{R}_{\\varepsilon ,M} .\n\\end{equation*}\nNow we use the horizontal part of \\eqref{eq:reg-wave}: \naveraging it with respect to the vertical variable and applying the operator ${\\rm curl}_h$, we find\n\\begin{equation*}\n\\varepsilon\\,\\partial_t{\\rm curl}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,+\\,{\\rm div}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle \\,=\\,\n\\varepsilon\\, {\\rm curl}_h\\langle \\vec f_{\\varepsilon ,M}^{h}\\rangle\\, .\n\\end{equation*}\nTaking the difference of this equation with the first one in \\eqref{eq:reg-wave}, we discover that\n\\begin{equation*}\n\\partial_t\\widetilde\\gamma_{\\varepsilon,M}\n\\,=\\,{\\rm curl}_h\\langle \\vec f_{\\varepsilon ,M}^{h}\\rangle\\,,\\qquad\\qquad \\mbox{ where }\\qquad\n\\widetilde\\gamma_{\\varepsilon, M}:={\\rm curl}_h\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle\\,-\\,\\langle \\varrho^{(1)}_{\\varepsilon ,M}\\rangle\\,.\n\\end{equation*}\nAn argument analogous to the one used after \\eqref{eq:gamma} above, based on Aubin-Lions Lemma, shows that\n$\\big(\\widetilde\\gamma_{\\varepsilon,M}\\big)_{\\varepsilon}$ is compact in \\textsl{e.g.} $L_{T}^{2}(L_{\\rm loc}^{2})$. Then, this sequence converges strongly (up to extraction of a suitable subsequence, not relabelled here) to a tempered distribution $\\widetilde\\gamma_M$ in the same space. \n\nUsing the previous property, we may deduce that\n\\begin{equation*}\n\\widetilde\\gamma_{\\varepsilon,M}\\,\\langle \\vec{V}_{\\varepsilon ,M}^{h}\\rangle^{\\perp}\\,\\longrightarrow\\, \\widetilde\\gamma_M\\, \\langle \\vec{V}_{M}^{h}\\rangle^{\\perp}\\qquad \\text{ in }\\qquad \\mathcal{D}^{\\prime}\\big(\\mathbb{R}_+\\times\\mathbb{R}^2\\big),\n\\end{equation*}\nwhere we have $\\langle \\vec{V}_{M}^{h}\\rangle=\\langle S_M\\vec{U}^{h}\\rangle$ and $\\widetilde\\gamma_M={\\rm curl}_h\\langle S_M \\vec{U}^{h}\\rangle-\\langle \\varrho^{(1)}_{M}\\rangle$.\n\nOwing to the regularity of the target velocity $\\vec U^h$, we can pass to the limit also for $M\\rightarrow+\\infty$, as detailed in Section \\ref{ss:limit} above. Thus, we find\n\\begin{equation} \\label{eq:limit_T1-1}\n\\int^T_0\\!\\!\\!\\int_{\\Omega}\\varrho_\\varepsilon\\,\\vec{u}_\\varepsilon\\otimes \\vec{u}_\\varepsilon: \\nabla_{x}\\vec{\\psi}\\, dx \\, dt\\,\\longrightarrow\\,\n\\int^T_0\\!\\!\\!\\int_{\\mathbb{R}^2}\\big(\\vec U^h\\otimes\\vec U^h:\\nabla_h\\vec\\psi^h\\,-\\, \\varrho^{(1)}\\,(\\vec U^h)^\\perp\\cdot\\vec\\psi^h\\big)\\,dx^h\\,dt,\n\\end{equation}\nfor all test functions $\\vec\\psi$ such that ${\\rm div}\\,\\vec\\psi=0$ and $\\partial_3\\vec\\psi=0$. Recall the convention $|\\mathbb{T}^1|=1$.\nNotice that, since $\\vec U^h=\\nabla_h^\\perp \\varrho^{(1)}$ when $m=1$ (keep in mind Proposition \\ref{p:limit_iso}), the last term in the integral on the right-hand side is actually zero.\n\n\n\\subsection{End of the proof} \\label{ss:limit_1}\nThanks to the previous analysis, we are now ready to pass to the limit in equation \\eqref{weak-mom}.\nFor this, we take a test-function $\\vec\\psi$ as in \\eqref{eq:test-2};\nnotice in particular that ${\\rm div}\\,\\vec\\psi=0$ and $\\partial_3\\vec\\psi=0$. Then, once again all the gradient terms and all the contributions coming from the vertical\ncomponent of the momentum equation vanish identically, when tested against such a $\\vec\\psi$. Recall that all the integrals will be performed in $\\mathbb{R}^2$. So, equation \\eqref{weak-mom} reduces\nto\n\\begin{align*\n\\int_0^T\\!\\!\\!\\int_{\\Omega} \\left( -\\vre \\ue \\cdot \\partial_t \\vec\\psi -\\vre \\ue\\otimes\\ue : \\nabla \\vec\\psi\n+ \\frac{1}{\\ep}\\vre\\big(\\ue^{h}\\big)^\\perp\\cdot\\vec\\psi^h+\\mathbb{S}(\\nabla_x\\vec\\ue): \\nabla_x \\vec\\psi\\right)\n =\\int_{\\Omega}\\vrez \\uez \\cdot \\vec\\psi(0,\\cdot)\\,.\n\\end{align*}\n\nFor the rotation term, we can test the first equation in \\eqref{eq:wave_m=1} against $\\phi$ to get\n\\begin{equation*}\n\\begin{split}\n-\\int_0^T\\!\\!\\!\\int_{\\mathbb{R}^2} \\left( \\langle \\varrho^{(1)}_{\\varepsilon}\\rangle\\, \\partial_{t}\\phi +\\frac{1}{\\varepsilon}\\, \\langle\\varrho_{\\varepsilon}\\ue^{h}\\rangle\\cdot \\nabla_{h}\\phi\\right)=\n\\int_{\\mathbb{R}^2}\\langle \\varrho^{(1)}_{0,\\varepsilon }\\rangle\\, \\phi (0,\\cdot ) \\, ,\n\\end{split}\n\\end{equation*}\nwhence we deduce that\n\\begin{align*}\n\\int_0^T\\!\\!\\!\\int_{\\Omega}\\frac{1}{\\ep}\\vre\\big(\\ue^{h}\\big)^\\perp\\cdot\\vec\\psi^h\\,&=\\,\\int_0^T\\!\\!\\!\\int_{\\mathbb{R}^2}\\frac{1}{\\ep}\\langle\\vre \\ue^{h}\\rangle \\cdot \\nabla_{h}\\phi\\,=\\,-\\,\\int_0^T\\!\\!\\!\\int_{\\mathbb{R}^2}\\langle \\varrho^{(1)}_\\varepsilon\\rangle\\, \\partial_t\\phi\\,-\\,\\int_{\\mathbb{R}^2}\\langle \\varrho^{(1)}_{0,\\varepsilon}\\rangle\\, \\phi(0,\\cdot )\\,. \n\\end{align*}\n\nIn addition, the convergence of the convective term has been performed in \\eqref{eq:limit_T1-1}. As for other terms, we can argue as in Section \\ref{ss:limit}.\nHence, letting $\\varepsilon \\rightarrow 0^+$ in the equation above, we get\n\\begin{align*}\n&-\\int_0^T\\!\\!\\!\\int_{\\mathbb{R}^2} \\left(\\vec{U}^{h}\\cdot \\partial_{t}\\nabla_{h}^{\\perp} \\phi+ \\vec{U}^{h}\\otimes \\vec{U}^{h}:\\nabla_{h}(\\nabla_{h}^{\\perp}\\phi )+\\varrho^{(1)}\\, \\partial_t \\phi \\right)\\, dx^h\\, dt\\\\\n&\\qquad\\qquad=-\\int_0^T\\!\\!\\!\\int_{\\mathbb{R}^2} \\mu \\nabla_{h}\\vec{U}^{h}:\\nabla_{h}(\\nabla_{h}^{\\perp}\\phi ) \\, dx^h\\, dt+\\int_{\\mathbb{R}^2}\\left(\\langle\\vec{u}_{0}^{h}\\rangle\\cdot \\nabla _{h}^{\\perp}\\phi (0,\\cdot )+\n\\langle \\varrho^{(1)}_{0}\\rangle\\phi (0,\\cdot )\\right) \\, dx^h\\, ,\n\\end{align*}\nwhich is the weak formulation of equation \\eqref{eq_lim:QG}. In the end, also Theorem \\ref{th:m=1} is proved.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}