For $l$-homogeneous linear differential operators $\mathcal{A}$ of constant rank, we study the implication $v_j\rightharpoonup v$ in $X$ and $\mathcal{A} v_j\rightarrow \mathcal{A} v$ in $W^{-l}Y$ implies $F(v_j)\rightsquigarrow F(v)$ in $Z$, where $F$ is an $\mathcal{A}$-quasiaffine function and $\rightsquigarrow$ denotes an appropriate type of weak convergence. Here $Z$ is a local $L^1$-type space, either the space $\mathscr{M}$ of measures, or $L^1$, or the Hardy space $\mathscr{H}^1$; $X,\, Y$ are $L^p$-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of $X,\,Y,\,Z$ are sharp. Analogous statements are also given in the case when $F(v)$ is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove $\mathscr{H}^p$-bounds for the sequence $(F(v_j))_j$, for appropriate $p<1$, and new convergence results in the dual of Hölder spaces when $(v_j)$ is $\mathcal{A}$-free and lies in a suitable negative order Sobolev space $W^{-β,s}$. The choice of these Hölder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.
We investigate conditions under which, for two sequences $(u_r)$ and $(v_r)$ weakly converging to $u$ and $v$ in $L^p(R^d;R^N)$ and $L^{q}(R^d;R^N)$, respectively, $1/p+1/q \leq 1$, a quadratic form $q(x;u_r,v_r)=\sum\limits_{j,m=1}^N q_{j m}(x)u_{j r} v_{m r}$ converges toward $q(x;u,v)$ in the sense of distributions. The conditions involve fractional derivatives and variable coefficients, and they represent a generalization of the known compensated compactness theory. The proofs are accomplished using a recently introduced $H$-distribution concept. We apply the developed techniques to a nonl…
This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in $\mathrm{L}^p$-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending at the same time the theories of (generalized) Young measures and H-measures. In $\mathrm{L}^p$-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, and thus MCFs allow to quantify the difference in compactness. The definition of MCFs involves a Fouri…
We introduce new variant of $H$-measures defined on spectra of general algebra of test symbols and derive the localization properties of such $H$-measures. Applications for the compensated compactness theory are given. In particular, we present new compensated compactness results for quadratic functionals in the case of general pseudo-differential constraints. The case of inhomogeneous second order differential constraints is also studied.
Method of compensated compactness is used to show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it Bardos et.al}\cite{MR542510}) of the corresponding scalar conservation laws in a bounded domain in $\mathbb{R}^{d}$, where the viscous term is of the form $\varepsilon div\left(B(u^{\varepsilon})\nabla u^{\varepsilon}\right)$.
We use the method of Compensated Compactness and Kinteic Formulation to show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it F. Otto}) of the corresponding scalar conservation laws on a bounded domain in $\mathbb{R}^{d}$, where the viscous term is of the form $\varepsilon\,div\left(B(u^{\varepsilon})\nabla u^{\varepsilon}\right)$.
We give a short proof of the fact that each homogeneous linear differential operator $A$ of constant rank admits a homogeneous potential operator $B$, meaning that $\ker A(x)=\mathrm{im\,}B(x)$ for $x\neq 0$. We make some refinements of the original result and some related remarks.
We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div-curl lemma can be employed to establish the global weak rigidity of the Gauss-Codazzi-Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.
In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order $t^{-4}$ in the Gauss curvature. Here we show that the decay of Hong $t^{-2-δ/2}$ where $δ\in(0,4)$ suffices.
In this paper, we are concerned with a model of polytropic gas flow, which consists the mass equation, the momentum equation and a varying entropy equation. First, a new technique, to set up a relation between the Riemann invariants of the isentropic system and the entropy variable $s$, coupled with the maximum principle, is introduced to obtain the a-priori $L^{\infty}$ estimates for the viscosity-flux approximation solutions. Second, the convergence framework from the compensated compactness theory on the system of isentropic gas dynamics is applied to prove the pointwise convergence of the …
A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal wi…
We prove the existence of a weak solution to a two-dimensional resonant 3x3 system of conservation laws with BV initial data. Due to possible resonance (coinciding eigenvalues), spatial BV estimates are in general not available. Instead, we use an entropy dissipation bound combined with the time translation invariance property of the system to prove existence based on a two-dimensional compensated compactness argument adapted from the paper of Tadmor, Rascle and Bagnerini, "Compensated compactness for 2D conservation laws", [JHDEs 2(3):697--712, 2005]. Existence is proved under the assumption …
Compensated compactness is an important method used to solve nonlinear PDEs. A simple formulation of a compensated compactness problem is to ask for conditions on a set $\mathcal{K}\subset M^{m\times n}$ such that $$ \lim_{n\rightarrow \infty} \mathrm{dist}(Du_n,\mathcal{K})\overset{L^p}{\rightarrow} 0\; \Rightarrow \{Du_{n}\}_{n}\text{ is precompact.} $$ Let $M_1,M_2,\dots, M_q$ denote the set of minors of $M^{m\times n}$. A sufficient condition for this is that any measure $μ$ supported on $\mathcal{K}$ satisfying $$ \int M_k(X) dμ(X)=M_k\left(\int X dμ(X)\right)\text{ for }k=1,2,\dots, q $$…
We present a systematic treatment of the theory of Compensated Compactness under Murat's constant rank assumption. We give a short proof of a sharp weak lower semicontinuity result for signed integrands, extending the results of Fonseca--Müller. The null Lagrangians are an important class of signed integrands, since they are the weakly continuous functions. We show that they are precisely the compensated compactness quantities with Hardy space integrability, thus proposing an answer to a question raised by Coifman-Lions-Meyer-Semmes. Finally we provide an effective way of computing the null La…
Consider a sequence of $C^4$ Lorentzian metrics $\{h_n\}_{n=1}^{+\infty}$ on a manifold $\mathcal M$ satisfying the Einstein vacuum equation $\mathrm{Ric}(h_n)=0$. Suppose there exists a smooth Lorentzian metric $h_0$ on $\mathcal M$ such that $h_n\to h_0$ uniformly on compact sets. Assume also that on any compact set $K\subset \mathcal M$, there is a decreasing sequence of positive numbers $λ_n \to 0$ such that $$\|\partial^α (h_n - h_0)\|_{L^{\infty}(K)} \lesssim λ_n^{1-|α|},\quad |α|\geq 4.$$ It is well-known that $h_0$, which represents a "high-frequency limit", is not necessarily a soluti…
The Div-Curl Lemma, which is the basic result of the compensated compactness theory in Sobolev spaces, was introduced by F. Murat (1978) with distinct proofs for the $L^2(Ω)$ and $L^p(Ω)$, $p \neq 2$, cases. In this note we present a slightly different proof, relying only on a Green-Gauss integral formula and on the usual Rellich-Kondrachov compactness properties.
We present a quantitative compensated compactness estimate for stochastic conservation laws, which generalises a previous result of Golse & Perthame (2013) for deterministic equations. With a stochastic modification of Kruzkov's interpolation lemma, this estimate provides bounds on the rate at which a sequence of vanishing viscosity solutions becomes compact. This contribution is for the Proceedings of HYP2022.
For any bounded, smooth domain $Ω\subset \R^2$, %(or $Ω=\R^2$), we will establish the weak compactness property of solutions to the simplified Ericksen-Leslie system for both uniaxial and biaxial nematics, and the convergence of weak solutions of the Ginzburg-Landau type nematic liquid crystal flow to a weak solution of the simplified Ericksen-Leslie system as the parameter tends to zero. This is based on the compensated compactness property of the Ericksen stress tensors, which is obtained by the $L^p$-estimate of the Hopf differential for the Ericksen-Leslie system and the Pohozaev type argu…
In this paper we consider the existence of global weak entropy solutions for a particualr nonsymmetric Keyfitz-Kranzer type system, by using the compensated compactness method we get bounde entropy weak solutions
We present several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. We first focus on the compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropy flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and…
In this paper we apply the Invariant Regions Theory and the Compensated Compactness Method, to study the singular limit of stiff relaxation and domain diffusion for the Cauchy problem of Aw-Rascle system.
This paper relates the lower semi-continuity of an integral functional in the compensated compactness setting of vector fields satisfying a constant-rank first-order differential constraint, to closed $\mathcal{A}$-$p$ quasiconvexity of the integrand. The lower semi-continuous envelope of relaxation is identified for continuous, but potentially extended real-valued integrands. We discuss the continuity assumption and show that when it is dropped our notion of quasiconvexity is still equivalent to lower semi-continuity of the integrand under an additional assumption on the characteristic cone o…
In this paper we deal with diffusive relaxation limits of nonlinear systems of Euler type modeling chemotactic movement of cells toward Keller--Segel type systems. The approximating systems are either hyperbolic--parabolic or hyperbolic--elliptic. They all feature a nonlinear pressure term arising from a \emph{volume filling effect} which takes into account the fact that cells do not interpenetrate. The main convergence result relies on compensated compactness tools and is obtained for large initial data under suitable assumptions on the approximating solutions. In order to justify such assump…
In the present paper we study the fast rotation limit for viscous incompressible fluids with variable density, whose motion is influenced by the Coriolis force. We restrict our analysis to two dimensional flows. In the case when the initial density is a small perturbation of a constant state, we recover in the limit the convergence to the homogeneous incompressible Navier-Stokes equations (up to an additional term, due to density fluctuations). For general non-homogeneous fluids, the limit equations are instead linear, and the limit dynamics is described in terms of the vorticity and the densi…
In this work, we study a new notion involving convergence of microstructures represented by matrices $B^ε$ related to the classical $H$-convergence of $A^ε$. It incorporates the interaction between the two microstructures. This work is about its effects on various aspects : existence, examples, optimal bounds on emerging macro quantities, application etc. Five among them are highlighted below : $(1)$ The usual arguments based on translated inequality, $H$-measures, Compensated Compactness etc for obtaining optimal bounds are not enough. Additional compactness properties are needed. $(2)$ Assum…
We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform $L^p$-bounded solution sequence for $p>2$, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any sequence of approximate solutions to this system which is uniformly bounded in $L^2$ and has reasonable bounds on the errors made in the approximation (the errors are confined in a c…
We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic $p$-Laplace equation. We show that, up to a subsequence, weak solutions of the perturbed problem converge uniformly-in-time to weak solutions of the original problem as the perturbed data approach the original data. We do not assume uniqueness or additional regularity of the solution. However, when uniqueness is known, our result demonstrates that the weak solution is uniformly temporally stable to pert…
In the setting of general initial data and whole space we perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma with capillarity tensor represented by the Navier Stokes Korteweg Poisson system. We shall provide a detailed mathematical description of the convergence process by analyzing the dispersion of the fast oscillating acoustic waves. However the standard acoustic wave analysis is not sufficient to control the high frequency oscillations in the electric field but it is necessary to estimates the dispersive properties induced by the capillarit…
