{
  "id": "1801.02912",
  "version": "v1",
  "published": "2018-01-09T12:26:11.000Z",
  "updated": "2018-01-09T12:26:11.000Z",
  "title": "Null Lagrangian Measures in planes, compensated compactness and conservation laws",
  "authors": [
    "Andrew Lorent",
    "Guanying Peng"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "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 $\\mu$ supported on $\\mathcal{K}$ satisfying $$ \\int M_k(X) d\\mu (X)=M_k\\left(\\int X d\\mu (X)\\right)\\text{ for }k=1,2,\\dots, q $$ is a Dirac measure. We call measures that satisfy the above equation \"Null Lagrangian Measures\" and we denote the set of Null Lagrangian Measures supported on $\\mathcal{K}$ by $\\mathcal{M}^{pc}(\\mathcal{K})$. For general $m,n$, a necessary and sufficient condition for triviality of $\\mathcal{M}^{pc}(\\mathcal{K})$ is unknown even in the case where $\\mathcal{K}$ is a linear subspace of $M^{m\\times n}$. We provide a condition that is sufficient for any linear subspace $\\mathcal{K}$ and also necessary in the case where $2\\leq\\min\\left\\{m,n\\right\\}\\leq 3$. A corollary is that two dimensional subspaces $\\mathcal{K}\\subset M^{m\\times n}$ support nontrivial Null Lagrangian Measures if and only if $\\mathcal{K}$ has Rank-$1$ connections. Using the ideas developed we are able to answer (up to first order) a question of Kirchheim, M\\\"{u}ller and Sverak on the Null Lagrangian measures arising in the study of a (one) entropy solution of a $2\\times 2$ system of conservation laws that arises in elasticity. Further the methods lead to a strategy to provide a more direct proof of DiPerna's well known result on existence of entropy solutions to the system. The strategy could potentially be applied to other systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-09T12:26:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J99",
      "28A25",
      "35L65"
    ],
    "keywords": [
      "compensated compactness",
      "conservation laws",
      "support nontrivial null lagrangian measures",
      "sufficient condition",
      "entropy solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}