Null Lagrangian Measures in subspaces, compensated compactness and conservation laws

• Published in: arXiv.org 2018-09
• Main Authors: Lorent, Andrew, Guanying Peng
• Format: Article
• Language: English
• Subjects: Conservation laws; Elasticity; Entropy of solution; Servers; Subspaces
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1801.02912

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 \(\mathcal{M}^{pc}(\mathcal{K})\) was an open question even in the case where \(\mathcal{K}\) is a linear subspace of \(M^{m\times n}\). We answer this question and provide a necessary and sufficient condition for any linear subspace \(\mathcal{K}\subset M^{m\times n}\). The ideas also allow us to show that for any \(d\in \left\{1,2,3\right\}\), \(d\)-dimensional subspaces \(\mathcal{K}\subset M^{m\times n}\) support non-trivial Null Lagrangian Measures if and only if \(\mathcal{K}\) has Rank-\(1\) connections. This is known to be false for \(d\ge 4\). 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.