# Null Lagrangian Measures in Subspaces, Compensated Compactness and Conservation Laws

• Andrew Lorent
• Guanying Peng
Article

## Abstract

Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set $${\mathcal {K}}\subset M^{m\times n}$$ such that
\begin{aligned}&\lim _{j\rightarrow \infty } \Vert \mathrm {dist}(Du_j,{\mathcal {K}})\Vert _{L^p}= 0 \text { and } \sup _{j}\Vert u_j\Vert _{W^{1,p}}<\infty \nonumber \\&\quad \Rightarrow \{Du_{j}\}_{j}\text { is precompact in } L^p. \end{aligned}
(1)
Let $$M_1,M_2,\dots , M_q$$ denote the set of all minors of $$M^{m\times n}$$. A sufficient condition for (1) is that any probability measure $$\mu$$ supported on $${\mathcal {K}}$$ satisfying
\begin{aligned} \int M_k(X) \hbox {d}\mu (X)=M_k\left( \int X \hbox {d}\mu (X)\right) \text { for all } k \end{aligned}
(2)
is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [21], we denote the set of Null Lagrangian Measures supported on $${\mathcal {K}}$$ by $${\mathcal {M}}^{pc}({\mathcal {K}})$$. For general mn, 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 $$d\ge 4$$ from [5]. Further using the ideas developed we are able to answer a question of Kirchheim et al. [18]. Let $$P_1(u,v):=\left( \begin{matrix} u &{} v\\ a(v) &{} u\\ ua(v) &{} \frac{1}{2}u^2+ F(v)\\ \end{matrix}\right)$$ and $${\mathcal {K}}_1 := \left\{ P_1(u,v): u,v\in \mathrm {I\!R}\right\}$$ for some function a and its primitive F. The set $${\mathcal {K}}_1$$ arises in the study of entropy solutions to the $$2\times 2$$ system of conservation laws
\begin{aligned} u_t=a(v)_x \quad \text { and }\quad v_t=u_x. \end{aligned}
In [18], the authors asked what are the conditions on the function a such that $${\mathcal {M}}^{pc}({\mathcal {K}}_1\cap U)$$ consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in $${\mathcal {K}}_1$$. Given $$\alpha =(\alpha _1,\alpha _2)\in \mathrm {I\!R}^2$$, if $$a'(\alpha _2)>0$$ then we construct non-trivial measures in $${\mathcal {M}}^{pc}({\mathcal {K}}_1\cap B_{\delta }\left( P_1(\alpha )\right) )$$ for any $$\delta >0$$. On the other hand if $$a'(\alpha _2)<0$$ then for sufficiently small $$\delta >0$$, we show that $${\mathcal {M}}^{pc}({\mathcal {K}}_1\cap B_{\delta }\left( P_1(\alpha )\right) )$$ consists of Dirac measures.

## Notes

### Acknowledgements

Andrew Lorent is very grateful to V. Šverák for many very helpful conversations during a two-week visit to Minnesota in November of 2016. These conversations essentially introduced us to this topic and led to some initial ideas for Theorem 2. Both authors are very grateful for a great deal of very helpful correspondence since then. We would also like to thank S. Müller for providing us with a proof of Lemma 26. The proof provided is considerably simpler and more elegant than our original proof. Guanying Peng would like to thank Y. Shi for helpful discussions on elementary algebraic geometry. Andrew Lorent acknowledes the support of the Simons Foundation, Grant #426900.

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