On the structure of solutions of nonlinear hyperbolicsystems of conservation laws

Gui-Qiang Chen; Monica  Torres

doi:10.3934/cpaa.2011.10.1011

Article Contents

2011, Volume 10, Issue 4: 1011-1036. Doi: 10.3934/cpaa.2011.10.1011

This issue Previous Article Next Article Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents

On the structure of solutions of nonlinear hyperbolic systems of conservation laws

Gui-Qiang Chen^1, and
Monica Torres^2,

1.
Mathematical Institute, University of Oxford, Oxford, OX1 3LB
2.
Department of Mathematics, Purdue University, 150 N. University Street 47907-2067

Received: March 2010

Revised: December 2010

Published: July 2011

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Abstract

We are concerned with entropy solutions $u$ in $L^\infty$ of nonlinear hyperbolic systems of conservation laws. It is shown that, given any entropy function $\eta$ and any hyperplane $t=const.$, if $u$ satisfies a vanishing mean oscillation property on the half balls, then $\eta(u)$ has a trace $H^d$-almost everywhere on the hyperplane. For the general case, given any set $E$ of finite perimeter and its inner unit normal $\nu: \partial^*E \to S^d$ and assuming the vanishing mean oscillation property of $u$ on the half balls, we show that the weak trace of the vector field $(\eta(u), q(u))$, defined in Chen-Torres-Ziemer [9], satisfies a stronger property for any entropy pair $(\eta, q)$. We then introduce an approach to analyze the structure of bounded entropy solutions for the isentropic Euler equations.

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Mathematics Subject Classification: Primary: 35L65, 35L50, 35L80, 35L67, 35L05, 76N10; Secondary: 26B12, 28C05, 76J20, 76L05.

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