Compensated Integrability in bounded domains ; Applications to gases

Abstract

An accurate functional inequality for Div-BV positive symmetric tensors $A$ in a bounded domain $U\subset {\mathbb{R}}^n$ arises whenever the tangential part of the normal trace ${\gamma }_{\nu }A\sim A\vec{\nu }$ is a finite measure over $\partial U$. The proof involves an extension operator to a neighbourhood of $\bar{U}$. The resulting inequality depends upon the domain only through the $C^3$-regularity of $\partial U$, some constant involving the curvature and its first derivatives.This abstract statement applies to several models of Gas Dynamics (Euler system, Hard Spheres dynamics), as the boundary condition (slip, or reflection) tells us that $A\vec{\nu }$ is parallel to $\vec{\nu }$, where $A$ is the mass-momentum tensor.