Compensated Integrability in bounded domains ; Applications to gases

TL;DR

This paper establishes a new functional inequality for symmetric tensors in bounded domains, with applications to gas dynamics models, depending on domain regularity and boundary conditions.

Contribution

It introduces a novel compensated integrability inequality for Div-BV tensors in bounded domains, applicable to gas dynamics models with specific boundary conditions.

Findings

Inequality depends on domain's $C^3$-regularity and curvature.

Applicable to Euler system and Hard Spheres dynamics.

Boundary conditions influence the tensor trace behavior.

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.