Mixed determinants, Compensated Integrability and new {\em a priori} estimates in Gas dynamics

TL;DR

This paper extends compensated integrability theory to derive new a priori estimates for inviscid gas flows, using multi-linearity of determinants to control defect measures and spatial correlations based on total mass and energy.

Contribution

It introduces novel a priori estimates in gas dynamics by leveraging the multi-linearity of determinants, expanding the scope of compensated integrability theory.

Findings

New bounds for defect measures in Boltzmann equation

Estimates for weighted spatial correlations in Euler system

Bounds depend only on total mass and energy

Abstract

We extend the scope of our recent Compensated Integrability theory, by exploiting the multi-linearity of the determinant map over ${\bf Sym}_n(\mathbb{R})$. This allows us to establish new {\em a priori} estimates for inviscid gases flowing in the whole space ${\mathbb R}^d$. Notably, we estimate the defect measure (Boltzman equation) or weighted spacial correlations of the velocity field (Euler system). As usual, our bounds involve only the total mass and energy of the flow.