# Divergence-free positive symmetric tensors and fluid dynamics

## TL;DR

This paper studies divergence-free positive symmetric tensors and derives sharp inequalities, applying them to fluid dynamics models to obtain new a priori estimates for quantities like density and pressure.

## Contribution

It introduces new inequalities for divergence-free positive symmetric tensors and applies them to various fluid models to derive novel a priori estimates.

## Key findings

- Established sharp inequalities for tensor determinants.
- Derived an a priori estimate for the integral of density and pressure.
- Applied results to multiple fluid dynamics models.

## Abstract

We consider $d\times d$ tensors $A(x)$ that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of $(\det A)^{\frac1{d-1}}$. We apply them to models of compressible inviscid fluids: Euler equations, Euler--Fourier, relativistic Euler, Boltzman, BGK, etc... We deduce an {\em a priori} estimate for a new quantity, namely the space-time integral of $\rho^{\frac1n}p$, where $\rho$ is the mass density, $p$ the pressure and $n$ the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.