# Divergence-free positive symmetric tensors and fluid dynamics

### Denis Serre

École Normale Supérieure de Lyon, U.M.P.A., UMR CNRS–ENSL # 5669, 46 allée d'Italie, 69364 Lyon cedex 07, France

A subscription is required to access this article.

## 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 $(\mathrm{\det }A)^{\frac{1}{d−1}}$. We apply them to models of compressible inviscid fluids: Euler equations, Euler–Fourier, relativistic Euler, Boltzman, BGK, etc. We deduce an *a priori* estimate for a new quantity, namely the space–time integral of $\rho ^{\frac{1}{n}}p$, where *ρ* is the mass density, *p* the pressure and *n* the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.

## Cite this article

Denis Serre, Divergence-free positive symmetric tensors and fluid dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, pp. 1209–1234

DOI 10.1016/J.ANIHPC.2017.11.002