Divergence-free positive symmetric tensors and fluid dynamics

Denis Serre (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We consider d × 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) 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 ρ 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.

Keywords: kinetic equations; functional inequalities; fluid dynamics; Stress-energy tensor; homogenization; Conservation laws; gas dynamics

Domains: Mathematics [math]; Analysis of PDEs [math.AP]; Physics [physics]; Mechanics [physics]; Fluid mechanics [physics.class-ph]

Denis Serre. Divergence-free positive symmetric tensors and fluid dynamics. 2017. ⟨ensl-01514880v4⟩