Divergence-free positive symmetric tensors and fluid dynamics

2017/04/30 by Serre, Denis · 4 citations
#Analysis of PDEs (math.AP) #Classical Physics (physics.class-ph) #FOS: Mathematics #FOS: Physical sciences

paper · doi:10.48550/arxiv.1705.00331

Abstract

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)^\frac1d-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 ρ^\frac1np, where ρ is the mass density, p the pressure and n the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.

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