Divergence-free positive symmetric tensors and fluid dynamics

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)1d−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 ρ1np, where ρ is the mass density, p the pressure and n the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.