  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.
