Divergence-free positive symmetric tensors and fluid dynamics

Serre, Denis

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1209-1234.

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 ${(\det A)}^{\frac{1}{d-1}}$. 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.

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