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We consider tensors that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of . 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 , 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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Denis Serre, Divergence-free positive symmetric tensors and fluid dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, pp. 1209–1234DOI 10.1016/J.ANIHPC.2017.11.002