JournalsaihpcVol. 35, No. 5pp. 1209–1234

Divergence-free positive symmetric tensors and fluid dynamics

  • Denis Serre

    École Normale Supérieure de Lyon, U.M.P.A., UMR CNRS–ENSL # 5669, 46 allée d'Italie, 69364 Lyon cedex 07, France
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Abstract

We consider d×dd \times d tensors A(x)A(x) that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of (detA)1d1(\mathrm{\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 ρ1np\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.

Cite this article

Denis Serre, Divergence-free positive symmetric tensors and fluid dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, pp. 1209–1234

DOI 10.1016/J.ANIHPC.2017.11.002