Global existence of weak solutions for the anisotropic compressible Stokes system

Bresch, D.; Burtea, C.

doi:10.1016/j.anihpc.2020.04.003

Bresch, D.¹ ; Burtea, C.²

¹ Univ. Grenoble Alpes, Univ. Savoie Mont-Blanc, CNRS, LAMA, Chambéry, France
² Université de Paris and Institut de mathématiques de Jussieu-Paris Rive Gauche, Bâtiment Sophie Germain, Bureau 727, 8 place Aurélie Nemours, 75013 Paris, France

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 6, pp. 1271-1297.

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Résumé

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.

MR Zbl

DOI : 10.1016/j.anihpc.2020.04.003

Classification : 35Q35, 35B25, 76T20
Mots-clés : Compressible quasi-stationary Stokes equations, Anisotropic viscous tensor, Global weak solutions

Affiliations des auteurs :

Bresch, D. ¹ ; Burtea, C. ²

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     title = {Global existence of weak solutions for the anisotropic compressible {Stokes} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1271--1297},
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Bresch, D.; Burtea, C. Global existence of weak solutions for the anisotropic compressible Stokes system. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 6, pp. 1271-1297. doi : 10.1016/j.anihpc.2020.04.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.04.003/

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