Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

By Didier Bresch, Pierre-Emmanuel Jabin. Annals of Mathematics, 2015

171 citations recorded by the source corpus.

We prove global existence of appropriate weak solutions for the compressible Navier--Stokes equations for more general stress tensor than those covered by P.-L. Lions and E. Feireisl's theory. More precisely we focus on more general pressure laws which are not thermodynamically stable; we are also able to handle some anisotropy in the viscous stress tensor. To give answers to these two longstanding problems, we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. These two cases open the theory to important physical applications, for instance to describe solar events (virial pressure law), geophysical flows (eddy viscosity) or biological situations (anisotropy).

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4abed7bb5b808897b0a7d2b07c93396ee462526e

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Citations in the corpus

Loss of Regularity for the Continuity Equation with Non-Lipschitz Velocity Field (2018, methodology): “More recently, in [6] preservation of “a logarithm of a derivative” for the solution of the continuity equation is established and used for a new theory of existence of solutions to the compressible Navier-Stokes equations.”
On mean-field limits and quantitative estimates with a large class of singular kernels: Application to the Patlak–Keller–Segel model (2019, methodology): “In some sense, the proposed modulated free energy may also be understood as introducing appropriate weights in the relative entropy used in [18] (in the spirit of what has been recently developed by some of the authors in [4] in an other framework) to cancel the most singular terms involving the div”