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We prove global existence of appropriate weak solutions for the compressibleNavier--Stokes equations for more general stress tensor than those covered byP.-L. Lions and E. Feireisl's theory. More precisely we focus on more generalpressure laws which are not thermodynamically stable; we are also able tohandle some anisotropy in the viscous stress tensor. To give answers to thesetwo longstanding problems, we revisit the classical compactness theory on thedensity by obtaining precise quantitative regularity estimates: This requires amore precise analysis of the structure of the equations combined to a novelapproach to the compactness of the continuity equation. These two cases openthe theory to important physical applications, for instance to describe solarevents (virial pressure law), geophysical flows (eddy viscosity) or biologicalsituations (anisotropy).

Global Existence of Weak Solutions for Compresssible Navier--Stokes  Equations: Thermodynamically unstable pressure and anisotropic viscous stress  tensor