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Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor. (English) Zbl 1405.35133

The authors extend P.-L. Lions and E. Feireisl’s theory concerning global existence of weak solutions for the compressible Navier-Stokes equations. This long paper first summarizes the classical theory: a priori estimates, heuristic presentation of the method by E. Feireisl and P.-L. Lions, the limitations of Lions-Feireisl’s theory, physical discussions on pressure laws and stress tensors. Next, the authors present their new results for the compressible Navier-Stokes system which include sketch of the new compactness method, the compactness criterion, compactness for linear transport equation, a rough sketch of the extension to compressible Navier-Stokes, stability results, renormalized equation and weights. The propagation of regularity on the transport equation. The authors discuss the control on the effective viscous flux, the coupling with the pressure law, the coupling with the pressure in the anisotropic case. The passage from the global existence of regularized system with added viscosity to the case no viscosity is presented. In Appendix, Besov spaces and Littlewood-Paley decomposition are described.

MSC:

35Q30 Navier-Stokes equations
35D30 Weak solutions to PDEs
54D30 Compactness
42B37 Harmonic analysis and PDEs
35Q86 PDEs in connection with geophysics
92B05 General biology and biomathematics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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