Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stability, and Longtime Behavior

This double-sized chapter contains two related themes that were supposed to be covered by two independent chapters of the handbook in the original project: (1) weak solutions of the Navier-Stokes equations in the barotropic regime and (2) weak solutions of the Navier-Stokes-Fourier system. We shall discuss for both systems: (1) Various notions of weak solutions, their relevance, and their mutual relations. (2) Global existence of weak solutions. (3) Notions of relative energy functional, dissipative solutions and relative energy inequality and its impact on the investigation of the stability analysis of compressible flows. (4) Weak strong uniqueness principle. (5) Longtime behavior of weak solutions. For physical reasons, we shall limit ourselves to the three-dimensional physical space.

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