Global existence and regularity of the weakly compressible Navier–Stokes system

Abstract

We construct the weakly nonlinear-dissipative approximate system for the general compressible Navier–Stokes system in a periodic domain. It was shown in Arch. Rational Mech. Anal. 201 (2011), 377–412, that because the Navier–Stokes system has an entropy structure, its approximate system will have Leray-like global weak solutions. These solutions decompose into an incompressible part governed by an incompressible Navier–Stokes system, and an acoustic part governed by a nonlocal quadratic equation which couples it to the incompressible part. We obtain regularity results for the acoustic part of the solution via a Littlewood–Paley decomposition that extend to this general setting results found by Masmoudi [Ann. Inst. H. Poincaré 18 (2001), 199–224] and Danchin [Amer. J. Math. 124 (2002), 1153–1219] in the γ-law barotropic setting.