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

We construct the weakly nonlinear-dissipative approximate system for the general compressible Navier-Stokes system in a periodic domain. It was shown in [11] 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 [18] and Danchin [6] in the γ-law barotropic setting.