Global existence and regularity of the weakly compressible Navier–Stokes system
Ning Jiang and C. David Levermore
Abstract: We construct the weakly nonlinear-dissipative approximate system for the general compressible Navier–Stokes system in a periodic domain. It was shown in  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  and Danchin  in the γ-law barotropic setting.
 R. Danchin, Zero Mach Number Limit for Compressible Flows with Periodic Boundary Conditions, Amer. J. Math. 124 (2002), 115––1219.
 N. Jiang and C.D. Levermore, Weakly Nonlinear-Dissipative Approximations of Hyperbolic-Parabolic Systems with Entropy, Indiana Univ. Math. (submitted 2009).
 N. Masmoudi, Incompressible, Inviscid Limit of the Compressible Navier–Stokes System, Ann. Inst. H. Poincaré 18 (2001), 199–224.