## A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations.(English)Zbl 0997.35050

The author studies periodic solutions of the weakly compressible Navier-Stokes equations for small Mach numbers. The main results of the paper are:
(i) If the initial velocity satisfies a smallness condition in terms of the viscosity, then the system has a smooth solution for all $$T > 0.$$
(ii) As the Mach number tends to zero, the solution of the compressible system approaches the solution of the incompressible Navier-Stokes equations plus a term which represents the fast waves.

### MSC:

 35Q30 Navier-Stokes equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35B10 Periodic solutions to PDEs
Full Text: