A remark on smooth solutions or the weakly compressible periodic Navier-Stokes equations

We investigate the limit of the periodic compressible Navier-Stokes equations, when the Mach number goes to zero, and the density goes to a constant. We prove long time existence results for smooth solutions of the weakly compressible Navier-Stokes equations, for small Mach numbers, under a smallness condition on the initial velocity field, which depends only on the viscosity (that smallness condition is imposed only on the compressible part of the velocity in the bidimensional case). We also prove the convergence of the solutions of the compressible equations to the incompressible equations, once the fast waves, which satisfy a fully parabolic equation, have been removed.