Zero Mach number limit for compressible flows with periodic boundary conditions

We address the question of convergence to the incompressible Navier-Stokes equations for slightly compressible viscous flows with ill-prepared initial data and periodic boundary conditions (the case of the whole space has been studied in an earlier paper by the author). The functional setting is very close to the one used by H. Fujita and T. Kato for incompressible flows. For arbitrarily large initial data, we show that the compressible flow with small Mach number exists as long as the incompressible one does. In particular, it exists globally if the corresponding incompressible solution exists for all time. We further state a convergence result for the slightly compressible solution filtered by the group of acoustics.