The time periodic problem of the Navier–Stokes equations in a bounded domain with moving boundary

Abstract

The time periodic problem of the Navier–Stokes equations on a non-cylindrical space–time domain is studied. Motivated by a recent result by Saal (2006) on maximal regularity for this kind of system we construct time periodic solutions in Lq-spaces provided the bounded domain moves periodically with small amplitude and the given periodic external force is small. The proof is based on new decay estimates for the solution operator of parabolic evolution equations corresponding to the non-cylindrical space–time domain problem.