Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains

Abstract

We investigate the time-periodic Stokes equations with non-homogeneous divergence data in the whole space, the half space, bent half spaces and bounded domains. The solutions decompose into a well-studied stationary part and a purely periodic part, for which we establish L^p estimates. For the whole space and the half space case we use a reduction of the Stokes equations to (n − 1) heat equations. Perturbation and localisation methods yield the result on bent half spaces and bounded domains. A one-to-one correspondence between maximal regularity for the initial value problem and time periodic maximal regularity is proven, providing a short proof for the maximal regularity of the Stokes operator avoiding the notion of ℛ-boundedness. The results are applied to a quasilinear model governing the flow of nematic liquid crystals.