The time periodic problem for the Navier-Stokes equations on half spaces with moving boundary: Linear theory

In this article, we develop a linear theory to deal with the time periodic problem for the Navier-Stokes equations on unbounded domains with moving boundary. Compared to the case of bounded domains the underlying modified time-dependent Stokes operators are no longer invertible, thus leading to a more sophisticated construction of the evolution operator. Moreover, Sobolev embeddings on $L^q$ spaces imply restrictions on q depending on geometric properties of the domain. The theory is focusing on the half space case, the construction and local-in-time estimates of the evolution operator and its adjoint in view of time periodic solutions.