Periodic and almost periodic motions of Navier–Stokes flows and their stability in a perturbed half-space

Consider the system of Navier–Stokes equations in a perturbed half-space. We study the existence and stability of periodic solutions and almost periodic solutions to this system. For the Stokes system we use \(L^p\text {-}L^q\) $L^p\text{-}{L}^q$ estimates in combination with interpolation spaces to show the existence of bounded (in time) mild solutions as well as the existence of periodic solutions. Moreover, the existence of almost periodic solutions is proved. Then the existence, uniqueness and stability of the periodic and almost periodic solutions to the Navier–Stokes system are proved by using fixed-point arguments and certain interpolation relations.