Asymptotically Almost Periodic and Almost Automorphic Solutions to the Non-autonomous Oseen-Navier-Stokes Equations

In this paper, we investigate the existence, uniqueness and stability of asymptotically almost periodic and almost automorphic solutions to the non-autonomous Oseen-Navier-Stokes Equations (ONSE) in an unbounded domain \(\varOmega \subset \mathbb {R}^{3}\) exterior to a rigid body D, i.e., \(\varOmega =\mathbb {R}^{3}\backslash D\), with the data belonging to L\(^{p}\)-spaces and with the asymptotically almost periodic or almost automorphic external forces, respectively. Our method is based on the L\(^{p}-L^{q}\) smoothness of the evolution family corresponding to linearized equations in combination with interpolation functors and fixed-point arguments.