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 \(\varOmega \subset \mathbb {R}^{3}\) $\Omega \subset {\mathbb{R}}^3$ exterior to a rigid body D, i.e., \(\varOmega =\mathbb {R}^{3}\backslash D\) $\Omega ={\mathbb{R}}^3\backslash D$ , with the data belonging to L \(^{p}\) $_{}^p$ -spaces and with the asymptotically almost periodic or almost automorphic external forces, respectively. Our method is based on the L \(^{p}-L^{q}\) $_{}^p-L^q$ smoothness of the evolution family corresponding to linearized equations in combination with interpolation functors and fixed-point arguments.