The Oseen-Navier-Stokes flow in the exterior of a rotating obstacle: The non-autonomous case

Consider the Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity – sometimes called an Oseen condition. By a suitable change of coordinates the problem is transformed to an non-autonomous problem with unbounded drift terms on a fixed exterior domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$. It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system $\{T_{\mathrm{\Omega }}(t,s){\}}_{t\ge s\ge 0}$ on $L_{\sigma }^p(\mathrm{\Omega })$ for $1<p<\mathrm{\infty }$. Moreover, $L^p$-$L^q$ smoothing properties and gradient estimates of $T_{\mathrm{\Omega }}(t,s)$, $0\le s\le t$, are obtained. These results are the key ingredients to show local in time existence of mild solutions to the full nonlinear problem for $p\ge d$ and initial value in $L_{\sigma }^p(\mathrm{\Omega })$.