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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 $Ω\subset \R^d$. It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system $\{T_Ω(t,s)\}_{t\geq s\geq0}$ on $L^p_σ(Ω)$ for $1<p<\infty$. Moreover, $L^p$-$L^q$ smoothing properties and gradient estimates of $T_Ω(t,s)$, $0\leq s \leq t$, are obtained. These results are the key ingredients to show local in time existence of mild solutions to the full nonlinear problem for $p\geq d$ and initial value in $L^p_σ(Ω)$.