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Consider the incompressible 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 a non-autonomous problem with unbounded drift terms on a fixed exterior domainΩ  ⊂    ℝ   d$\Omega \subset {\mathbb {R}}^d$   . It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system{   T  Ω    (  t  ,  s  )   }    t  ≥  s  ≥  0$\lbrace T_\Omega (t,s)\rbrace _{t\ge s\ge 0}$    onL  σ  p    (  Ω  )$L^p_\sigma (\Omega )$    for1  &lt;  p  &lt;  ∞    $1&amp;lt;p&amp;lt;\infty $   . Moreover,  L p  - L q   smoothing properties and gradient estimates ofT  Ω    (  t  ,  s  )$T_\Omega (t,s)$   ,0  ≤  s  ≤  t    $0\le s \le t$   , are obtained. These results are the key ingredients to show local in time existence of mild solutions to the full nonlinear problem for initial value inL  σ  p    (  Ω  )$L^p_\sigma (\Omega )$   ,  p  ≥  d .

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