Navier--Stokes flow in the exterior of a moving obstacle with a Lipschitz boundary

Consider the three-dimensional Navier–Stokes flow past a moving rigid body $\mathcal{O}\subset {\mathbb{R}}^3$ with prescribed translational and angular velocities, where $\mathcal{O}$ stands for a bounded Lipschitz domain. We prove that the solution to the linearized problem is governed by a $C_0$-semigroup on solenoidal $L^q$-vector spaces with the $L^q$-$L^r$ estimates provided that $\mathrm{\mid }1\mathrm{/}q-1\mathrm{/}2\mathrm{\mid }<1\mathrm{/}6+\epsilon$ with some $\epsilon >0$, where $r\ge q$ may be taken arbitrary large. As an application, we prove the existence and uniqueness of global mild solutions to the Navier–Stokes problem if the translational and angular velocities as well as the initial are sufficiently small.