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 O ⊂ R³ with prescribed translational and angular velocities, where O stands for a bounded Lipschitz domain. We prove that the solution to the linearized problem is governed by a C₀-semigroup on solenoidal Lq-vector spaces with the Lq-Lʳ estimates provided that |1/q-1/2|<1/6+ε with some ε>0, where r ≥ 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.