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

Abstract

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 $|1/q-1/2|<1/6+\epsilon$ 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.

32 pages