Large time behavior of a generalized Oseen evolution operator, with applications to the Navier–Stokes flow past a rotating obstacle

Abstract

Consider the motion of a viscous incompressible fluid in a 3D exterior domain D when a rigid body $$\mathbb R^3{\setminus } D$$ moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $$L^q$$ - $$L^r$$ smoothing action near $$t=s$$ as well as generation of the evolution operator $$\{T(t,s)\}_{t\ge s\ge 0}$$ was shown by Hansel and Rhandi (J Reine Angew Math 694:1–26, 2014) under reasonable conditions. In this paper we develop the $$L^q$$ - $$L^r$$ decay estimates of the evolution operator T(t, s) as $$(t-s)\rightarrow \infty $$ and then apply them to the Navier–Stokes initial value problem.