On the Rate of Decay of the Oseen Semigroup in Exterior Domains and its Application to Navier–Stokes Equation

Abstract

We prove L p -L q estimates of the Oseen semigroup in n-dimensional exterior domains $$(n\,\geqslant\, 3),$$ which refine and improve those obtained by Kobayashi and Shibata [15]. As an application, we give a globally in time stability theory for the stationary Navier–Stokes flow whose velocity at infinity is a non-zero constant vector. We thus extend the result of Shibata [21]. In particular, we find an optimal rate of convergence of solutions of the non-stationary problem to those of the corresponding stationary problem.