Spatial stability of 3D exterior stationary Navier-Stokes flows

In this paper, we study the stability of stationary solutions w for the Navier-Stokes flows in an exterior domain with zero velocity at infinity. With suitable assumptions of w, by the works of Chen (1993), Kozono-Ogawa (1994) and Borchers-Miyakawa (1995), if u(0) - w is an element of L-r (Omega) boolean AND L-3 (Omega) then one can obtain parallel to u(t) - w parallel to(p) = O(t(-3/2(1/r - 1/p)))) for 1 < r < p < infinity, parallel to del(u(t) - w)parallel to(p) = O(t(-3/2(1/r - 1/p)-1/2)) for 1 < r < p < 3, where u(x, t) is a solution of the Navier-Stokes equations with the initial condition u(0). In this paper, we will prove that for any 0 < alpha < 3 if vertical bar x vertical bar(alpha) (u(0) - w) belongs to L-r (Omega) then one has parallel to vertical bar x vertical bar(alpha) (u((t)) - w)parallel to(Lp) = O(t(-3/2(1/r - 1/p)+alpha/2)) for p > 3r/3-r alpha (C) 2012 Elsevier Inc. All rights reserved.