On L3,∞-stability of the Navier-Stokes system in exterior domains

This paper studies the stability of a stationary solution of the Navier-Stokes system with a constant velocity at infinity in an exterior domain. More precisely, this paper considers the stability of the Navier-Stokes system governing the stationary solution which belongs to the weak L-3-space L-3,L-infinity Under the condition that the initial datum belongs to a solenoidal L-3,L-infinity-space, we prove that if both the L-3,L-infinity-norm of the initial datum and the L-3,L-infinity-norm of the stationary solution are sufficiently small then the system admits a unique global-in-time strong L-3,L-infinity-solution satisfying both L-3,L-infinity-asymptotic stability and L-3,L-infinity-asymptotic stability. Moreover, we investigate L-3,L-infinity-asymptotic stability of the global-in-time solution. Using LP-Lq type estimates for the Oseen semigroup and applying an equivalent norm on the Lorentz space are key ideas to establish both the existence of a unique global-in-time strong (or mild) solution of our system and the stability of our solution. (C) 2016 Elsevier Inc. All rights reserved.