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 ∈ L r ( Ω ) ∩ L 3 ( Ω ) then one can obtain ‖ u ( t ) − w ‖ p = O ( t − 3 2 ( 1 r − 1 p ) ) for 1 < r < p < ∞ , ‖ ∇ ( u ( t ) − w ) ‖ 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 < α < 3 if | x | α ( u 0 − w ) belongs to L r ( Ω ) then one has ‖ | x | α ( u ( t ) − w ) ‖ L p = O ( t − 3 2 ( 1 r − 1 p ) + α 2 ) for p > 3 r 3 − r α .