Stability for the 3D Navier-Stokes Equations with Nonzero far Field Velocity on Exterior Domains

Concerning to the non-stationary Navier-Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier-Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7: 339-367, 2005), in L-p spaces for p >= 3. In this article, we first extend their result to the case 3/2 < p by modifying the method in Bae and Jin (J Math Fluid Mech 10: 423-433, 2008) that was used to obtain weighted estimates for the Navier-Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier-Stokes flow with a nonzero velocity at infinity.