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 L3-space L3,∞. Under the condition that the initial datum belongs to a solenoidal L3,∞-space, we prove that if both the L3,∞-norm of the initial datum and the L3,∞-norm of the stationary solution are sufficiently small then the system admits a unique global-in-time strong L3,∞-solution satisfying both L3,∞-asymptotic stability and L∞-asymptotic stability. Moreover, we investigate L3,r-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.