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This paper studies the stability of a stationary solution of theNavier-Stokes system with a constant velocity at infinity in an exteriordomain. More precisely, this paper considers the stability of the Navier-Stokessystem governing the stationary solution which belongs to the weak $L^3$-space$L^{3,\infty}$. Under the condition that the initial datum belongs to asolenoidal $L^{3 , \infty}$-space, we prove that if both the$L^{3,\infty}$-norm of the initial datum and the $L^{3,\infty}$-norm of thestationary solution are sufficiently small then the system admits a uniqueglobal-in-time strong $L^{3,\infty}$-solution satisfying both$L^{3,\infty}$-asymptotic stability and $L^\infty$-asymptotic stability.Moreover, we investigate $L^{3,r}$-asymptotic stability of the global-in-timesolution. Using $L^p$-$L^q$ type estimates for the Oseen semigroup and applyingan equivalent norm on the Lorentz space are key ideas to establish both theexistence of a unique global-in-time strong (or mild) solution of our systemand the stability of our solution.

On $L^{3,\infty}$-stability of the Navier-Stokes system in exterior domains