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

TL;DR

This paper proves the stability and existence of unique global solutions for the Navier-Stokes system in exterior domains with initial data in weak $L^3$-spaces, extending stability results to these critical Lorentz spaces.

Contribution

It establishes the $L^{3, abla}$-stability of stationary solutions in exterior domains for the Navier-Stokes equations with initial data in weak $L^3$-spaces, using $L^p$-$L^q$ estimates and Lorentz space norms.

Findings

Existence of unique global-in-time strong solutions under small $L^{3, abla}$-norm conditions.

Proven $L^{3, abla}$-asymptotic stability of solutions.

Extended stability analysis to $L^{3,r}$-spaces.

Abstract

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 $L^{3,\infty}$. Under the condition that the initial datum belongs to a solenoidal $L^{3 , \infty}$-space, we prove that if both the $L^{3,\infty}$-norm of the initial datum and the $L^{3,\infty}$-norm of the stationary solution are sufficiently small then the system admits a unique global-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-time solution. Using $L^p$-$L^q$ 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.