Asymptotic stability for the 3D Navier-Stokes equations in $L^3$ and nearby spaces

Abstract

We provide a short proof of $L^3$-asymptotic stability around vector fields that are small in weak-$L^3$, including small Landau solutions. We show that asymptotic stability also holds for vector fields in the range of Lorentz spaces strictly between $L^3$ and weak-$L^3$, as well as in the closure of the test functions in weak-$L^3$. To provide a comprehensive perspective on the matter, we observe that asymptotic stability of Landau solutions does not generally extend to weak-$L^3$ via a counterexample.