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

TL;DR

This paper proves asymptotic stability of solutions to the 3D Navier-Stokes equations in $L^3$ and related spaces, including weak-$L^3$ and Lorentz spaces, with implications for Landau solutions.

Contribution

It offers a concise proof of $L^3$-stability and extends stability results to a broader class of function spaces, clarifying limitations via counterexamples.

Findings

Asymptotic stability holds for small vector fields in weak-$L^3$.

Stability extends to Lorentz spaces between $L^3$ and weak-$L^3$.

Counterexample shows stability does not extend to all weak-$L^3$.

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.