Asymptotic stability of homogeneous solutions of incompressible stationary Navier-Stokes equations

TL;DR

This paper investigates the asymptotic stability of certain homogeneous solutions to the incompressible stationary Navier-Stokes equations, extending previous results on Landau solutions to a broader class of solutions.

Contribution

It proves the asymptotic stability of the least singular axisymmetric no-swirl solutions, other than Landau solutions, under $L^2$-perturbations.

Findings

Landau solutions are asymptotically stable under $L^2$-perturbations.

Least singular solutions other than Landau solutions are also asymptotically stable.

Classification of $(-1)$-homogeneous axisymmetric solutions was used in the analysis.

Abstract

It was proved by Karch and Pilarzyc that Landau solutions are asymptotically stable under any $L^2$-perturbation. In our earlier work with L. Li, we have classified all $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any $L^2$-perturbation.