On the asymptotic stability of steady flows with nonzero flux in two-dimensional exterior domains

TL;DR

This paper proves the asymptotic stability of certain steady solutions to the 2D Navier-Stokes equations in exterior domains, including flux carriers with small flux and symmetric solutions, under $L^2$-perturbations.

Contribution

It establishes stability results for steady flows with nonzero flux in exterior domains, extending previous work to include critically decaying flux carriers and symmetric solutions.

Findings

Stability proven for flux carriers with flux $|\

Stability holds for symmetric solutions with smallness conditions.

Applicable to a broad class of steady solutions with nonzero flux.

Abstract

The Navier-Stokes equations in a two-dimensional exterior domain are considered. The asymptotic stability of stationary solutions satisfying a general hypothesis is proven under any $L^2$-perturbation. In particular the general hypothesis is valid if the steady solution is the sum of the critically decaying flux carrier with flux $|\Phi| < 2\pi$ and a small subcritically decaying term. Under the central symmetry assumption, the general hypothesis is also proven for any critically decaying steady solutions under a suitable smallness condition.