Stability of planar exterior stationary flows with suction

TL;DR

This paper proves the nonlinear stability of a specific class of stationary flows in a 2D exterior domain with suction, addressing an open problem in fluid dynamics by analyzing the Navier-Stokes system.

Contribution

It demonstrates the nonlinear stability of critical decay stationary flows with suction in the exterior of a disk, a problem previously unresolved.

Findings

Nonlinear stability established for flows with negative flux coefficients

Stability proven for initial disturbances in L^2 space

Partially resolves an open problem in flow stability theory

Abstract

We consider the two-dimensional Navier-Stokes system in a domain exterior to a disk. The system admits a stationary solution with critical decay $O(|x|^{-1})$ written as a linear combination of the pure rotating flow and the flux carrier. We prove its nonlinear stability in large time for initial disturbances in $L^2$ under smallness conditions, assuming that there is suction across the boundary, namely that the sign of coefficients of the flux carrier is negative. This result partially solves an open problem in the literature.