Stability of two-dimensional steady Euler flows with concentrated vorticity

TL;DR

This paper investigates the nonlinear stability of 2D steady Euler flows with highly concentrated vorticity, demonstrating stability when vorticity is confined to small regions, using energy maximization and compactness arguments.

Contribution

It establishes the nonlinear stability of concentrated vorticity flows as local energy maximizers, extending understanding of flow stability in bounded domains.

Findings

Flows with vorticity in one small region are stable.

Flows with two small opposite-signed vorticity regions are stable.

The flows form a compact, isolated set of local energy maximizers.

Abstract

In this paper, we study the stability two-dimensional (2D) steady Euler flows with sharply concentrated vorticity in a simply-connected bounded domain. These flows are obtained as maximizers of the kinetic energy subject to the constraint that the vorticity is compactly supported in a finite number of disjoint regions of small diameter. We prove the nonlinear stability of these flows when the vorticity is concentrated in one small region, or in two small regions with opposite signs. The proof is achieved by showing that these flows constitute a compact and isolated set of local maximizers of the kinetic energy on an isovortical surface. The separation property of the stream function plays a crucial role in validating the isolatedness.