Stable plane Euler flows with concentrated and sign-changing vorticity

TL;DR

This paper constructs stable steady solutions to the 2D Euler equations with vorticity concentrated in two regions, converging to opposite point vortices, without requiring a local minimum of the Kirchhoff-Routh function.

Contribution

It introduces a new variational method to construct stable solutions with sign-changing vorticity, avoiding the need for an isolated minimum of the Kirchhoff-Routh function.

Findings

Solutions are Lyapunov stable in $L^p$ norm.

Vorticity concentrates in two small regions converging to point vortices.

Method applies to general bounded domains without special assumptions.

Abstract

We construct a family of steady solutions to the two-dimensional incompressible Euler equation in a general bounded domain, such that the vorticity is supported in two well-separated regions of small diameter and converges to a pair of point vortices with opposite signs. Compared with previous results, we do not need to assume the existence of an isolated local minimum point of the Kirchhoff-Routh function. Moreover, due to their variational nature, the solutions obtained are Lyapunov stable in $L^p$ norm of the vorticity. The proofs are achieved by maximizing the kinetic energy over an appropriate family of rearrangement classes of sign-changing functions and studying the limiting behavior of the maximizers.