Steady double vortex patches with opposite signs in a planar ideal fluid

TL;DR

This paper constructs and analyzes steady double vortex patches with opposite signs in a planar ideal fluid, demonstrating their stability and energy-maximizing properties near local minima of the Kirchhoff-Routh function.

Contribution

It introduces a variational method to construct steady vortex patches with opposite signs and proves their stability and energy-maximizing nature.

Findings

Vortex patches concentrate at local minima of the Kirchhoff-Routh function.

Steady solutions are local energy maximizers among isovortical patches.

Constructed solutions are stable and unique in their energy-maximizing class.

Abstract

In this paper we consider steady vortex flows for the incompressible Euler equations in a planar bounded domain. By solving a variational problem for the vorticity, we construct steady double vortex patches with opposite signs concentrating at a strict local minimum point of the Kirchhoff-Routh function with $k=2$. Moreover, we show that such steady solutions are in fact local maximizers of the kinetic energy among isovortical patches, which correlates stability to uniqueness.