Steady vortex patches with opposite rotation directions in a planar ideal fluid

Abstract

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