Steady Contiguous Vortex-Patch Dipole Solutions of the 2D Incompressible Euler Equation

TL;DR

This paper constructs the first steady vortex-patch dipole solutions for the 2D incompressible Euler equations, using a novel fixed-point method to determine the boundary of the vortex patches.

Contribution

It introduces a new fixed-point approach to rigorously construct steady vortex-patch dipole solutions, extending the understanding of vortex structures in fluid dynamics.

Findings

First steady vortex-patch dipole solutions constructed

Patch boundary determined as a fixed point of a nonlinear map

Properties and smoothness of the vortex boundary established

Abstract

We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known Lamb-Chaplygin dipole. Our construction is based on a novel fixed-point approach that determines the patch boundary as the fixed point of a certain nonlinear map. Smoothness and other properties of the patch boundary are also obtained.