# Steady vortex patches near a nontrivial irrotational flow

## TL;DR

This paper investigates steady vortex patches in an ideal fluid within a bounded domain, demonstrating their concentration near extremum points of a harmonic function associated with irrotational flow, including multiple vortex configurations.

## Contribution

It establishes the existence and asymptotic behavior of steady vortex patches near extremum points of harmonic functions in bounded domains, extending understanding of vortex configurations.

## Key findings

- Vortex patches concentrate near extremum points of harmonic functions.
- Existence of steady multiple vortex patches approaching strict extremum points.
- Analysis of the limiting behavior of vortex patches via minimization problems.

## Abstract

In this paper, we study the vortex patch problem in an ideal fluid in a planar bounded domain. By solving a certain minimization problem and studying the limiting behavior of the minimizer, we prove that for any harmonic function $q$ corresponding to a nontrivial irrotational flow, there exists a family of steady vortex patches approaching the set of extremum points of $q$ on the boundary of the domain. Furthermore, we show that each finite collection of strict extreme points of $q$ corresponds to a family of steady multiple vortex patches approaching it.