Steady vortex patches near a rotating flow with constant vorticity in a planar bounded domain

TL;DR

This paper constructs steady vortex patch solutions to the 2D Euler equations in a bounded domain, where patches concentrate near maximum points of a related elliptic problem as their total vorticity diminishes.

Contribution

It provides a method to generate steady vortex patches concentrating at specific points in a bounded domain, extending understanding of vortex structures in fluid dynamics.

Findings

Vortex patches can be constructed near maximum points of a related elliptic problem.

Vortex patches shrink to specified points as total vorticity approaches zero.

The solutions are explicitly characterized by the characteristic functions of shrinking sets.

Abstract

In this paper, we study steady vortex patch solutions to the incompressible Euler equations in a planar bounded domain $D$. Let $\psi_0$ be the solution of the elliptic problem $-\Delta \psi _{0} =1$ in $D$; $\psi_0=0$ on $\partial D$. We prove that for any finite collection of isolated maximum points of $\psi_0$, say $\{x_1,\cdot\cdot\cdot,x_k\},$ and any $k$-tuple $\vec{\kappa}=(\kappa_1,\cdot,\cdot,\cdot,\kappa_k)$ with $\kappa_i>0$ and $|\vec{\kappa}|:=\sum_{i=1}^k\kappa_i<<1,$ there exists a steady solution of the Euler equations such that the vorticity has the form $\omega^{\vec{\kappa}}=1-I_{\cup_{i=1}^k A^{\vec{\kappa}}_i}$, where $I$ denotes the characteristic function, $|A^{\vec{\kappa}}_i|=\kappa_i$ and $A^{\vec{\kappa}}_i$ "shrinks" to $x_i$ as $|\vec{\kappa}|\to 0$.