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

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 $|\vec{\kappa}|\to 0$.