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 ${\omega }^{\vec{\kappa }}=1-I_{{\cup }_{i=1}^k{A}_i^{\vec{\kappa }}}$, where $I$ denotes the characteristic function, $|A_i^{\vec{\kappa }}|={\kappa }_i$ and $A_i^{\vec{\kappa }}$ "shrinks" to $x_i$ as $|\vec{\kappa }|\to 0$.

15 pages