Local uniqueness of vortices for 2D steady Euler flow in a bounded domain

Abstract

We study the 2D Euler equation in a bounded simply-connected domain, and establish the local uniqueness of flow whose stream function ${\psi }_{\epsilon }$ satisfies \begin{equation*} \begin{cases} -\varepsilon^2Δψ_\varepsilon=\sum\limits_{i=1}^k \mathbf1_{B_δ(z_{0,i})}(ψ_\varepsilon-μ_{\varepsilon,i})_+^γ,\ \ \ & \text{in} \ Ω, ψ_\varepsilon=0,\ \ \ & \text{on} \ Ω, \end{cases} \end{equation*} with $\epsilon \to 0^{+}$ the scale parameter of vortices, $\gamma \in (0,\infty )$, $\Omega \subset {\mathbb{R}}^2$ a bounded simply connected Lipschitz domain, $z_{0,i}\in \Omega$ the limiting location of $i^{\text{th}}$ vortex, and ${\mu }_{\epsilon ,i}$ the flux constants unprescribed. Our proof is achieved by a detailed description of asymptotic behavior for ${\psi }_{\epsilon }$ and Pohozaev identity technique. For $k=1$, we prove the nonlinear stability of corresponding vorticity in $L^p$ norm, provided $z_{0,1}$ is a non-degenerate minimum point of Robin function. This stability result can be generalized to the case $k\ge 2$, and $(z_{0,1},\cdots ,z_{0,k})\in {\Omega }^k$ being a non-degenerate minimum point of the Kirchhoff-Routh function.

38 pages