We study the 2D Euler equation in a bounded simply-connected domain, and establish the local uniqueness of flow whose stream function $\psi_\varepsilon$ satisfies \begin{ equation*} \begin{ cases} -\varepsilon^2\Delta \psi_\varepsilon=\sum\limits_{ i=1}^k \mathbf1_{ B_\delta(z_{ 0,i})}(\psi_\varepsilon-\mu_{ \varepsilon,i})_+^\gamma,\ \ \ & \text{ in} \ \Omega, \psi_\varepsilon=0,\ \ \ & \text{ on} \ \Omega, \end{ cases} \end{ equation*} with $\varepsilon\to 0^+$ 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 $\mu_{ \varepsilon,i}$ the flux constants unprescribed. Our proof is achieved by a detailed description of asymptotic behavior for $\psi_\varepsilon$ 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.
Categories: math.AP