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 $\psi_\varepsilon$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2004.13512
- arXiv:
- arXiv:2004.13512
- Bibcode:
- 2020arXiv200413512C
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- 38 pages