arXiv 2004.135122020-04math.AP

二维定常有界Euler流中涡的局部唯一性

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

Daomin Cao, Weilin Yu, Changjun Zou

备注：: 38 pages

摘要：

本文研究了有界单连通区域上的二维Euler方程，建立了流函数$ψ_\varepsilon$满足\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*}的流动的局部唯一性，其中$\varepsilon\to 0^+$为涡的尺度参数，$γ\in(0,\infty)$，$Ω\subset \mathbb R^2$为有界单连通Lipschitz区域，$z_{0,i}\inΩ$为$i^{\text{th}}$涡的极限位置，$μ_{\varepsilon,i}$为未指定的通量常数.我们的证明是通过详细描述的渐近行为的$ψ_\varepsilon$和Pohozaev身份技术。对于$k=1$，只要$z_{0,1}$是Robin函数的非退化极小点，我们证明了相应涡量在$L^p$范数下的非线性稳定性.这个稳定性结果可以推广到$k\ge 2$和$(z_{0,1},\cdots,z_{0,k})\in Ω^k$是Kirchhoff-Routh函数的非退化极小点的情况。

英文摘要：

We study the 2D Euler equation in a bounded simply-connected domain, and establish the local uniqueness of flow whose stream function $ψ_\varepsilon$ 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 $\varepsilon\to 0^+$ the scale parameter of vortices, $γ\in(0,\infty)$, $Ω\subset \mathbb R^2$ a bounded simply connected Lipschitz domain, $z_{0,i}\inΩ$ the limiting location of $i^{\text{th}}$ vortex, and $μ_{\varepsilon,i}$ the flux constants unprescribed. Our proof is achieved by a detailed description of asymptotic behavior for $ψ_\varepsilon$ 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 $(z_{0,1},\cdots,z_{0,k})\in Ω^k$ being a non-degenerate minimum point of the Kirchhoff-Routh function.

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摘要：

英文摘要：

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