论文标题
在有限域中2D稳定Euler流的涡流局部唯一性
Local uniqueness of vortices for 2D steady Euler flow in a bounded domain
论文作者
论文摘要
我们在一个有限的简单连接域中研究2D Euler方程，并建立流的局部唯一性，其流函数$ψ_\ Varepsilon $满足\ begin {equination*} \ begin {case} - \ varepsilon^2Δψ_\ varepsilon = \ sum \ limits_ {i = 1}^k \ mathbf1_ {b_Δ（z__ {0，i}）}（ψ_\arepsilon-μ__ ω， ψ_\ varepsilon = 0，\ \ \ \＆\ text {on} \ω， \ end {cases} \ end {equation*}带有$ \ varepsilon \ to 0^+$涡流的比例参数，$γ\ in（0，\ infty）$，$ω\ subset \ subset \ subbb r^2 $ i^{\ text {th}} $ vortex和$μ_ {\ varepsilon，i} $ flux常数未经处方。我们的证明是通过对$ψ_\ Varepsilon $和Pohozaev身份技术的渐近行为的详细描述来实现的。对于$ k = 1 $，我们证明了$ l^p $ norm中相应涡度的非线性稳定性，前提是$ z_ {0,1} $是罗宾功能的非排分最小点。该稳定性结果可以推广到$ k \ ge 2 $，而$（z_ {0,1}，\ cdots，z_ {0，k}）\ inω^k $是kirchhoff-routh函数的非脱位最小点。
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.
