论文标题
2D Euler方程的紧凑型稳定解的对称结果
Symmetry results for compactly supported steady solutions of the 2D Euler equations
论文作者
论文摘要
在本文中，我们证明了2D Euler方程的紧凑型稳定解的对称性。假设$ω= \ {x \ in \ mathbb {r}^2：\ u（x）\ neq 0 \} $是一个环形域，我们证明了流的流线是圆形的。如果我们在$ \ partialω$上对$ u $施加规律性和非平稳性假设，我们还可以删除$ω$的拓扑条件。该证据使用表明，相应的流函数解决了边界处的椭圆形半连接问题$ -Δϕ = f（ϕ）$，$ \ nabla ϕ = 0 $。我们研究的主要困难之一是$ f $不是在边界值附近的Lipschitz连续。但是，$ f（ϕ）$在边界值上消失，然后我们可以应用F. Brock的局部对称性结果来得出结论。 在情况下，$ \partial_νu \ neq 0 $ at $ \partialΩ$不可能。在这种情况下，尽管可能缺乏$ f $的规律性，但我们能够使用移动平面方案表现出对称性。我们认为，这种结果本身就是有趣的，并且将被陈述和证明，以实现更高的维度。该证明需要研究具有单数系数的椭圆线性算子的最大原理，HOPF引理和锯齿蛋白角引理。
In this paper we prove symmetry of compactly supported steady solutions of the 2D Euler equations. Assuming that $Ω= \{x \in \mathbb{R}^2:\ u(x) \neq 0\}$ is an annular domain, we prove that the streamlines of the flow are circular. We are also able to remove the topological condition on $Ω$ if we impose regularity and nondegeneracy assumptions on $u$ at $\partial Ω$. The proof uses that the corresponding stream function solves an elliptic semilinear problem $-Δϕ= f(ϕ)$ with $\nabla ϕ=0$ at the boundary. One of the main difficulties in our study is that $f$ is not Lipschitz continuous near the boundary values. However, $f(ϕ)$ vanishes at the boundary values and then we can apply a local symmetry result of F. Brock to conclude. In the case $\partial_ν u \neq 0$ at $\partial Ω$ this argument is not possible. In this case we are able to use the moving plane scheme to show symmetry, despite the possible lack of regularity of $f$. We think that such result is interesting in its own right and will be stated and proved also for higher dimensions. The proof requires the study of maximum principles, Hopf lemma and Serrin corner lemma for elliptic linear operators with singular coefficients.
