SYMMETRY IN STATIONARY AND UNIFORMLY ROTATING SOLUTIONS OF ACTIVE SCALAR EQUATIONS
SYMMETRY IN STATIONARY AND UNIFORMLY ROTATING SOLUTIONS OF ACTIVE SCALAR EQUATIONS
复制标题
DOI：
10.1215/00127094-2021-0002
复制
发表时间：
2021-09-15
影响因子：
2.5
通讯作者：
Yao, Yao
中科院分区：
文献类型：
--
作者：
Gomez-Serrano, Javier;Park, Jaemin;Yao, Yao
我们研究了二维Euler方程和gSQG方程的静止解和均匀旋转解的径向对称性，包括光滑解和分片解。对于二维Euler方程，我们证明了任何具有紧支集和非负涡度的光滑定常解必须是径向的，而不需要对支集或水平集的连通性作任何假设。对于二维欧拉方程中的补丁设置，我们表明，每个均匀旋转的补丁D角速度Ω = 1/82必须是径向的，其中两个边界是尖锐的。对于gSQG方程，我们得到了类似的对称性结果，Ω = Ω（α）。(with边界是尖锐的），在额外的假设下，补丁是简单连接的。这些结果解决了几个悬而未决的问题所提出的Hmidi，德拉Hoz，Hassainia，和Mateu均匀旋转补丁。沿着的方式，我们关闭了一个问题的Choksi，Neumayer和Topaloglu的超定问题的分数拉普拉斯，这可能是独立的利益。主要的新思想来自变分法的观点。
We study the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler and gSQG equations, both in the smooth setting and the patch setting. For the 2D Euler equation, we show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial, without any assumptions on the connectedness of the support or the level sets. For the 2D Euler equation in the patch setting, we show that every uniformly rotating patch D with angular velocity Omega = 1/82 must be radial, where both bounds are sharp. For the gSQG equation, we obtain a similar symmetry result for Omega = Omega(alpha) . (with the bounds being sharp), under the additional assumption that the patch is simply connected. These results settle several open questions posed by Hmidi, de la Hoz, Hassainia, and Mateu on uniformly rotating patches. Along the way, we close a question by Choksi, Neumayer, and Topaloglu on overdetermined problems for the fractional Laplacian, which may be of independent interest. The main new ideas come from a calculus-of-variations point of view.