Circular flows for the Euler equations in two-dimensional annular domains, and related free boundary problems
Circular flows for the Euler equations in two-dimensional annular domains, and related free boundary problems
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二维环形域中欧拉方程的环形流以及相关的自由边界问题
DOI：
10.4171/jems/1177
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发表时间：
2019
影响因子：
2.6
通讯作者：
N. Nadirashvili
中科院分区：
文献类型：
--
作者：
F. Hamel;N. Nadirashvili
本文考虑了二维有界环空、外圆域、穿孔盘和穿孔平面中的稳定欧拉流。我们总是假设刚性壁面边界条件。我们证明了，如果流动没有任何滞止点，并且在外域无穷远处，在穿孔盘或穿孔面中心处满足进一步的条件，则流动是圆的，即流线是同心圆。换句话说，流继承了区域的径向对称性。通过研究流的运动轨迹和流函数梯度的正交运动轨迹，证明了流函数在整个域上满足半线性椭圆方程。在外部或穿孔区域，将移动平面的方法应用于位于流的流线之间的一些近似圆形区域，并通过极限论证证明了流函数的径向对称性。本文还在具有自由边界的单连通或双连通有界区域上得到了两个serrin型结果。在这里，进一步假设流动在边界的每个连接分量上具有恒定范数，然后证明域是圆盘或环空。
In this paper, we consider steady Euler flows in two-dimensional bounded annuli, as well as in exterior circular domains, in punctured disks and in the punctured plane. We always assume rigid wall boundary conditions. We prove that, if the flow does not have any stagnation point, and if it satisfies further conditions at infinity in the case of an exterior domain or at the center in the case of a punctured disk or the punctured plane, then the flow is circular, namely the streamlines are concentric circles. In other words, the flow then inherits the radial symmetry of the domain. The proofs are based on the study of the trajectories of the flow and the orthogonal trajectories of the gradient of the stream function, which is shown to satisfy a semilinear elliptic equation in the whole domain. In exterior or punctured domains, the method of moving planes is applied to some almost circular domains located between some streamlines of the flow, and the radial symmetry of the stream function is shown by a limiting argument. The paper also contains two Serrin-type results in simply or doubly connected bounded domains with free boundaries. Here, the flows are further assumed to have constant norm on each connected component of the boundary and the domains are then proved to be disks or annuli.