A Liouville Theorem for the Euler Equations in the Plane
A Liouville Theorem for the Euler Equations in the Plane
复制标题
平面上欧拉方程的刘维尔定理
DOI：
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发表时间：
2017
影响因子：
2.5
通讯作者：
N. Nadirashvili
中科院分区：
文献类型：
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作者：
F. Hamel;N. Nadirashvili
本文研究的是二维平面上无驻点的理想不可压缩流体的有界稳定流的定性性质 R2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt}例如{文档}$${mathbb{R}^2}$$end{文档}。我们证明任何这样的流动都是剪切流，也就是说，它平行于某个常数向量。这个刘维尔型结果的证明首先基于对流函数水平曲线的几何性质的研究，其次基于对大球中流动参数的至多对数增长的一些估计的推导。这些估计得出的结论是，流动的流线都是平行线。
This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane R2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathbb{R}^2}$$end{document}. We show that any such flow is a shear flow, that is, it is parallel to some constant vector. The proof of this Liouville-type result is firstly based on the study of the geometric properties of the level curves of the stream function and secondly on the derivation of some estimates on the at-most-logarithmic growth of the argument of the flow in large balls. These estimates lead to the conclusion that the streamlines of the flow are all parallel lines.