二维环形区域内的定常非均匀不可压缩 Euler 方程的刚性——献给陈化教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 819-839.

二维环形区域内的定常非均匀不可压缩 Euler 方程的刚性——献给陈化教授 70 寿辰

王天怡^*(), 余正阳()

武汉理工大学数学与统计学院武汉 430070

收稿日期:2026-03-13 修回日期:2026-03-20 出版日期:2026-04-26 发布日期:2026-04-27
通讯作者: 王天怡 E-mail:tianyiwang@whut.edu.cn;347460@whut.edu.cn
作者简介:余正阳, Email：347460@whut.edu.cn
基金资助:
国家自然科学基金(12371223)

Rigidity of Steady Inhomogeneous Incompressible Euler Equations in Two-Dimensional Annular Domains

Tianyi Wang^*(), Zhengyang Yu()

School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070

Received:2026-03-13 Revised:2026-03-20 Online:2026-04-26 Published:2026-04-27
Contact: Tianyi Wang E-mail:tianyiwang@whut.edu.cn;347460@whut.edu.cn
Supported by:
NSFC(12371223)

摘要/Abstract

摘要：

该文研究二维环形区域内的定常非均匀不可压缩 Euler 流的刚性问题. 在流体无驻点且满足滑移边界条件的前提下, 对无界区域和去心区域分别附加无穷远处和原点附近的渐近性条件, 证明了: 光滑流体运动必为环形剪切流, 将此前均匀情形的刚性定理成功推广至非均匀情形. 首先通过建立流线与流函数梯度轨迹的几何性质, 将原方程转化为含梯度项的半线性椭圆方程. 进而, 运用移动平面法, 在相应区域中建立比较原理, 由此证明流函数与流线的径向对称性. 最后, 对于自由边值问题, 证明了非均匀情形的 Serrin 型定理, 并在此基础上建立了接触间断解的刚性定理.

关键词: Euler 方程, 环流, 半线性椭圆方程, 自由边界问题

Abstract:

This paper considers the rigidity of steady inhomogeneous incompressible Euler flows in two-dimensional annular domains. Under the assumption of no stagnation points and the slip boundary condition, with additional asymptotic conditions at infinity for unbounded domains and near the origin for punctured domains, the smooth fluids are proved to be circular shear flows, which extends the rigidity theorem for the homogeneous case to the inhomogeneous case. First, by establishing geometric properties of streamlines and the gradient of the stream function, the original system is transferred to a semilinear elliptic equation depending on the gradient terms. Then, by the moving plane method, the comparison principles are established in the corresponding domains, from which the radial symmetry properties of the stream function and streamlines are derived. Finally, for free boundary problems, a Serrin-type theorem for the inhomogeneous case is proved, based on which the rigidity theorem for contact discontinuity solutions is established.

Key words: Euler equations, annular flow, semilinear elliptic equations, free boundary problem

中图分类号:

O175.29

王天怡, 余正阳. 二维环形区域内的定常非均匀不可压缩 Euler 方程的刚性——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 819-839.

Tianyi Wang, Zhengyang Yu. Rigidity of Steady Inhomogeneous Incompressible Euler Equations in Two-Dimensional Annular Domains[J]. Acta mathematica scientia,Series A, 2026, 46(2): 819-839.

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