发表机构
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Sciences, Dalian University of Technology（中国科学院数学与系统科学研究院; 大连理工大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对不可压缩欧拉方程，通过开发新的李雅普诺夫机制，证明了有界平面区域中近径向集中涡的形状与位置可全局受控，还建立了一对异号涡的类似结果。
AI 中文摘要
本文研究有界平面区域中不可压缩欧拉方程的近径向集中涡演化问题。我们证明：若单个涡初始时集中在区域Robin函数的严格局部极小点附近，且经平移后接近其对称递减重排，则其形状与位置在所有时间内均保持一致受控。我们还对一对异号涡在对应Kirchhoff-Routh函数的严格局部极小点附近建立了类似结果。区域与初始数据均未施加对称性，且初始数据无需接近任何定态。为证明这些结果，我们针对由欧拉方程支配的涡度演化，从此类初始数据出发开发了一种新的李雅普诺夫机制，提供涡形状与位置的定量控制。具体而言，我们将动能守恒、涡度等测度性与若干固定时间估计相结合以得到条件估计，再利用集值首出论证将其全局时间传播。
英文摘要
In this paper, we study the evolution of nearly radial concentrated vortices for the incompressible Euler equation in a bounded planar domain. We prove that if a single vortex is initially concentrated near a strict local minimum point of the Robin function of the domain and is close, up to translation, to its symmetric decreasing rearrangement, then both its shape and location remain uniformly controlled for all time. We also establish an analogous result for a pair of opposite-sign vortices near a strict local minimum point of the corresponding Kirchhoff--Routh function. No symmetry is imposed on the domain or the initial data, and the initial data need not be close to any steady state. To prove these results, we develop a new Lyapunov mechanism for the evolution of vorticity governed by the Euler equation starting from such initial data, providing quantitative control of both vortex shape and location. Specifically, we combine kinetic-energy conservation and vorticity equimeasurability with several fixed-time estimates to obtain a conditional estimate, and then use a set-valued first-exit argument to propagate it globally in time.
Comments40 pages
