Lamb偶极子在无符号条件下的稳定性
Lamb dipole stability without sign condition
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中文总结 AI 辅助
本文针对半平面二维不可压缩Euler方程，在涡量无符号条件下证明了Lamb偶极子的Lyapunov稳定性，通过拉格朗日自举方法处理混合符号涡量带来的加权范数增长难题，为更复杂涡旋构型稳定性分析奠定基础。
中文摘要 AI 辅助
对于半平面上的二维不可压缩Euler方程，我们在涡量无符号条件下建立了Lamb偶极子的Lyapunov稳定性。主要挑战源于混合符号涡量的$x_{2}$加权范数的增长，而在单符号涡量情形下，该范数是守恒的并负责偶极子的稳定性。这一困难通过拉格朗日自举论证来处理，该论证基于将解仔细划分为若干部分并分析它们之间的相互作用，其中最微妙的相互作用涉及涡量在垂直方向漂移至无穷远。我们期望这里发展的策略将为更一般涡旋构型的稳定性分析提供起点，包括在混合符号涡量情形下多个涡旋偶极子之间以及多个涡环之间的相互作用。
英文摘要
For the 2D incompressible Euler equations on the half-plane, we establish Lyapunov stability of the Lamb dipole without sign condition on the vorticity. The main challenge arises from growth of the $x_{2}$ weighted norm for mixed sign vorticities, whereas this norm is conserved and responsible for dipole stability in the single signed vorticity case. This difficulty is handled by a Lagrangian bootstrap argument, based on carefully dividing the solution into several pieces and analyzing interactions among them, with the most delicate interaction involving vorticity drifting to infinity in the vertical direction. We expect that the strategy developed here will provide a starting point for the stability analysis of more general vortex configurations, including interactions among multiple vortex dipoles and among multiple vortex rings in settings involving mixed-sign vorticity.
发表机构
- Osaka Metropolitan University（大阪公立大学）
- Ulsan National Institute of Science and Technology（蔚山科学技术院）
- Korea Institute for Advanced Study（韩国高等研究院）
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。
