旋转涡片：刚性、分岔与统一结构
Rotating vortex patches: rigidity, bifurcation, and unified structures
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- New York University Abu Dhabi（纽约大学阿布扎比分校）
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中文总结 AI 辅助
该专著系统阐述旋转涡片理论，通过柯西积分与共形映射统一分析，并基于分岔理论构建统一框架，涵盖欧拉及多种主动标量模型。
中文摘要 AI 辅助
本专著系统阐述了二维欧拉方程及相关主动标量模型中旋转涡片（即V态）理论的若干经典与近期进展。其目的兼具阐述性与结构性：我们重新审视该主题的一些基础结果，提供详细证明与替代表述，并在统一的分析框架内呈现近期成果。我们首先介绍欧拉涡片及刚性旋转解的轮廓动力学表述。随后，我们发展两种互补的V态方程方法，分别基于柯西积分与共形映射，并讨论它们与位势理论和Faber多项式的联系。这些工具被用于重新审视经典例子，包括Rankine涡与Kirchhoff椭圆，以及刚性与分类结果。相当篇幅致力于通过分岔理论构造非圆形V态的Burbea构造。最后部分为一大类不可压缩主动标量方程发展统一的旋转涡片方法。相互作用核的结构性质，特别是完全单调性及由此产生的谱分解，产生了一个统一的分岔框架，涵盖欧拉、广义表面准地转、准地转浅水及相关模型。
英文摘要
This monograph presents a systematic account of several classical and recent developments in the theory of rotating vortex patches, or V-states, for the two-dimensional Euler equations and related active scalar models. Its purpose is both expository and structural: we revisit some of the foundational results of the subject, provide detailed proofs and alternative formulations, and present more recent results within a common analytical framework. We begin with Euler vortex patches and the contour dynamics formulation of rigidly rotating solutions. We then develop two complementary approaches to the V-state equation, based on Cauchy integrals and conformal mappings, and discuss their connections with potential theory and Faber polynomials. These tools are used to revisit classical examples, including Rankine vortices and Kirchhoff ellipses, as well as rigidity and classification results. A substantial part is devoted to Burbea construction of noncircular V-states via bifurcation theory. The final part develops a unified approach to rotating patches for a broad class of incompressible active scalar equations. Structural properties of the interaction kernel, in particular complete monotonicity and the resulting spectral factorization, yield a common bifurcation framework encompassing the Euler, generalized surface quasi-geostrophic, quasi-geostrophic shallow-water, and related models.
