A Criterion for the Equivalence of the Birkhoff-Rott and Euler Descriptions of Vortex Sheet Evolution

Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show that initially smooth self-avoiding vortex sheet remains smooth for all times under the alpha-regularized dynamics, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure.

Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-alpha equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-alpha equatio…

非衰减、非周期涡片的速度场与Birkhoff-Rott积分

The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic vortex sheets

针对非衰减、非周期涡片问题，本文提出统一的Birkhoff-Rott积分公式，推广了衰减与周期情形，并验证其对应不可压缩无旋速度场且切向速度存在跳跃，同时给出实例并指出方法适用于三维流体。

Co-rotating and traveling vortex sheets for the 2D incompressible Euler equation

We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.

对称对数螺旋涡片的线性不稳定性

Linear instability of symmetric logarithmic spiral vortex sheets

本文研究对称对数螺旋涡片（Alexander螺线，$M\geq 3$支）作为Birkhoff-Rott方程解的线性不稳定性，通过在对数变量中分析傅里叶模式，证明其在$L^\infty$意义下存在随时间多项式增长的不稳定解。

Cusp formation for a nonlocal evolution equation

In this paper, we introduce a nonlocal evolution equation inspired by the Córdoba-Córdoba-Fontelos nonlocal transport equation. The Córdoba-Córdoba-Fontelos equation can be regarded as a model for the 2D surface quasigeostrophic equation or the Birkhoff-Rott equation. We prove blowup in finite time, and more importantly, investigate conditions under which the solution forms a cusp in finite time.