单涡环的全局动力学

三维不可压缩欧拉方程的跳跃式涡环

Leapfrogging vortex rings for the 3-dimensional incompressible Euler equations

针对不可压缩三维欧拉方程，构造光滑解严格证明了亥姆霍兹观察到的同轴涡环反复穿越的跳跃式运动现象。

Desingularization of vortex rings and shallow water vortices by semilinear elliptic problem

Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and showed to tend asymptotically to singular vortex filaments. The construction is based on the asymptotic study of solutions to a semilinear elliptic problem.

稳态环形涡面

Steady Ring-Shaped Vortex Sheets

本文构造两相欧拉方程中带涡面的薄环形行波解，利用隐函数定理保证局部唯一性，并推广开尔文-希克斯公式以包含表面张力。

On the steady axisymmetric vortex rings for 3-D incompressible Euler flows

In this paper, we study nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler flows. We construct a family of steady vortex rings (with and without swirl) which constitutes a desingularization of the classical circular vortex filament in $\mathbb{R}^3$. The construction is based on a study of solutions to the similinear elliptic problem \begin{equation*} -\frac{1}{r}\frac{\partial}{\partial r}\Big(\frac{1}{r}\frac{\partialψ^\varepsilon}{\partial r}\Big)-\frac{1}{r^2}\frac{\partial^2ψ^\varepsilon}{\partial z^2}=\frac{1}{\varepsilon^2}\left(g(ψ^\varepsilon)…

小截面定常涡环的存在性、唯一性与稳定性

Existence, uniqueness and stability of steady vortex rings of small cross-section

本文针对理想流体中核心位涡为常数的小截面定常涡环，通过流函数在 Rankine 涡附近的泰勒展开、Lyapunov-Schmidt 约化及 Arnol'd 对偶变分原理，系统证明了其存在性、唯一性与非线性轨道稳定性，解决了一个长期未解问题。

三维欧拉方程中的时间周期性跳跃环 vortex 环

Time-periodic leapfrogging vortex rings in the 3D Euler equations

研究证明了三维不可压缩欧拉方程中时间周期性跳跃环 vortex 环的存在，首次严谨地实现了海姆霍兹在1858年提出的现象。通过构造两个相互作用的 vortex 纤维的去奇异化，得到近似同心环的哈密顿描述，克服了线性传输动力学中的奇异小分母问题，构造了非平凡的时间周期性解。

用几乎刚体近似涡环的模型

A model for the approximation of vortex rings by almost rigid bodies

本文提出用牛顿力学中几乎刚体的运动近似轴对称三维欧拉方程中的涡环，证明其可化为位置常微分方程，并在细丝极限下收敛于点涡系统，且能呈现跳跃式运动。

非线性尺度局部涡环几何变形的光滑欧拉流中的研究：通过贝叶斯优化与伴随方法

Nonlinear Scale-Local Geometric Deformations of Vortex Rings in Smooth Euler Flows via Bayesian Optimization and Adjoint Methods

本文通过贝叶斯优化与伴随方法，研究了涡环在光滑欧拉流中非线性尺度局部变形的机制，提出了一种新的波方程来描述 swirling particles 的轴线运动。

On the Cauchy problem for axi-symmetric vortex rings

We consider the classical Cauchy problem for the 3d Navier-Stokes equation with the initial vorticity $ω_0$ concentrated on a circle, or more generally, a linear combination of such data for circles with common axis of symmetry. We show that natural approximations of the problem obtained by smoothing the initial data satisfy good a-priori estimates which enable us to conclude that the original problem with the singular initial distribution of vorticity has a solution. We impose no restriction on the size of the initial data.

Generalized Adler-Moser Polynomials and Multiple vortex rings for the Gross-Pitaevskii equation

New finite energy traveling wave solutions with small speed are constructed for the three dimensional Gross-Pitaevskii equation \begin{equation*} iΨ_t= ΔΨ+(1-|Ψ|^2)Ψ, \end{equation*} where $Ψ$ is a complex valued function defined on ${\mathbb R}^3\times{\mathbb R}$. These solutions have the shape of $2n+1$ vortex rings, far away from each other. Among these vortex rings, $n+1$ of them have positive orientation and the other $n$ of them have negative orientation. The location of these rings are described by the roots of a sequence of polynomials with rational coefficients. The polynomials found…

Vanishing viscosity limit for concentrated vortex rings

We study the time evolution of a viscous incompressible fluid with axial symmetry without swirl, when the initial vorticity is very concentrated in $N$ disjoint rings. We show that in a suitable joint limit, in which both the thickness of the rings and the viscosity tend to zero, the vorticity remains concentrated in $N$ disjointed rings, each one of them performing a simple translation along the symmetry axis with constant speed.

Time evolution of vortex rings with large radius and very concentrated vorticity

We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on $N$ annuli of radii $\approx$ $r_0$ and thickness $ε$. We prove that when $r_0= |\log ε|^α, \,\, α>2$, the vorticity field of the fluid converges as $ε\to 0$ to the point vortex model, at least for a small but positive time. This result generalizes a previous paper that assumed a power law for the relation between $r_0$ and $ε$.

作为欧拉方程标度极限的跳蛙式涡环

Leapfrogging vortex rings as scaling limit of Euler Equations

该研究针对无旋轴对称不可压缩流体，证明当小厚度涡环的厚度趋于零时，其运动收敛于已知动力系统，且双大半径涡环情形下结果可覆盖多次超越，严格推导了跳蛙现象。