Ken Abe
We construct traveling wave solutions to the 3d Euler equations by axisymmetric Beltrami fields with a non-constant proportionality factor. They form a vortex ring with nested invariant tori consisting of level sets of the proportionality factor.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Strong Beltrami fields have long played a key role in fluid mechanics and magnetohydrodynamics. In particular, they are the kind of stationary solutions of the Euler equations where one has been able to show the existence of vortex structures (vortex tubes and vortex lines) of arbitrarily complicated topology. On the contrary, there are very few results about the existence of generalized Beltrami fields, that is, divergence-free fields whose curl is the field itself times a non-constant function. In fact, generalized Beltrami fields (which are also stationary solutions to the Euler equations) …
Localization of Beltrami fields: global smooth solutions and vortex reconnection for the Navier-Stokes equations
本研究构造了贝尔特拉米场局部化得到的无散向量场，既用于构建三维Navier-Stokes方程临界空间大初值下的全局光滑解，也借助双曲临界点这一拓扑不变量给出了全空间上的涡重联解析示例。
Let S be a finite union of (pairwise disjoint but possibly knotted and linked) closed curves and tubes in the round sphere S^3 or in the flat torus T^3. In the case of the torus, S is further assumed to be contained in a contractible subset of T^3. In this paper we show that for any sufficiently large odd integer λthere exists a Beltrami field on S^3 or T^3 satisfying curl u = λu and with a collection of vortex lines and vortex tubes given by S, up to an ambient diffeomorphism.
We study desingularization of steady vortex rings in three-dimensional axisymmetric incompressible Euler fluids with swirl. Using the variational method, we construct a two-parameter family of steady vortex rings, which constitute a desingularization of the classical circular vortex filament, in several kinds of domains. The precise localization of the asymptotic singular vortex filament is shown to depend on the circulation and the velocity at far fields of the vortex ring and the geometry of the domains. We also discuss other qualitative and asymptotic properties of these vortices.
In this paper, we are concerned with nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler fluids. We focus on the case when the vorticity function has a simple discontinuity, which corresponding to a jump in vorticity at the boundary of the cross-section of the vortex ring. Using the vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in several kinds of domains. The precise localization of the asymptotic singular vortex filament is proved to depend on the circulati…
Uniqueness and stability of steady vortex rings for 3D incompressible Euler equation
本文通过变分原理和集中紧致性方法，证明了三维欧拉方程中特殊涡环的唯一性和非线性稳定性。
In this paper, we are concerned with nonlinear desingularization of steady vortex rings in $\mathbb{R}^3$ with given general nonlinearity $f$. Using the improved vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in the whole space. The requirements on $f$ are very general, and $f$ may have a simple discontinuity at the origin. Some qualitative and asymptotic properties are also established.
Global dynamics of a single vortex ring
本文研究了单涡环在三维不可压缩欧拉方程下的全局动力学，证明了涡度集中传播及涡丝猜想的首次全局验证，并揭示了涡丝支撑的线性时间伸展机制。
Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation.
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。
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.
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。
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)…
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。
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。
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。
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。
We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in R^3. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in R^3, we show that they can be transformed with a C^m-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted vortex tubes…
Recently, the Nash-style convex integration has been becoming the main scheme for the mathematical study of turbulence, and the main building block of it has been either Beltrami flow (finite mode) or Mikado flow (compactly supported in the physical side). On the other hand, in physics, it is observed that turbulence is composed of a hierarchy of scale-by-scale vortex stretching. Thus our mathematical motivation in this study is to find another type of building blocks accompanied by vortex stretching and scale locality (possibly finitely many Fourier modes). In this paper, we give a complete l…
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…
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.
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 $ε$.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。