S. Gustafson, I. M. Sigal
We study the linearized stability of n-vortex solutions of the magnetic Ginzburg-Landau (or Abelian-Higgs) equations. We prove that the fundamental vortices (n=1,-1) are stable for all values of the coupling constant, k, and we prove that the higher-degree vortices (|n| > 1) are stable for k < 1 and unstable for k > 1. This resolves a long-standing conjecture.
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In the analysis of the long-time behaviour of two-dimensional incompressible viscous fluids, Oseen vortices play a major role as attractors of any homogeneous solution with integrable initial vorticity. As a first step in the study of the density-dependent case, the present paper establishes the asymptotic stability of Oseen vortices for slighty inhomogeneous fluids with respect to localized regular perturbations.
We study Ginzburg-Landau equations for a complex vector order parameter. We consider the Dirichlet problem in the disk in the plane with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find that the stability of the degree-one equivariant solution depends on both the Ginzburg-Landau parameter as well as the sign of the interaction term in the energy.
We study solutions of Ginzburg-Landau-type evolution equations (both dissipative and Hamiltonian) with initial data representing collections of widely-spaced vortices. We show that for long times, the solutions continue to describe collections of vortices, and we identify (to leading order in the vortex separation) the dynamical system describing the motion of the vortex centres (effective dynamics).
We study vortex nucleation for minimizers of a Ginzburg-Landau energy with discontinuous constraint. For applied magnetic fields comparable with the first critical field of vortex nucleation, we determine the limiting vorticities.
This paper is devoted to an analysis of vortex-nucleation for a Ginzburg-Landau functional with discontinuous constraint. This functional has been proposed as a model for vortex-pinning, and usually accounts for the energy resulting from the interface of two superconductors. The critical applied magnetic field for vortex nucleation is estimated in the London singular limit, and as a by-product, results concerning vortex-pinning and boundary conditions on the interface are obtained.
Burgers vortices are stationary solutions of the three-dimensional Navier-Stokes equations in the presence of a background straining flow. These solutions are given by explicit formulas only when the strain is axisymmetric. In this paper we consider a weakly asymmetric strain and prove in that case that non-axisymmetric vortices exist for all values of the Reynolds number. In the limit of large Reynolds numbers, we recover the asymptotic results of Moffatt, Kida and Ohkitani (1994). We also show that the asymmetric vortices are stable with respect to localized two-dimensional perturbations.
In this paper we establish rigorously that the family of Burgers vortices of the three-dimensional Navier-Stokes equation is stable for small Reynolds numbers. More precisely, we prove that any solution whose initial condition is a small perturbation of a Burgers vortex will converge toward another Burgers vortex as time goes to infinity, and we give an explicit formula for computing the change in the circulation number (which characterizes the limiting vortex completely.) We also give a rigorous proof of the existence and stability of non-axisymmetric Burgers vortices provided the Reynolds nu…
Asymptotic linear stability of columnar vortices driven by Coriolis force
本文研究了柯氏力驱动的三维轴对称欧拉方程中柱状涡旋的渐近线性稳定性，通过构造扭曲傅里叶基底和分解c-ξ平面区域，揭示了涡旋的稳定性机制。
We investigate spectral stability of vortex solutions of the Gross-Pitaevskii equation, a mean-field approximation for Bose-Einstein condensates (BEC) in an effectively two-dimensional axisymmetric harmonic trap. We study eigenvalues of the linearization both rigorously and through computation of the Evans function, a sensitive and robust technique whose use we justify mathematically. The absence of unstable eigenvalues is justified a posteriori through use of the Krein signature of purely imaginary eigenvalues, which also can be used to significantly reduce computational effort. In particular…
The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the whole plane, and assume that the initial vorticity is a finite measure. This general setting includes vortex patches, vortex sheets, and point vortices. We first prove the existence of a unique global solution, for any value of the viscosity parameter, and we investigate its long-time behavior. We next consider the particular case where the initial flow is a…
Existence and stability of capillary-gravity wave-borne vortices
本文证明有限深二维水体中毛细重力波可携带点涡旋（亚临界弗劳德数、超临界邦德数下存在且条件轨道稳定）及空腔涡旋，波速均为O(1)。
Vorticity plays a prominent role in the dynamics of incompressible viscous flows. In two-dimensional freely decaying turbulence, after a short transient period, evolution is essentially driven by interactions of viscous vortices, the archetype of which is the self-similar Lamb-Oseen vortex. In three dimensions, amplification of vorticity due to stretching can counterbalance viscous dissipation and produce stable tubular vortices. This phenomenon is illustrated in a famous model originally proposed by Burgers, where a straight vortex tube is produced by a linear uniaxial strain field. In real f…
Long time confinement of multiple concentrated vortices
研究流体中多重近圆集中涡的稳定性，证明在分离状态下，涡体在时间尺度上保持集中，提出新的对数相互作用能稳定性估计。
The mathematical theory of hydrodynamic stability started in the middle of the 19th century with the study of model examples, such as parallel flows, vortex rings, and surfaces of discontinuity. We focus here on the equally interesting case of columnar vortices, which are axisymmetric stationary flows where the velocity field only depends on the distance to the symmetry axis and has no component in the axial direction. The stability of such flows was first investigated by Kelvin in 1880 for some particular velocity profiles, and the problem benefited from important contributions by Rayleigh in…
Equivariant stability of vortices in Manton's Chern-Simons-Schrödinger system on the hyperbolic plane
本文在双曲平面上研究Manton的Chern-Simons-Schrödinger系统，通过非线性Darboux变换和椭圆算子分析，证明了任意度数m的等变涡旋的渐近稳定性。
Arnold's variational principle and its application to the stability of planar vortices
该研究围绕二维不可压缩Euler方程稳态解的Arnold稳定性判据相关变分原理展开，明确了对应二次型可用于稳定性分析的条件，给出Oseen涡旋的新几何稳定性证明，并分析了无黏情形及高斯涡旋的黏性影响。
Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability under either planar or axisymmetric perturbations. The purpose of this paper is to show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particu…
Using a Ginzburg-Landau model, we study the vortex behavior of a rectangular thin film superconductor subjected to an applied current fed into a portion of the sides and an applied magnetic field directed orthogonal to the film. Through a center manifold reduction we develop a rigorous bifurcation theory for the appearance of periodic solutions in certain parameter regimes near the normal state. The leading order dynamics yield in particular a motion law for kinematic vortices moving up and down the center line of the sample. We also present computations that reveal the co-existence and period…
Boundary vortices in the presence of a magnetic field
本文研究外加面内恒定磁场下薄铁磁薄膜中边界涡旋主导的微磁能量行为，提出重整化能量的新定义并证明其二阶Γ收敛及能量集中现象，同时给出数值模拟验证。
We consider a homogenization problem for magnetic Ginzburg-Landau functional in domains with large number of small holes. For sufficiently strong magnetic field, a large number of vortices is formed and they are pinned by the holes. We establish a scaling relation between sizes of holes and the magnitude of the external magnetic field when pinned vortices are multiple and their homogenized density is described by a hierarchy of variational problems. This stands in sharp contrast with homogeneous superconductors, where all vortices are known to be simple. The proof is based on $Γ$-convergence a…
Burgers vortices are explicit stationary solutions of the Navier-Stokes equations which are often used to describe the vortex tubes observed in numerical simulations of three-dimensional turbulence. In this model, the velocity field is a two-dimensional perturbation of a linear straining flow with axial symmetry. The only free parameter is the Reynolds number $Re = Γ/ν$, where $Γ$ is the total circulation of the vortex and $ν$ is the kinematic viscosity. The purpose of this paper is to show that Burgers vortex is asymptotically stable with respect to general three-dimensional perturbations, fo…
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