Sagun Chanillo, Michael Kiessling
For certain nonlinear elliptic PDE problems in two dimensions, the classical isoperimetric inequality produces a sharp inequality that violates a Pohozaev identity except for radial symmetric, decreasing solutions. A generalized version of this technique is used here to prove radial symmetry of curl-free Ginzburg-Landau vortices.
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This is a review about a series of results on vortices in the Ginzburg-Landau model of superconductivity on the one hand, and point patterns in Coulomb gases on the other hand, as well as the connections between the two topics.
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.
We study the asymptotic behavior of solutions for $n$-component Ginzburg-Landau equations as $\ve \to 0$. We prove that the minimizers converges locally in any $C^k$-norm to a solution of a system of generalized harmonic map equations.
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.
Entanglement of vortices in the Ginzburg--Landau equations for superconductors
本文构造了 Ginzburg--Landau 方程的解，证明了高温超导体中相邻涡旋线可相互纠缠，且能展现任意拓扑复杂性。
We consider the time-dependent 2D Ginzburg-Landau equation in the whole plane with terms modeling impurities and applied currents. The Ginzburg-Landau vortices are then subjected to three forces: their mutual repulsive Coulomb-like interaction, the applied current pushing them in a fixed direction, and the pinning force attracting them towards the impurities. The competition between the three is expected to lead to complicated glassy effects. We rigorously study the limit in which the number of vortices $N_ε$ blows up as the inverse Ginzburg-Landau parameter $ε$ goes to $0$, and we derive via …
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.
These are the lecture notes of a "Nachdiplomvorlesung" course taught at ETH Zurich in the Spring of 2013. They appeared in the EMS series Zurich Lectures in Advanced Mathematics.
We investigate two settings of Ginzburg-Landau posed on a manifold where vortices are unstable. The first is an instability result for critical points with vortices of the Ginzburg-Landau energy posed on a simply connected, compact, closed 2-manifold. The second is a vortex annihilation result for the Ginzburg-Landau heat flow posed on certain surfaces of revolution with boundary.
We study a complex Ginzburg-Landau equation in the plane, which has the form of a Gross-Pitaevskii equation with some dissipation added. We focus on the regime corresponding to well-prepared unitary vortices and derive their asymptotic motion law.
We prove some improved estimates for the Ginzburg-Landau energy (with or without magnetic field) in two dimensions, relating the asymptotic energy of an arbitrary configuration to its vortices and their degrees, with possibly unbounded numbers of vortices. The method is based on a localisation of the ``ball construction method" combined with a mass displacement idea which allows to compensate for negative errors in the ball construction estimates by energy ``displaced" from close by. Under good conditions, our main estimate allows to get a lower bound on the energy which includes a finite orde…
We consider the Ginzburg-Landau heat flow on the two-dimensional flat torus, starting from an initial data with a finite number of nondegenerate zeros -- but possibly very high initial energy. We show that the initial zeros are conserved and the flow rapidly enters a logarithmic energy regime, from which the evolution of vortices can be described by the works of Bethuel, Orlandi and Smets.
The anisotropic Ginzburg-Landau system \[ Δu+δ\, \nabla (\mathrm{div}\: u) +δ\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u\colon\mathbb R^2\to\mathbb R^2$ and $δ\in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescribed topological degree $d\leq -1$ as $|x|\to\infty$, for small values of the anisotropy parameter $|δ| < δ_0(d)$. Unlike the isotropic case $δ=0$, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzbu…
We consider the mixed Ginzburg-Landau flow that is supplemented with convective derivatives of the unknown function. We show that the associated vortex motion law is the mixed flow of the renormalized energy with new nonlinear forcing terms. These terms are uniquely determined by the extra terms in the initial PDE. Our proof relies on the assumption that the initial data are close to optimal.
It is proved that the vortices are attracted by impurities or inhomogeities in the superconducting materials. The strong H^1-convergence for the corresponding Ginzburg-Landau system is also proved.
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…
In the theory of $2D$ Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a $2D$ map $u$. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a $2D$ Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of $u$ at the boundar…
We consider the dissipative heat flow and conservative Gross-Pitaevskii dynamics associated with the Ginzburg-Landau energy posed on a Riemannian 2-manifold M. We show the limiting vortices of the solutions to these two problems evolve according to the gradient flow and Hamiltonian point-vortex flow respectively, associated with the renormalized energy on M. For the heat flow, we then specialize to the case where M is a sphere and study the limiting system of ODE's and establish an annihilation result. Finally, for the Ginzburg-Landau heat flow on a sphere, we derive some weighted energy ident…
In this paper, the two-component Ginzburg-Landau (TCGL) model, which is an important research object in mathematics and physics of two scalar fields with $U(1)\times U(1)$ symmetry, is studied in detail by the shooting method and the fixed point theorem, which contains two cases: single vacuum expectation value (1VEV) and two vacuum expectation values (2VEV). This represents two different physical states: superconductivity-normal state and superconductivity superconductivity state, and we prove the existence, uniqueness, asymptotic properties and quantization of the solutions to these two boun…
We study Ginzburg--Landau equations for a complex vector order parameter Psi=(psi_+,psi_-). We consider symmetric (equivariant) vortex solutions in the plane R^2 with given degrees n_\pm, and prove existence, uniqueness, and asymptotic behavior of solutions for large r. We also consider the monotonicity properties of solutions, and exhibit parameter ranges in which both vortex profiles |psi_+|, |psi_i| are monotone, as well as parameter regimes where one component is non-monotone. The qualitative results are obtained by means of a sub- and supersolution construction and a comparison theorem fo…
Nondegeneracy and Morse Index of Ginzburg--Landau Vortices
本文证明金兹堡-朗道方程的标准二度、三度涡旋解非退化并计算其莫尔斯指数，所用方法可推广至更高阶解的研究。
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 solutions of the 2D Ginzburg-Landau equation -Δu+\frac{1}{\ve^2}u(|u|^2-1)=0 subject to "semi-stiff" boundary conditions: the Dirichlet condition for the modulus, |u|=1, and the homogeneous Neumann condition for the phase. The principal result of this work shows there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small epsilon. For the Dirichlet bondary condition ("stiff" problem), the existence of stable solutions with vortices, whose energy blows up as epsilon goes to 0, is well known. By contr…
This paper consists of three results on pattern formation of Ginzburg-Landau $m$-armed vortex solutions and spiral waves in circular and spherical geometries. First, we completely describe the global bifurcation diagram of vortex equilibria. Second, we prove persistence of all bifurcation curves under perturbations of parameters, which yields the existence of spiral waves for the complex Ginzburg-Landau equation. Third, we explicitly construct the global attractor of $m$-armed vortex solutions. Our main tool is a new shooting method that allows us to prove hyperbolicity of vortex equilibria in…
We study the structure of vortex solutions in a Ginzburg-Landau system for two complex valued order parameters. We consider the Dirichlet problem in the disk in R^2 with symmetric, degree-one boundary condition, as well as the associated degree-one entire solutions in all of R^2. Each problem has degree-one equivariant solutions with radially symmetric profile vanishing at the origin, of the same form as the unique (complex scalar) Ginzburg-Landau minimizer. We find that there is a range of parameters for which these equivariant solutions are the unique locally energy minimizing solutions for …
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.
We study a variational model which combines features of the Ginzburg-Landau model in 2D and of the Mumford-Shah functional. As in the classical Ginzburg-Landau theory, a prescribed number of point vortices appear in the small energy regime; the model allows for discontinuities, and the energy penalizes their length. The novel phenomenon here is that the vortices have a fractional degree $1/m$ with $m\geq 2$ prescribed. Those vortices must be connected by line discontinuities to form clusters of total integer degrees. The vortices and line discontinuities are therefore coupled through a topolog…
We consider the variant of a stochastic parabolic Ginzburg-Landau equation that allows for the formation of point defects of the solution. The noise in the equation is multiplicative of the gradient type. We show that the family of the Jacobians associated to the solution is tight on a suitable space of measures. Our main result is the characterization of the limit points of this family. They are concentrated on finite sums of delta measures with integer weights. The singular set of the solution coincides with the points at which the delta measures are centered.
We study a class of solutions to the parabolic Ginzburg-Landau equation in dimension 2 or higher, with ill-prepared infinite energy initial data. We show that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke's weak formulation. Then, we prove that in the plane, point vortices do not move in the original time scale. These results extend the work of Bethuel, Orlandi and Smets [8, 9] for infinite energy data, they allow to consider the point vortices on a lattice (in dimension 2), or filament vortices of infinite length (in dimension 3).
We consider minimizers of a Ginzburg-Landau energy with a discontinuous and rapidly oscillating pinning term, subject to a Dirichlet boundary condition of degree $d > 0$. The pinning term models an unbounded number of small impurities in the domain. We prove that for strongly type II superconductor with impurities, minimizers have exactly d isolated zeros (vortices). These vortices are of degree 1 and pinned by the impurities. As in the standard case studied by Bethuel, Brezis and Hélein, the macroscopic location of vortices is governed by vortex/vortex and vortex/ boundary repelling effects. …
We study a mixed heat and Schrödinger Ginzburg-Landau evolution equation on a bounded two-dimensional domain with an electric current applied on the boundary and a pinning potential term. This is meant to model a superconductor subjected to an applied electric current and electromagnetic field and containing impurities. Such a current is expected to set the vortices in motion, while the pinning term drives them toward minima of the pinning potential and "pins" them there. We derive the limiting dynamics of a finite number of vortices in the limit of a large Ginzburg-Landau parameter, or $\ep \…
Let $Ω$ be a smooth bounded domain in $\mathbb{R}^2$. For $ε>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $-Δu=\frac{1}{ε^2}(1-|u|^2)u$ in $Ω$ such that on $\partial Ω$ u satisfies $|u|=1$ and $u\wedge \partial_νu=0$. These boundary conditions are called semi-stiff and are intermediate between the Dirichlet and the homogeneous Neumann boundary conditions. In order to construct such solutions we use a variational reduction method very similar to the one used by del Pino-Kowalczyk-Musso to find solutions of the Ginzburg-Landau equations with Dirichlet and homoge…
Mountain pass for the Ginzburg-Landau energy in a strip: solitons and solitonic vortices
研究在无限条带中应用相印刻的吉本斯-兰道能量临界点，发现临界宽度下孤子解为山峰解，临界宽度以上形成孤子涡，并探讨其在不同宽度下的极小值解行为。
On sections of complex line bundles over surfaces minimizing a Ginzburg-Landau energy
该研究将二维黎曼流形上切向量场的Ginzburg-Landau涡旋相关结果推广至复埃尔米特线丛场景，明确了球面上Q-张量及其更高秩类似物的涡旋位置。
Quantization effects for multi-component Ginzburg-Landau vortices
本文研究了 $n$ 分量 Ginzburg-Landau 方程，通过引入扩散常数证明其非平凡性，并将单分量结果推广至多分量，揭示了势能的量子化积分及扩散系数对非平凡解的关键作用。
We prove that in a certain asymptotic regime, solutions of the Gross-Pitaevskii equation converge to solutions of the incompressible Euler equation, and solutions to the parabolic Ginzburg-Landau equations converge to solutions of a limiting equation which we identify. We work in the setting of the whole plane (and possibly the whole three-dimensional space in the Gross-Pitaevskii case), in the asymptotic limit where $ε$, the characteristic lengthscale of the vortices, tends to $0$, and in a situation where the number of vortices $N$ blows up as $ε\to 0$. The requirements are that $N$ should b…
This paper studies questions related to the dynamic transition between local and global minimizers in the Ginzburg-Landau theory of superconductivity. We derive a heuristic equation governing the dynamics of vortices that are close to the boundary, and of dipoles with small inter vortex separation. We consider a small random perturbation of this equation, and study the asymptotic regime under which vortices nucleate.
We derive the asymptotical dynamical law for Ginzburg-Landau vortices in an inhomogeneous background density under the Schrödinger dynamics, when the Ginzburg-Landau parameter goes to zero. New ingredients involve across the cores lower bound estimates and approximations.
Dynamics of Ginzburg-Landau vortices for vector fields on surfaces
本文研究曲面向量场上Ginzburg-Landau涡旋的动力学，严格证明涡旋按重整化能量的梯度流运动，即ε→0时原能量梯度流的极限行为。
In this paper we prove novel lower bounds for the Ginzburg-Landau energy with or without magnetic field. These bounds rely on an improvement of the "vortex balls construction" estimates by extracting a new positive term in the energy lower bounds. This extra term can be conveniently estimated through a Lorentz space norm, on which it thus provides an upper bound. The Lorentz space $L^{2,\infty}$ we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results.
We study the variational convergence of a family of two-dimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity, as the Ginzburg-Landau parameter epsilon tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotat…
We consider the gradient flow of a Ginzburg-Landau functional of the type \[ F_\varepsilon^{\mathrm{extr}}(u):=\frac{1}{2}\int_M \left|D u\right|_g^2 + \left|\mathscr{S} u\right|^2_g +\frac{1}{2\varepsilon^2}\left(\left|u\right|^2_g-1\right)^2\mathrm{vol}_g \] which is defined for tangent vector fields (here $D$ stands for the covariant derivative) on a closed surface $M\subseteq\mathbb{R}^3$ and includes extrinsic effects via the shape operator $\mathscr{S}$ induced by the Euclidean embedding of~$M$. The functional depends on the small parameter $\varepsilon>0$. When $\varepsilon$ is small it…
We consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. The applied magnetic field varies smoothly and is allowed to vanish non-degenerately along a curve. Assuming that the strength of the applied magnetic field varies between two characteristic scales, and the Ginzburg-Landau parameter tends to $+\infty$, we determine an accurate asymptotic formula for the minimizing energy and show that the energy minimizers have vortices. The new aspect in the presence of a variable magnetic field is that the density of vortices in t…
We study a variational Ginzburg-Landau type model depending on a small parameter $\varepsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian manifold $S$. As $\varepsilon\to 0$, these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus $\mathfrak{g}$ of $S$ is different than $1$. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorti…
In this work, we study thin-film limits of the full three-dimensional Ginzburg-Landau model for a superconductor in an applied magnetic field oriented obliquely to the film surface. We obtain Gamma-convergence results in several regimes, determined by the asymptotic ratio between the magnitude of the parallel applied magnetic field and the thickness of the film. Depending on the regime, we show that there may be a decrease in the density of Cooper pairs. We also show that in the case of variable thickness of the film, its geometry will affect the effective applied magnetic field, thus influenc…
In this paper we study the time-dependent Ginzburg-Landau equations on a smooth, bounded domain $Ω\subset \Rn{2}$, subject to an electrical current applied on the boundary. The dynamics with an applied current are non-dissipative, but via the identification of a special structure in an interaction energy, we are able to derive a precise upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit $\ep \to 0$. We first consider the original time scale, in which the vortices do not move and the solutions undergo a "phase relaxation." …
On the first critical field in the 3D Ginzburg-Landau model of superconductivity
本文研究三维Ginzburg-Landau超导模型，在大参数下估算第一临界场，刻画无涡旋的Meissner解的全局最小能量特性，并证明超临界场特定范围内存在局部最小能量的唯一Meissner型解。
We study a variational model from micromagnetics involving a nonlocal Ginzburg-Landau type energy for S^1-valued vector fields. These vector fields form domain walls, called Neel walls, that correspond to one-dimensional transitions between two directions within the unit circle S^1. Due to the nonlocality of the energy, a Neel wall is a two length scale object, comprising a core and two logarithmically decaying tails. Our aim is to determine the energy differences leading to repulsion or attraction between Neel walls. In contrast to the usual Ginzburg-Landau vortices, we obtain a renormalised …
A central focus of Ginzburg-Landau theory is the understanding and characterization of vortex configurations. On a bounded domain $Ω\subseteq \mathbb{R}^2,$ global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit $ε\to 0,$ where $ε>0$ is the inverse of the Ginzburg-Landau parameter. The presence of an applied magnetic field of strength $h_{ex}\gg 1$ makes possible the existence of stable vortex states. A notable open problem is whether there are solutions of the Ginzburg-Landau equation for any number of vortices below…
We construct local minimizers to the Ginzburg-Landau functional of superconductivity whose number of vortices N is prescribed and blows up as the parameter epsilon, inverse of the Ginzburg-Landau parameter kappa, tends to zero. We treat the case of N as large as log epsilon, and a wide range of intensity of external magnetic field. The vortices of our solutions arrange themselves with uniform density over a subregion of the domain bounded by a "free boundary" determined via an obstacle problem, and asymptotically tend to minimize the "Coulombian renormalized energy" W introduced in [14]. The m…
We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the {\it pinning domains}. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\v\to0$; here, $\v$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $Ω\subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\d Ø$, with topological degree ${\rm deg}_{\d Ø…
Linearized dynamic stability for vortices of Ginzburg-Landau evolutions
本文研究相对论 Ginzburg-Landau n-涡旋的共转扰动，通过畸变 Fourier 基与奇异 Schrödinger 谱分析建立波动和 Klein-Gordon 衰减估计，并证明无不稳定谱及嵌入特征值位于 (1.332,2)。
We study the structure of symmetric vortices in a Ginzburg-Landau model based on S. C. Zhang's SO(5) theory of high temperature superconductivity and antiferromagnetism. We consider both a full Ginzburg-Landau theory (with Ginzburg-Landau scaling parameter kappa) and a high-kappa limiting model. In all cases we find that the usual superconducting vortices (with normal phase in the central core region) become unstable (not energy minimizing) when the chemical potential crosses a threshold level, giving rise to a new vortex profile with antiferromagnetic ordering in the core region. We show that…
We establish vortex dynamics for the time-dependent Ginzburg-Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids we prove that sequences of solutions converge to the hydrodynamic limit.
A Ginzburg-Landau problem on a circular cone
本文对圆锥上切向量场的Ginzburg-Landau模型进行渐近分析，确定了涡旋的度数与位置，并证明其中一个涡旋位于圆锥尖端。
We analyze Ginzburg--Landau minimization problems in two dimensions with either a strong or weak" tangential boundary condition. These problems are motivated by experiments in liquid crystal with boundary defects. In the singular limit when the correlation length tends to zero, we show that boundary defects will be observed for weak anchoring, while both boundary and interior vortices are possible for strong anchoring in the first order limit.
We study a variational Ginzburg-Landau type model depending on a small parameter $ε>0$ for (tangent) vector fields on a $2$-dimensional Riemannian surface. As $ε\to 0$, the vector fields tend to be of unit length and will have singular points of a (non-zero) index, called vortices. Our main result determines the interaction energy between these vortices as a $Γ$-limit (at the second order) as $ε\to 0$.
The correlation term, introduced in [13] to describe the interaction between very far apart vortices, governs symmetry-breaking for the Ginzburg-Landau equation in R^2 or bounded domains. It is a homogeneous function of degree (-2), and then for 2π/N-symmetric vortex configurations can be expressed in terms of the so-called correlation coefficient. Ovchinnikov and Sigal [13] have computed it in few cases and conjectured its value to be an integer multiple of π/4. We will disprove this conjecture by showing that the correlation coefficient always vanishes, and will discuss some of its consequen…
A priori estimates and $η-$compactness for anisotropic Ginzburg-Landau minimizers with tangential anchoring
本文研究二维区域上带切向锚定和散度或旋度惩罚的各向异性 Ginzburg-Landau 极小元，证明了一致的 L∞ 先验估计及 Lipschitz 常数的 ε⁻¹ 爆破速率，并得到收敛到具有内部点缺陷或边界半缺陷的 S¹ 值映射的紧性结果，同时证明散度惩罚情形下无边界涡旋。
The Asymptotic Theory of Ginzburg--Landau Critical Points on Domains with Corners
本文研究带角域上Ginzburg--Landau临界点的渐近行为，通过加权分裂和重整化能量，证明极限映射的涡旋位置为重整化能量的临界点。
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