Evelyne Miot
We consider a complex Ginzburg-Landau equation, corresponding to a Gross-Pitaevskii equation with a small dissipation term. We study an asymptotic regime for long-wave perturbations of constant maps of modulus one. We show that such solutions never vanish and we derive a damped wave dynamics for the perturbation.
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Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains
研究复吉辛堡-兰道方程在可能无界域上的解，包含奇异和饱和非线性阻尼项，通过复时间导数预因子连接非线性薛定谔方程和耗散抛物动力学，建立全局解的存在性和唯一性。
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 consider the stationary Ginzburg-Landau equations in $\R^d$, $d=2,3$. We exhibit a class of applied magnetic fields (including constant fields) such that the Ginzburg-Landau equations do not admit finite energy solutions.
In this paper, we prove a global Carleman estimate for the complex Ginzburg- Landau operator with a cubic nonlinear term in a bounded domain of Rn, n = 2, 3. As applications, we study state observation problems for the Ginzburg-Landau equation.
Symmetry of local minimizers for the three dimensional Ginzburg-Landau functional
对满足自然能量界的$H^{1}_{loc}(R^3;R^3)$中的标准Ginzburg-Landau泛函的非常数整局部极小元进行了分类.直到平移和旋转，Ginzburg-Landau系统的这些解在正交群的作用下由显式解等变给出。
In this paper, we give the boundeness of solutions to Fractional Laplacian Ginzburg-Landau equation, which extends the Brezis theorem into the nonlinear Fractional Laplacian equation. A related linear fractional Schrodinger equation is also studied.
Dynamics of vortices for the Complex Ginzburg-Landau equation
本文研究了平面上的复Ginzburg-Landau方程，它具有Gross-Pitaevskii方程的形式，并加入了一些耗散项。我们专注于制度相应的准备好酉涡，并推导出其渐近运动规律。
We study the inviscid limit for the Cauchy problem of derivative Ginzburg-Landau equation in higher dimension space n>2. We show that it is global well-posed and its solution will converge to that of derivative Schrodinger equation.
We prove the existence of time periodic solution to the 3D Ginzburg-Landau equation in weighted Sobolev spaces. We consider the cubic Ginzburg-Landau equation with an external force $g$ satisfying the oddness condition $g(-x,t)=-g(x,t)$. The existence of the periodic solution is proved for small time-periodic external force. The stability of the time periodic solution is also considered.
We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg-Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg-Landau energy is well-defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross-Pitaevskii equation, the natural Hamiltonian evolution equation associate…
The limit theory of the energy-critical complex Ginzburg-Landau equation
本文基于能量方法，严格证明了3维和4维下能量临界复Ginzburg-Landau方程向能量临界非线性热方程的零色散极限，以及4维聚焦情形下向能量临界非线性Schrödinger方程的无粘极限理论。
Threshold solutions of the energy-critical complex Ginzburg-Landau equation
研究三维与四维能量临界复 Ginzburg-Landau 方程，刻画初始数据能量等于稳态解能量这一阈值情形下的解动力学。
We prove that a local minimizer of the Ginzburg-Landau energy in ${\mathbb R}^3$ satisfying the condition $\liminf_{R\to\infty}E(u;B_R)/RlnR < 2π$ must be constant. The main tool is a new sharp eta-ellipticity result for minimizers in dimension three that might be of independent interest.
We consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. We determine an accurate asymptotic formula for the minimizing energy when the Ginzburg-Landau parameter and the magnetic field are large and of the same order. As a consequence, it is shown how bulk superconductivity decreases in average as the applied magnetic field increases.
We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $ϕ_t = e^{iθ} Δϕ+ e^{iγ} |ϕ|^αϕ$ with periodic boundary conditions. Our result includes all values of $θ$ and $γ$ for which $\cos θ\cos γ>0$, but requires that $α>0$ be sufficiently small.
The blow-up of solutions for the Cauchy problem of fractional Ginzburg-Landau equation with non-positive nonlinearity is shown by an ODE argument. Moreover, in one dimensional case, the optimal lifespan estimate for size of initial data is obtained.
In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg-Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg-Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem.
Saddle solution to the complex Ginzburg-Landau system in three dimensions
本文构造了三维非磁性复Ginzburg-Landau系统的整体鞍解，刻画了其零集为两条垂直相交直线的并集，并证明了blow-down以重数1集中于这些直线。
We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile.
In this paper, we study the asymptotic behaviour of minimizing solutions of a Ginzburg-Landau type functional with a positive weight and with convex potential near $0$ and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis-Merle-Rivière.
We study the Ginzburg-Landau energy of a superconductor with a variable magnetic field and a pinning term in a bounded smooth two dimensional domain $Ω$. Supposing that the Ginzburg-Landau parameter and the intensity of the magnetic field are large and of the same order, we determine an accurate asymptotic formula for the minimizing energy. This asymptotic formula displays the influence of the pinning term. Also, we discuss the existence of non-trivial solutions and prove some asymptotics of the third critical field.
Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization
研究复杂吉恩斯-兰道方程在可能无界域上的解，包含奇异和饱和非线性阻尼项，通过复时间导数预因子连接非线性薛定谔方程与耗散抛物动力学，证明强稳定化性质，特别是非线性饱和机制导致的有限时间消散。
This paper is concerned with the stochastic generalized Ginzburg-Landau equation driven by a multiplicative noise of jump type. By a prior estimate, weak convergence and monotonicity technique, we prove the existence and uniqueness of the solution of an initial-boundary value problem with homogeneous Dirichlet boundary condition. However, for the generalized Ginzburg-Landau equation, such a locally monotonic condition of the nonlinear term can not be satisfied in a straight way. For this, we utilize the characteristic structure of nonlinear term and refined analysis to overcome this gap.
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.
We study the time-dependent Ginzburg--Landau equations in a three-dimensional curved polyhedron (possibly nonconvex). Compared with the previous works, we prove existence and uniqueness of a global weak solution based on weaker regularity of the solution in the presence of edges or corners, where the magnetic potential may not be in $L^2(0,T;H^1(Ω)^3)$.
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.
In this paper, we consider the complex Ginzburg--Landau equation $u_t = e^{iθ} [Δu + |u|^αu] + γu$ on ${\mathbb R}^N $, where $α>0$, $γ\in \R$ and $-π/2<θ<π/2$. By convexity arguments we prove that, under certain conditions on $α,θ,γ$, a class of solutions with negative initial energy blows up in finite time.
We consider a two-dimensional Ginzburg-Landau problem on an arbitrary domain with a finite number of vanishingly small circular holes. A special choice of scaling relation between the material and geometric parameters (Ginzburg-Landau parameter vs hole radius) is motivated by a recently dsicovered phenomenon of vortex phase separation in superconducting composites. We show that, for each hole, the degrees of minimizers of the Ginzburg-Landau problems in the classes of $\mathbb S^1$-valued and $\mathbb C$-valued maps, respectively, are the same. The presence of two parameters that are widely se…
Carleman estimate for full-discrete approximations of the complex Ginzburg-Landau equation with dynamic boundary conditions and applications to controllability
本文为带动态边界条件的一维Ginzburg-Landau方程全离散逼近建立离散Carleman估计，进而推导松弛可观测性不等式，最终证明其可控性。
In this article, we review finite-time blowup criteria for the family of complex Ginzburg-Landau equations $u_t = e^{ iθ} [Δu + |u|^αu] + γu$ on ${\mathbb R}^N $, where $0 \le θ\le \frac {π} {2}$, $α>0$ and $γ\in {\mathbb R} $. We study in particular the effect of the parameters $θ$ and $γ$, and the dependence of the blowup time on these parameters.
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 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.
We introduce a framework to study the occurrence of vortex filament concentration in $3D$ Ginzburg-Landau theory. We derive a functional that describes the free-energy of a collection of nearly-parallel quantized vortex filaments in a cylindrical $3$-dimensional domain, in certain scaling limits; it is shown to arise as the $Γ$-limit of a sequence of scaled Ginzburg-Landau functionals. Our main result establishes for the first time a long believed connection between the Ginzburg-Landau functional and the energy of nearly parallel filaments that applies to many mathematically and physically rel…
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…
On solutions to the Ginzburg-Landau equations in higher dimensions
建立了$n > 2$维Ginzburg-Landau方程的一个胶合定理.为此，我们考虑一个余维为2的非退化极小子流形，并构造一个单参数的Ginzburg-Landau方程的解族，使得能量密度集中在这个子流形附近。证明是基于构造适当的近似解和隐函数定理。
Towards an asymptotic analysis of the anisotropic Ginzburg-Landau model
本文开发了一套数学工具，用于在具有负度 Dirichlet 边界数据的容许竞争者中，对各向异性 Ginzburg-Landau 泛函极小值进行渐近分析。
In this paper, we prove the exponential stabilization of solutions for complex Ginzburg-Landau equations using finite-parameter feedback control algorithms, which employ finitely many volume elements, Fourier modes or nodal observables (controllers). We also propose a feedback control for steering solutions of the Ginzburg-Landau equation to a desired solution of the non-controlled system. In this latter problem, the feedback controller also involves the measurement of the solution to the non-controlled system.
Quantized vortex dynamics of the complex Ginzburg-Landau equation on torus
该研究严格推导了环面上复金兹堡-朗道方程在涡旋核尺寸趋于零时的量子涡旋约化动力学定律，揭示其由重整化能量驱动的梯度流与哈密顿流混合支配，并探讨了对应的首次积分与解析解。
We consider the Aharonov-Bohm magnetic potential and study the transition from normal to superconducting solutions within the Ginzburg-Landau model of superconductivity. We obtain oscillations consistent with the Little-Parks effect. We study the same problem but for a regularization of the Aharonov-Bohm potential, which leads to an interesting Aharonov-Bohm like magnetic field, and we prove that the transition between superconducting and normal solutions is not monotone too. Our results show a mechanism to derive the Aharonov-Bohm magnetic potential starting from a step magnetic field thereby…
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.
Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II
我们研究了具有间断约束的Ginzburg-Landau能量极小值的涡旋成核。对于与涡旋成核的第一临界场相当的外加磁场，我们确定了极限涡度。
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…
Optimal Uniform Elliptic Estimates for the Ginzburg-Landau System
我们重新考虑椭圆估计的磁运营商在二维和三维用于与金斯堡-朗道理论。此外，我们讨论了所谓的爆破技术，以获得最优估计的极限情况下。
The vortex dynamics of a Ginzburg-Landau system under pinning effect
证明了超导材料中的杂质或不均匀性对涡旋的吸引作用。还证明了相应的Ginzburg-Landau系统的强H^1收敛性。
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 consider the Ginzburg-Landau functional defined over a bounded and smooth three dimensional domain. Supposing that the strength of the applied magnetic field varies between the first and second critical fields, in such a way that $H_{C_1}\ll H\ll H_{C_2}$, we estimate the ground state energy to leading order as the Ginzburg-Landau parameter tends to infinity.
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.
In Ginzburg-Landau Theory of superconductivity, the density and location of the superconducting electrons are measured by a complex-valued wave function, the order parameter. In this paper, when the intensity of the applied magnetic field is close to the second critical field, and when the order parameter minimizes the Ginzburg-Landau functional defined over a two dimensional domain, the leading order approximation of its $L^2$-norm in `small' squares is given as the Ginzburg-Landau parameter tends to infinity.
Ginzburg-Landau fields are the solutions of the Ginzburg-Landau equations which depend on two positive parameters, $α$ and $β$. We give conditions on $α$ and $β$ for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in $\mathbb{R}^2$, spheres, tori, etc.) with de Gennes-Neumann boundary conditions. We also prove that, for each such manifold and all positive $α$ and $β$, the Ginzburg-Landau free energy is a Palais-Smale function on the space of gauge equivalence classes, Ginzburg-Land…
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 the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a simplified branched transportation functional. We derive the simplified functional from the full Ginzburg-Landau model rigorously via $Γ$-convergence. The detailed analysis of the limiting procedure and the study of the limiting functional lead to a precise understanding of the multiple scales contained in the model.
Limiting jump conditions for Josephson junctions in Ginzburg-Landau theory
我们考虑一个超导/正常/超导Josephson结模型通过Ginzburg-Landau理论。当正常的材料是足够薄的和所施加的磁场是低于临界场的涡旋成核，我们证明了领先的顺序，跳边界条件的类型预测的de Gennes满足整个结。
We provide a quantitative three-dimensional vortex approximation construction for the Ginzburg-Landau functional. This construction gives an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the 2D ones, and valid for the first time at the $\varepsilon$-level. These tools allow for a new approach to analyze the behavior of global minimizers for the Ginzburg-Landau functional below and near the first critical field in 3D, followed in two forthcoming papers. In addition, they allow to obtain an $\varepsilon$-quantitative product estimat…
Curl-Free Ginzburg-Landau Vortices
对于某些二维非线性椭圆型偏微分方程问题，经典的等周不等式产生一个尖锐的不等式，除了径向对称的，递减的解决方案，违反Pohozaev恒等式.本文用这种方法的一个推广版本来证明无旋金-朗道涡旋的径向对称性。
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。
A new sharp estimate on the dimension of the attractor for the Dirichlet problem of the complex Ginzburg-Landau equation
利用A。D. Melas（Proc. Amer. Math. Soc. \textbf{131}（2003）631-636）中，我们报告了一个新的和尖锐的估计与补充Dirichlet边界条件的复Ginzburg-Landau方程相关联的全局吸引子的维数。
We study the infimum of the Ginzburg-Landau functional in the case of a vanishing external magnetic field in a two dimensional simply connected domain. We obtain an energy asymptotics which is valid when the Ginzburg-Landau parameter is large and the strength of the external field is comparable with the third critical field. Compared with the known results when the external magnetic field does not vanish, we show in this regime a concentration of the energy near the zero set of the external magnetic field. Our results complete former results obtained by K. Attar and X-B. Pan--K-H.~Kwek.
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 prove the existence of novel, nonminimal and irreducible solutions to the (self-dual) Ginzburg-Landau equations on closed surfaces. To our knowledge these are the first such examples on nontrivial line bundles, that is, with nonzero total magnetic flux. Our method works with the 2-dimensional, critically coupled Ginzburg-Landau theory and uses the topology of the moduli space. The method is nonconstructive, but works for all values of the remaining coupling constant. We also prove the instability of these solutions.
This article analyzes well-definedness and regularity of renormalized powers of Ornstein-Uhlenbeck processes and uses this analysis to establish local existence, uniqueness and regularity of strong solutions of stochastic Ginzburg-Landau equations with polynomial nonlinearities in two space dimensions and with quadratic nonlinearities in three space dimensions.
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