ON THE **GINZBURG**-**LANDAU** ENERGY WITH WEIGHT
(Sur l’´energie de **Ginzburg**-**Landau** avec poids)
C˘ at˘ alin LEFTER and Vicent¸iu D. R ˘ ADULESCU
Abstract. This paper gives a solution to an open problem raised by Bethuel, Brezis and H´elein. We study the **Ginzburg**-**Landau** energy with weight. We find the expression of the renormalized energy and we show that the finite configuration of singularities of the limit is a minimum point of this functional. We find a vanishing gradient type property and then we obtain the renormalized energy by Bethuel, Brezis and H´elein’s shrinking holes method.

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ON A FRACTIONAL **GINZBURG**-**LANDAU** EQUATION AND 1/2-HARMONIC MAPS INTO SPHERES
VINCENT MILLOT AND YANNICK SIRE
A BSTRACT . This paper is devoted to the asymptotic analysis of a fractional version of the **Ginzburg**-**Landau** equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular limit ε → 0, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a countably H n−1 -rectifiable closed set of finite (n − 1)-Hausdorff measure. The proof relies on the

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Th´ eor` eme (Th´eor`eme 5.3). Soient N ≥ 1 et u 0 ∈ E. L’´equation (CGL) admet une
solution globale u ∈ C(R + , H loc 1 ) telle que u(0) = u 0 . L’´energie de **Ginzburg**-**Landau** de u
est finie ` a temps positif et d´ecroissante.
La deuxi`eme partie du Chapitre 5, qui concerne exclusivement le cas bidimensionnel N = 2, est consacr´ee ` a l’´etude du probl`eme de Cauchy dans un espace « proche » de l’espace d’´energie finie E. Ce cadre de r´esolution fut introduit par Bethuel et Smets [44] pour l’´equation de Gross-Pitaevskii afin de traiter des donn´ees typiques 17 , d’´energie ´even- tuellement infinie, d’un r´egime que nous ´etudierons ult´erieurement. Il s’agit de l’espace V + H 1 (R 2 ), o` u V est l’ensemble

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Other approaches, rooted in a direct coarse-graining of the microscopic dynam- ics [12, 13, 14, 15, 40, 16, 41], on the other hand, are able to determine most transport coefficients and their functional dependence on the fields. The “Boltzmann-**Ginzburg**- **Landau**” (BGL) framework, which we detail below, stems from the early Boltzmann approach of [13] to the polar case, and combines it with the traditional **Ginzburg**- **Landau** weakly nonlinear analysis. We argue below that it offers better overall control, something needed since some confusion remains: the equations obtained by different methods often differ not only in details but also in structure. For instance, the equa- tions for rods derived by Baskaran and Marchetti from a Smoluchowski equation contain more and different terms than those derived in the BGL framework. The Chapman-Enskog formalism put forward in [16] for the polar case yields many more terms than those retained in other approaches, and their effect on the dynamics, especially at the nonlinear level, remains unclear.

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ENERGY DATA
DELPHINE CÔTE AND RAPHAËL CÔTE
A BSTRACT . 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, the 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 works of Bethuel, Orlandi and Smets [8, 9] to infinite energy data; they allow to consider point vortices on a lattice (in dimension 2), or filament vortices of infinite length (in dimension 3).

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with x i the vortex locations and d i their degrees (this corresponds to the so-called Jacobian estimates, a
notion we will come back to in the course of the paper). We wish to show that N ε −1 j ε converges as ε ↓ 0 to a
velocity field v solving a limiting PDE, which as in [82] is assumed to be regular enough. Note that solutions of the limiting equations are studied in [37] and shown to be global and regular enough if the initial data is. The method of the proof in [82] is based on a “modulated energy” technique, which originates in the relative entropy method first designed by DiPerna [31] and Dafermos [23, 24] to establish weak-strong stability principles for some hyperbolic systems. Such a relative entropy method was later rediscovered by Yau [90] for the hydrodynamic limit of the **Ginzburg**-**Landau** lattice model, was introduced in kinetic theory by Golse [12] for the convergence of suitably scaled solutions of the Boltzmann equation towards solutions of the incompressible Euler equations (see e.g. [71] for the many recent developments on the topic), and first took the form of a modulated energy method in the work by Brenier [15] on the quasi-neutral limit of the Vlasov-Poisson system. In the present situation, the method consists in defining a “modulated energy”, which without pinning takes the form

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Let us give an idea of the method used to prove the results. We construct the blow- up solution with the profile in Theorem 1, by following the method of [MZ97], [BK94], though we are far from a simple adaptation, since we are studying the critical problem, which make the technical details harder to elaborate. This kind of methods has been applied for various nonlinear evolution equations. For hyperbolic equations, it has been successfully used for the construction of multi-solitons for semilinear wave equation in one space dimension (see [CZ13]). For parabolic equations, it has been used in [MZ08] and [Zaa01] for the Complex **Ginzburg** **Landau** equation with no gradient structure, the critical harmonic heat flow in [RS13], the two dimensional Keller-Segel equation in [RS14] and the nonlinear heat equation involving nonlinear gradient term in [EZ11] and [TZ15]. Recently, this method has been applied for a non variational parabolic system in [NZ15b] for a logarithmically perturbed nonlinear equation in [NZ15a].

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Let us give an idea of the method used to prove the results. We construct the blow- up solution with the profile in Theorem 1, by following the method of [MZ97], [BK94], though we are far from a simple adaptation, since we are studying the critical problem, which make the technical details harder to elaborate. This kind of methods has been applied for various nonlinear evolution equations. For hyperbolic equations, it has been successfully used for the construction of multi-solitons for semilinear wave equation in one space dimension (see [CZ13]). For parabolic equations, it has been used in [MZ08] and [Zaa01] for the Complex **Ginzburg** **Landau** (CGL) equation with no gradient structure, the critical harmonic heat flow in [RS13], the two dimensional Keller-Segel equation in [RS14] and the nonlinear heat equation involving nonlinear gradient term in [EZ11], [TZ19]. Recently, this method has been applied for various non variational parabolic system in [NZ15] and [GNZ17, GNZ18b, GNZ18a, GNZ19], for a logarithmically perturbed nonlinear equation in [NZ16, Duo19b, Duo19a, DNZ19]. We also mention a result for a higher order parabolic equation [GNZ20], two more results for equation involving non local terms in [DZ19, AZ19].

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β η, A(0) = A(L) = 0, (1) where η is a white noise in space and time, < η(x, t)η(x ′ , t ′ ) >= δ(t −t ′ )δ(x −x ′ ) [51, 52]. It has several names:
it is often called the Allen–Cahn equation in mathematics and solid state physics [1] and the Chaffee–Infante in climate science. In non-linear physics, it is one of the several instances of **Ginzburg**–**Landau** equation [53]. In the study of the deterministic front propagation in biology, it is a particular case of the Fisher–Kolmogorov equation [54]. After change of temporal, spatial and amplitude scales, there are only two free parameters, the nondimensional inverse temperature β and the nondimensional size of the domain L. With this scaling choice, the size of the boundary layers, fronts and coherent length is of order one. We recall an important property of the Allen–Cahn equation which is essential for theoretical considerations: it is a gradient system (for the L 2 scalar product) with respect to the potential V

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To analyze the vortices, authors have developed tools, in particular the ball construc- tion method and Jacobian estimates. The first one was introduced independently by Jerrard [Jer99] and Sandier [San98]. It allows one to obtain universal lower bounds for two-dimensional **Ginzburg**-**Landau** energies in terms of the topology of the vortices. These lower bounds capture the fact that vortices of degree d cost at least an order π|d| log 1 ε of energy. The second tool, that has been widely used in the analysis of the **Ginzburg**-**Landau** model in any dimension after the work by Jerrard and Soner [JS02], is the Jacobian (or vorticity) estimate, which allows one to relate the vorticity µ(u, A) with, roughly speak- ing, Dirac masses supported on co-dimension 2 singularities. When n = 2, these masses are supported on points naturally derived from the ball construction. The following result presents an optimal version of these estimates in 2D.

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Keywords: uniqueness, minimisers, **Ginzburg**-**Landau**. MSC: 35A02, 35B06, 35J50.
This note is based on the article [3] of the authors and represents the talk of the ﬁrst author (Radu Ignat) given at the Workshop “Nonlinear Data: Theory and Algorithms” in Oberwolfach, 22 April – 28 April 2018. It will be included in the volume Oberwolfach Reports No. 20/2018 dedicated to that workshop.

Its stability will be analyzed in a forthcoming paper [ 27 ]. All these results are presented in Fig. 3 .
To summarize, using a ‘‘Boltzmann-**Ginzburg**-**Landau**’’ controlled expansion scheme, we derived a set of minimal yet complete nonlinear field equations from the Vicsek model with nematic alignment studied in Ref. [ 16 ]. This simple setting allowed for a comprehensive analysis of the linear and nonlinear dynamics of the field equations obtained because our approach automatically yields a ‘‘meaningful manifold’’ parametrized by global density and noise strength in the high-dimensional space spanned by all coefficients of the continuous equations. Excellent agreement was found (at a qualitative level) with the simu- lations of the original microscopic model. The banded solutions were studied analytically. Their existence domain was found different from the region of linear instability of the homogeneous ordered phase, stressing the importance of a nonlinear analysis. More work, beyond the scope of this Letter, is needed to obtain a better understanding of the chaotic regimes observed. To this aim, we plan to study the linear stability of the band solutions in two dimensions.

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with 0 ≤ m, m 0 < + ∞ and m 6= m 0 can not hold. Our Theorem even states that if we have
two straight line vortices in the half plane {z < 0} meeting at 0, then they must be in a plane
orthogonal to {z = 0}. At the opposite of [CH], our approach is based on equation (1) only,
whereas the London equation is the limit equation for the current (see (22) below), which is the second equation of the **Ginzburg**-**Landau** equation with magnetic field.

1. Introduction
The **Ginzburg**-**Landau** functional is a model describing the response of a superconducting ma- terial to an applied magnetic field through the qualitative behavior of the minimizing/critical configurations. The mathematically rigorous analysis of such configurations led to a vast litera- ture and to many mathematically challenging questions, with the aim to recover what physicists have already observed through experiments or heuristic computations. (See [17] for an introduc- tion to the physics of superconductivity, and the two monographs [8, 32] for the mathematical progress on this subject).

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We present new estimates on the two-dimensional **Ginzburg**-**Landau** energy of a type-II superconductor in an applied magnetic field varying between the second and third critical fields. In this regime, superconductivity is restricted to a thin layer along the boundary of the sample. We provide new energy lower bounds, proving that the **Ginzburg**-**Landau** energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. Estimates relating the density of the **Ginzburg**- **Landau** order parameter to that of the 1D problem follow. In the particular case of a disc sample, a refinement of our method leads to a pointwise estimate on the **Ginzburg**-**Landau** order parameter, thereby proving a strong form of uniformity of the surface superconductivity layer, related to a conjecture by Xing-Bin Pan.

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µ , which prevents us to apply directly the existing approximation results by functions with a smooth jump set, see e.g. [CT99, BCP96, DPFP17, BCG14]). Our approach uses a (new) regularization technique (see Lemma 3.17) which is somehow reminiscent of [AT90] and could be of independent interest. Another difficulty comes from the optimal profile problem defining the core energy γ m . The underlying minimization problem involves the **Ginzburg**-**Landau** energy of N -valued maps, and one has to find almost minimizers which can be lifted into C-valued maps in SBV 2 , see Section 3.2.

[Vie09, ACO09] for related results). In [Gol17], the optimal microstructures for a two- dimensional analogue of I(µ) are exactly computed.
The paper is organized as follows. In section 2, we set some notation and recall some no- tions from optimal transport theory. In Section 3, we recall the definition of the **Ginzburg**- **Landau** functional together with various important quantities such as the superconducting current. We also introduce there the anisotropic rescaling leading to the functional e E T . In

We compute the analytical location of critical points of a directional field defined on the unit disk. The calculations 221[r]

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Our analysis, in particular the estimate of the subleading order of the GL energy, requires some quantitative control on the variations of the optimal 1D energy, phase and density when t[r]