Parabolic Ginzburg-Landau Equation Research Articles

Structural descriptions of limits of the parabolic Ginzburg-Landau equation on closed manifolds

In the setting of a compact Riemannian manifold of dimension $N\ge3$, we provide a structural description of the limiting behavior of the energy measures of solutions to the parabolic Ginzburg-Landau equation. In particular, we provide a decomposition of the limiting energy measure into a diffuse part, which is absolutely continuous with respect to the volume measure, and a concentrated part supported on a codimension $2$ rectifiable subset. We also demonstrate that the time evolution of the diffuse part is determined by the heat equation while the concentrated part evolves according to a Brakke flow. This paper extends the work of Bethuel, Orlandi, and Smets from [8].

Vortices in a stochastic parabolic Ginzburg-Landau equation

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.

Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations

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 $\epsilon$, the characteristic lengthscale of the vortices, tends to $0$, and in a situation where the number of vortices $N$ blows up as $\epsilon \to 0$. The requirements are that $N$ should blow up faster than $|\log \epsilon|$ in the Gross-Pitaevskii case, and at most like $|\log \epsilon|$ in the parabolic case. Both results assume a well-prepared initial condition and regularity of the limiting initial data, and use the regularity of the solution to the limiting equation. In the case of the parabolic Ginzburg-Landau equation, the limiting mean-field dynamical law that we identify coincides with the one proposed by Chapman-Rubinstein-Schatzman and E in the regime $N\ll |\log \epsilon|$, but not if $N$ grows faster.

Limiting Motion for the Parabolic Ginzburg–Landau Equation with Infinite Energy Data

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).

Motion of vortex-filaments for superconductivity

In this paper, we study the asymptotic behavior of solutions to the simplified Ginzburg–Landau model for superconductivity. We prove that, asymptotically, vortex-filaments evolves according to the mean curvature flow in the sense of weak formulation. This can be seen as a first attempt to understand the nature of the motion of vortex filaments in three dimensions with magnetic field. On the other hand, this paper revisits the pioneering work of Bethuel–Orlandi–Smets [F. Bethuel, G. Orlandi, D. Smets, Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature, Ann. of Math. 163 (2006) 37–163] in a slightly relaxed setting.

Relaxation of the curve shortening flow via the parabolic Ginzburg -Landau equation

In this paper we study how to find solutions $$u_\epsilon$$ to the parabolic Ginzburg–Landau equation that as $$\epsilon \to 0$$ have as interface a given curve that evolves under curve shortening flow. Moreover, for compact embedded curves we find a uniform profile for the solution $$u_\epsilon$$ up the extinction time of the curve. We show that after the extinction time the solution converges uniformly to a constant.

Dynamics of Multiple Degree Ginzburg-Landau Vortices

For the two dimensional complex parabolic Ginzburg-Landau equation we prove that, asymptotically, vortices evolve according to a simple ordinary differential equation, which is a gradient flow of the Kirchhoff-Onsager functional. This convergence holds except for a finite number of times, corresponding to vortex collisions and splittings, which we describe carefully. The only assumption is a natural energy bound on the initial data.

Quantization and Motion Law for Ginzburg–Landau Vortices

We study the vortex trajectories for the two-dimensional complex parabolic Ginzburg–Landau equation without a well-preparedness assumption. We prove that the trajectory set is rectifiable, and satisfies a weak motion law. In the case of degree ± 1 vortices, the motion law is satisfied in the classical sense. Moreover, dissipation occurs only at a finite number of times. Away from these times, possible collisions and splittings of vortices are constrained by algebraic equations involving their topological degrees.Quantization properties of the energy and potential densities play a central role in the proofs.

Dynamics of multiple degree Ginzburg–Landau vortices

For the two-dimensional complex parabolic Ginzburg–Landau equation we prove that, asymptotically, vortices evolve according to a simple ordinary differential equation, which is a gradient flow of the Kirchhoff–Onsager functional. This convergence holds except for a finite number of times, corresponding to vortex collisions and splittings, which we describe carefully. The only assumption is a natural energy bound on the initial data. To cite this article: F. Bethuel et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature

For the complex parabolic Ginzburg-Landau equation, we prove that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke?s weak formulation. The only assumption is a natural energy bound on the initial data. In some cases, we also prove convergence to enhanced motion in the sense of Ilmanen.

Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics

In this article, we describe a natural framework for the vortex dynamics in the complex-valued parabolic Ginzburg-Landau equation in R2. This general setting does not rely on any assumption of well-preparedness and has the advantage of being valid even after collision times. We carefully analyze collisions leading to annihilation. A new phenomenon is identified, the phase-vortex interaction, which is related to the persistence of low-frequency oscillations and leads to an unexpected drift in the motion of vortices

Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature

We present some new results for the asymptotic behavior of the complex parabolic Ginzburg–Landau equation. In particular, we establish that, as the parameter ε tends to 0, vorticity evolves according to motion by mean curvature in Brakke's weak formulation. The only assumption we make is a natural energy bound on the initial data. In some cases, we also prove convergence to enhanced motion in the sense of Ilmanen. To cite this article: F. Bethuel et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).