Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations

Spirn, Daniel

doi:10.1002/cpa.3018

Cited by 34 publications

(45 citation statements)

References 35 publications

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“…Although (1) provides a fertile ground to test the mathematics of the Gorkov-Eliashberg equations, the more physical problem entails looking at the hydrodynamic limit of (4). For the Gorkov-Eliashberg equations (4), corresponding proofs of the vortex motion law are due to the second author [49] for O(1) fields and Sandier and Serfaty [41] for larger fields, following the formal asymptotic work of [38]. Formally, it was shown by Chapman, Rubinstein, and Schatzman [7] that the hydrodynamic limit of the associated ODE arising from the vortex motion law of (4) converges to a weak solution of…”

Section: Results In the Following We Letmentioning

confidence: 99%

Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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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.

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Section: Results In the Following We Letmentioning

confidence: 99%

Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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“…Statement ii) in Proposition 1 was already proved in [11,12,16,18] in the case of "wellprepared" initial data, i.e. having l vortices of degree +1 and −1 and an energy E ε (u 0 ε ) = π|log ε| + O(1): this is actually the minimal energy required for such a vortex configuration.…”

Section: Proposition 1 I)mentioning

confidence: 70%

Quantization and Motion Law for Ginzburg–Landau Vortices

Smets

Béthuel

Orlandi

2006

Arch Rational Mech Anal

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We study the vortex trajectories for the 2D parabolic Ginzburg-Landau equation without well-preparedness assumption. We prove that the trajectory set is rectifiable, satisfies a weak motion law and that dissipation occurs only at a finite number of times. Moreover, away from these times, 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.

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“…the gradient-flow of (1), for large applied fields (the result for bounded applied fields had been obtained by Spirn [52]). …”

Section: Dynamical Law For Ginzburg-landau With Magnetic Fieldmentioning

confidence: 83%

Vortices in the Ginzburg–Landau model of superconductivity

Serfaty¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We review some mathematical results on the Ginzburg-Landau model with and without magnetic field. The Ginzburg-Landau energy is the standard model for superconductivity, able to predict the existence of vortices (which are quantized, topological defects) in certain regimes of the applied magnetic field. We focus particularly on deriving limiting (or reduced) energies for the Ginzburg-Landau energy functional, depending on the various parameter regimes, in the spirit of -convergence. These passages to the limit allow to perform a sort of dimension-reduction and to deduce a rather complete characterization of the behavior of vortices for energy-minimizers, in agreement with the physics results. We also describe the behavior of energy critical points, the stability of the solutions, the motion of vortices for solutions of the gradient-flow of the Ginzburg-Landau energy, and show how they are also governed by those of the limiting energies. Mathematics Subject Classification (2000). Primary 00A05; Secondary 00B10.

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Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations

Cited by 34 publications

References 35 publications

Vortex Liquids and the Ginzburg–landau Equation

Vortex Liquids and the Ginzburg–landau Equation

Quantization and Motion Law for Ginzburg–Landau Vortices

Vortices in the Ginzburg–Landau model of superconductivity

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