Ginzburg-Landau Vortices

The following sections are included:

• Preface

• Contents

This course is dedicated to the study of bifurcation diagrams in the framework of the Ginzburg-Landau theory. The aim is to characterize the properties of solutions according to the values of the different free parameters: existence of non trivial solutions, number of solutions, stability and symmetry.

In the first two sections, we study the case of a superconducting material placed in a magnetic field for two different geometries. In the Ginzburg-Landau model, the state of the system can be described by a system of two coupled PDE's. In Section 1, for the infinite slab, the equations get scalar and reduce to a system of two coupled ODE's. Section 2 deals with the infinite cylinder, which is a 2d problem.

The 3rd section addresses a related problem: vortices in rotating Bose-Einstein condensates. The physical phenomenon is modeled by a nonlinear Schrodinger equation with a trapping and rotating term. Analogies are made with Ginzburg-Landau and symmetry and breaking of symmetry of 3d vortices are analyzed.

The following sections are included:

• Introduction
  • History
  • The Ginzburg-Landau model
  • Adimensionalizing
  • Dimension reduction
  • Notation

• Minimization of the Ginzburg-Landau Functional
  • Gauge invariance
  • The Coulomb gauge
  • Minimization of the Ginzburg-Landau energy
    • Restriction to Ω
    • Minimization of J
  • Euler-Lagrange equations
    • Properties of critical points

• Introduction to Critical Fields ℝ²
  • Pure states
  • Critical line H_c2
  • Critical line H_c1
    • Approximate vortex
    • The energy of the approximate vortex
    • The critical line H_c1
  • Phase diagram

• Mathematical Preliminaries
  • Degree theory
    • Definition
    • Properties
  • The co-area formula
  • Sard's theorem

• The Ball Construction Method
  • 𝕊¹-valued maps on perforated domains
    • Radius of a set
    • Main theorem
    • Variants
  • Vortex balls
    • Using the co-area formula
    • Main result

• The Jacobian Estimate
  • Main theorem
  • The case |u| = 1 on Ω \ ∪_iB(a_i, r_i)
  • Reduction to the previous case
  • A particular case

• Computing H_c1
  • Meissner solution
  • Energy in terms of vortices
  • Value of H_c1
    • Lower bound
    • Upper bound
  • Conclusion

• References

The following sections are included:

• Introduction

• The Ginzburg-Landau energy with g : ∂G → S¹

• The Ginzburg-Landau energy with g : ∂G → ℝ² \ {0}

• The Ginzburg-Landau energy when g has zeros

• On potentials which are functions of the distance to a curve

• The general “circular-well” potential

• References

The present survey is essentially the contents of the lectures given in the CIMPA school and the ISFMA symposium on the topics of Ginzburg-Landau vortices held at Fudan Uinversity, in the November of 2002. It is a review of some contributions to the existence results of Ginzburg-Landau equations in recent years. In the first place we recall some known results concerning the asymptotic behavior of minimizers of a simplified model studied by Bethuel-Brezis-Hélein. In the second place we deal with the renormalized energy: we study the topological similarity between the level sets and using variational methods to prove the multiplicity of solutions with Dirichlet boundary conditions. Finally, existence results with other boundary conditions are presented.

In this paper, we give a survey on Ginzburg-Landau vortices of superconducting thin films.

In this paper, we study the hydro-dynamic limit of the finite Ginzburg-Landau wave vortices, which was established in [6]. We prove here that the densities approximated by the vortex blob method associated with the Ginzburg-Landau wave vortices tend to the solutions of the pressure-less compressible Euler-Poisson equations before the formation of singularities in the limit system as the blob sizes and the grid sizes tend to zero in appropriate rates.

The following sections are included:

• Introduction

• References

A free boundary problem is derived for type I superconductivity from Ginzburg-Landan theory in superconductor, and the solvability of the free boundary problem is considered in any dimensional case.

We prove the existence of solutions to the static Landau-Lifshitz equation corresponding to the homotopy classes of continuous functions from Ω → S² ∩ {u₃ = 0}, here Ω is a non-simply connected bounded domain in ℝ³. The stability of the solutions to the time-dependent Landau-Lifshitz equation is also obtained. We prove that any non-constant smooth static solution to Landau-Lifshitz equation is unstable, provided that the domain is convex.