## 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 Γ-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.

Original language | English (US) |
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Pages | 267-290 |

Number of pages | 24 |

State | Published - 2006 |

Event | 25th International Congress of Mathematicians, ICM 2006 - Madrid, Spain Duration: Aug 22 2006 → Aug 30 2006 |

### Other

Other | 25th International Congress of Mathematicians, ICM 2006 |
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Country/Territory | Spain |

City | Madrid |

Period | 8/22/06 → 8/30/06 |

## Keywords

- Ginzburg-Landau equations
- Superconductivity
- Variational methods
- Vortices
- Γ-convergence

## ASJC Scopus subject areas

- Mathematics(all)