### 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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Title of host publication | International Congress of Mathematicians, ICM 2006 |

Pages | 267-290 |

Number of pages | 24 |

Volume | 3 |

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 |
---|---|

Country | Spain |

City | Madrid |

Period | 8/22/06 → 8/30/06 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Congress of Mathematicians, ICM 2006*(Vol. 3, pp. 267-290)

**Vortices in the Ginzburg-Landau model of superconductivity.** / Serfaty, Sylvia.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*International Congress of Mathematicians, ICM 2006.*vol. 3, pp. 267-290, 25th International Congress of Mathematicians, ICM 2006, Madrid, Spain, 8/22/06.

}

TY - GEN

T1 - Vortices in the Ginzburg-Landau model of superconductivity

AU - Serfaty, Sylvia

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Γ-convergence

KW - Ginzburg-Landau equations

KW - Superconductivity

KW - Variational methods

KW - Vortices

UR - http://www.scopus.com/inward/record.url?scp=84878096939&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878096939&partnerID=8YFLogxK

M3 - Conference contribution

VL - 3

SP - 267

EP - 290

BT - International Congress of Mathematicians, ICM 2006

ER -