Vortices in the Magnetic Ginzburg-Landau Model

Etienne Sandier, Sylvia Serfaty (auth.)0817643168, 9780817643164

With the discovery of type-II superconductivity by Abrikosov, the prediction of vortex lattices, and their experimental observation, quantized vortices have become a central object of study in superconductivity, superfluidity, and Bose–Einstein condensation. This book presents the mathematics of superconducting vortices in the framework of the acclaimed two-dimensional Ginzburg-Landau model, with or without magnetic field, and in the limit of a large Ginzburg-Landau parameter, kappa.

This text presents complete and mathematically rigorous versions of both results either already known by physicists or applied mathematicians, or entirely new. It begins by introducing mathematical tools such as the vortex balls construction and Jacobian estimates. Among the applications presented are: the determination of the vortex densities and vortex locations for energy minimizers in a wide range of regimes of applied fields, the precise expansion of the so-called first critical field in a bounded domain, the existence of branches of solutions with given numbers of vortices, and the derivation of a criticality condition for vortex densities of non-minimizing solutions. Thus, this book retraces in an almost entirely self-contained way many results that are scattered in series of articles, while containing a number of previously unpublished results as well.

The book also provides a list of open problems and a guide to the increasingly diverse mathematical literature on Ginzburg–Landau related topics. It will benefit both pure and applied mathematicians, physicists, and graduate students having either an introductory or an advanced knowledge of the subject.

Table of contents :
Front Matter….Pages i-xii
Introduction….Pages 1-24
Physical Presentation of the Model—Critical Fields….Pages 25-38
First Properties of Solutions to the Ginzburg-Landau Equations….Pages 39-58
The Vortex-Balls Construction….Pages 59-81
Coupling the Ball Construction to the Pohozaev Identity and Applications….Pages 83-115
Jacobian Estimate….Pages 117-125
The Obstacle Problem….Pages 127-154
Higher Values of the Applied Field….Pages 155-163
The Intermediate Regime….Pages 165-206
The Case of a Bounded Number of Vortices….Pages 207-218
Branches of Solutions….Pages 219-242
Back to Global Minimization….Pages 243-251
Asymptotics for Solutions….Pages 253-282
A Guide to the Literature….Pages 283-297
Open Problems….Pages 299-302
Back Matter….Pages 303-322