Table Of ContentISNM
International Series of Numerical Mathematics
Vol. 134
Managing Editors:
K.-H. Hoffmann, Munchen
D. Mittelmann, Tempe
Associate Editors:
R. E. Bank, La Jolla
H. Kawarada, Chiba
R. J. LeVeque, Seattle
C. Verdi, Milano
Honorary Editor:
J.Todd,Pasadena
Ginzburg-Landau
Phase Transition Theory
and Superconductivity
K.-H. Hoffmann
Q. Tang
Springer Basel AG
Authors:
Karl-Heinz Hoffmann
Caesar
Friedensplatz 16
53111 Bonn
Germany
and
Qi Tang
SMS
University of Sussex
Brighton BN2 8LR
UK
2000 Mathematics Subject Classification 35-02, 35Qxx
A CIP catalogue record for this book is available from the Library of Congress, Washington D,C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Hoffmann, Karl-Heinz:
Ginzburg-Landau phase transition theory and superconductivity /
K,-H. Hoffmann; Q. Tang. -Basel; Boston; Berlin: Birkhauser, 2001
(International series of numerical mathematics; Vol. 134)
ISBN 978-3-0348-9499-9 ISBN 978-3-0348-8274-3 (eBook)
DOI 10.1007/978-3-0348-8274-3
ISBN 978-3-0348-9499-9
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other
ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.
© 2001 Springer Basel AG
Originally published by Birkhiiuser Verlag in 2001
Softcover reprint ofthe hardcover 1st edition 2001
Printed on acid-free paper produced of chlorine-free pulp. TCF
00
ISBN 978-3-0348-9499-9
Contents
Preface ..... Xl
1 Introduction
1.1 Brief history . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Meissner effect - diamagnetism . . . . . . . . . . 3
1.1.2 The London equation and the penetration depth 4
1.1.3 The coherence length ..... . 6
1.1.4 Classification of superconductors 6
1.1.5 Vortices .......... . 7
1.1.6 Summary............. 9
1.2 The G-L phenomenological theory ... 10
1.2.1 The free energy and the G-L equations. 10
1.2.2 Rescaling and the values of the constants 13
1.2.3 Gauge invariance . . . . . . . . . . . ... 15
1.3 Some considerations arising from scaling. . . . . 16
1.3.1 The two characteristic lengths ~(T) and >'(T) 16
1.3.2 The validity of the G-L theory . 17
1.4 The evolutionary G-L system - 2-d case 21
1.4.1 The system . . . . . . . . . . . . 21
1.4.2 Mathematical scaling ...... . 21
1.4.3 The G-L functional as a Lyapunov functional 23
1.4.4 Gauge invariance . . . . . . . . 24
1.4.5 A uniform bound on I?/II .... 25
1.5 Exterior evolutionary Maxwell system 26
1.5.1 Review of the Maxwell system 27
1.5.2 The G-L superconductivity model 28
1.5.3 The setting of the problem .... 28
1.6 Exterior steady-state Maxwell system . . 31
1. 7 Surface energy, superconductor classification 32
«
1.7.1 The sign of Uns when", 1 . 34
»
1. 7.2 The sign of Uns when", 1 ..... . 35
VI Contents
1.7.3 The case K, = 1/V2 ...... . 35
1. 7.4 Conclusion ........... . 36
1.8 Difference between 2-d and 3-d models. 36
1.9 Bibliographical remarks 38
2 Mathematical Foundation
2.1 Co-dimension one phase transition problems 40
2.1.1 Steady state problems 40
2.1.2 Evolutionary problems . 42
2.1.3 Long time behaviour . 44
2.2 Co-dimension two phase transition problems 46
2.2.1 Steady state problems on bounded domains 46
2.2.2 Steady state problems on JR? 47
2.2.3 Evolutionary problems . 48
2.2.4 Long time behaviour . 49
2.3 Mathematical description of vortices in JR2 49
2.4 Asymptotic methods for describing vortices in JR2 53
2.4.1 Steady state case in JR2: . 53
2.4.2 Evolutionary case in JR2 - Introduction: 54
2.4.3 Evolutionary case in JR2 - far field expansion: . 56
2.4.4 Evolutionary case in JR2 - local structure of the far field
solution near a vortex 56
2.4.5 Evolutionary case in JR2 - Core expansion . 58
2.4.6 Evolutionary case in JR2 - Matching of the core and far field
expansions 58
2.4.7 Vortex motion equation 59
2.5 Asymptotic methods for describing vortices in JR3 61
2.5.1 Steady state case in JR3 61
2.5.2 Evolutionary case in JR3 64
2.6 Bibliographical remarks. 65
3 Asymptotics Involving Magnetic Potential
3.1 Basic facts concerning fluid vortices 67
3.2 Asymptotic analysis 70
3.2.1 2-D steady state case . 70
3.2.2 Evolutionary case. 71
3.2.3 Far field 72
3.2.4 Core region 75
3.3 Asymptotic analysis of densely packed vortices 80
3.3.1 Outer region - a mean field model 81
3.3.2 Intermediate region 82
3.3.3 Core region 83
3.4 Bibliographical remarks 85
Contents vii
4 Steady State Solutions
4.1 Existence of steady state solutions . . . . . . . . . . . . . . . . .. 87
4.1.1 The outside field is a given function, 2-d case . . . . . . .. 87
4.1.2 The outside field is governed by the Maxwell system, 3-d case 90
4.2 Stability and mapping properties of solutions 91
4.2.1 Non-existence of local maxima ....... . 91
4.2.2 Boundedness of the order parameter .... . 91
4.2.3 Constant solutions and mixed state solutions 92
4.3 Co-dimension two vortex domain 93
4.4 Breakdown of superconductivity 97
4.5 A linearized problem. . 104
4.6 Bibliographical remarks 115
5 Evolutionary Solutions
5.1 2-d solutions with given external field 118
5.1.1 Mathematical setting ..... . 119
5.1.2 Existence and uniqueness of solutions 123
5.1.3 Proof of Theorem 1.2 ..... . 123
5.1.4 Proof of Theorem 1.1 129
5.2 Existence of 3-d evolutionary solutions. 133
5.3 The existence of an w-limit set as t ----> 00 141
5.4 An abstract theorem on global at tractors 147
5.5 Global atractor for the G-L sstem . . . . 149
5.6 Physical bounds on the global attractor . 152
5.7 The uniqueness of the long time limit of the evolutionary G-L so-
lutions. . . . . . . . . . 158
5.8 Bibliographical remarks . . . . . . . . . 159
6 Complex G-L Type Phase Transition Theory
6.1 Existence and basic properties of solutions 162
6.2 BBH type upper bound for energy of minimizers 164
6.3 Global estimates ............ . 166
6.4 Local estimates . . . . . . . . . . . . . . 171
6.5 The behaviour of solutions near vortices 174
6.6 Global E-independent estimates .... 185
6.7 Convergence of the solutions as E ----> 0 192
6.8 Main results on the limit functions 196
6.9 Renormalized energies . 205
6.10 Bibliographical remarks ..... . 219
viii Contents
7 The Slow Motion of Vortices
7.1 Introduction ..... . 221
7.2 Preliminaries . . . . . . . . . . . . . . . 224
7.3 Estimates from below for the mobilities 234
7.4 Estimates from above for the mobilities 244
7.5 Bibliographical remarks 250
8 Thin Plate/Film G-L Models
8.1 The outside Maxwell system - steady state case 252
8.1.1 The energy bound .. .......... . 253
8.1.2 Convergence properties of the rescaled variables. 255
8.1.3 Passing to the limit ......... . 257
8.2 The outside field is given - evolutionary case 259
8.2.1 Existence and uniqueness of solutions 259
8.2.2 The limit when E ----t 0 265
8.2.3 Some estimates . . . . ....... . 268
8.2.4 The convergence . ..... .. . . . 273
8.3 The outside field is given - formal analysis 278
8.3.1 Variational formulation ...... . 279
8.3.2 Formal asymptotic analysis when E ----t 0 279
8.4 Bibliographical remarks . . . . . . . . . . . . . 281
9 Pinning Theory
9.1 Local Pohozaev-type identity 284
9.2 Estimate the energy of minimizers 288
9.3 Local estimates. . . . . . . . . . . 290
9.4 Global Estimates. . . . . . . . . . . . . . . . . . . . . . 294
9.5 Convergence of solutions and the term 012 In (I1/!o 12 - 1) 2 294
9.6 Properties of (1/!*,A*) . . . . . . . . . . . . 301
9.7 Renormalized energy. . . . . . . . . . . . . . . . . . . . . 305
9.8 Pinning of vortices in other circumstances. . . . . . . . . 321
9.8.1 G-L model subject to thermo-perturbation or large horizon-
tal field . . . . . . . . . . 321
9.8.2 An anisotropic G-L model 323
9.8.3 A thin film G-L model 324
9.9 Bibliographical remarks 325
10 Numerical Analysis
10.1 Introduction . 327
10.2 Discretization. . . . . . . 331
10.2.1 Weak formulation 331
10.2.2 Discretization. 332
10.3 Stability estimates 335
10.4 Error estimates . .. . 339
Contents ix
10.5 A numerical example ......... . 350
10.6 Discretization using variable step length 351
10.7 A dual problem. . . . . . . . . . . . 353
10.7.1 Stability estimates ..... . 354
10.7.2 Error representation formula 359
10.8 A posteriori error analysis .. 361
10.8.1 Residuals ...... . 362
10.8.2 Proof of Theorem 4.1 369
10.9 Numerical implementation . 370
10.9.1 Comparison of the schemes 370
10.10 Bibliographical remarks . . . . . . 374
References 375
Index ... 384
Preface
The theory of complex Ginzburg-Landau type phase transition and its applica
tions to superconductivity and superfluidity has been a topic of great interest to
theoretical physicists and has been continuously and persistently studied since the
1950s. Today, there is an abundance of mathematical results spread over numer
ous scientific journals. However, before 1992, most of the studies concentrated on
formal asymptotics or linear analysis. Only isolated results by Berger, Jaffe and
Taubes and some of their colleagues touched the nonlinear aspects in great detail.
In 1991, a physics seminar given by Ed Copeland at Sussex University inspired
Q. Tang, the co-author of this monograph, to study the subject. Independently in
Munich, K.-H. Hoffmann and his collaborators Z. Chen and J. Liang started to
work on the topic at the same time. Soon it became clear that at that time, groups
of mathematicians at Oxford and Virginia Tech had already studied the subject
for a couple of years. They inspired experts in interface phase transition problems
and their combined effort established a rigorous mathematical framework for the
Ginzburg-Landau system.
At the beginning Q. Tang collaborated with C.M. Elliott and H. Matano.
When they met at the University of Sussex in the Summer of 1991, they discussed
what kind of problems they should study. They found that the mathematical justi
fication of the vortex structure and the relation between the applied magnetic field
and the number of vortices were the two fundamental mathematical problems that
the physicists failed to address, despite the long period they had been working on
the subject. The trio also intended to look at a problem which was mathematically
interesting: the limit behavior of the energy minimizers of the following Landau
problem
as E ---t O. Here Bl is the unit ball in JR2 and n is any integer. But during Matano's
stay at Sussex, they didn't find a solution to the problem.
Back in Tokyo, Matano talked to H. Brezis, who visited Japan at the time,
about the limit problem (in a Japanese cafe, according to H. Brezis), and that led
