This monograph compiles, rearranges, and refines recent research results in the complex G-L theory with or without immediate applications to the theory of superconductivity. An authoritative reference for applied mathematicians, theoretical physicists and engineers interested in the quantitative description of superconductivity using Ginzburg-Landau theory.

Inhalt

1 Introduction.- 1.1 Brief history.- 1.1.1 Meissner effect - diamagnetism.- 1.1.2 The London equation and the penetration depth.- 1.1.3 The coherence length.- 1.1.4 Classification of superconductors.- 1.1.5 Vortices.- 1.1.6 Summary.- 1.2 The G-L phenomenological theory.- 1.2.1 The free energy and the G-L equations.- 1.2.2 Rescaling and the values of the constants.- 1.2.3 Gauge invariance.- 1.3 Some considerations arising from scaling.- 1.3.1 The two characteristic lengths ?(T) and À(T).- 1.3.2 The validity of the G-L theory.- 1.4 The evolutionary G-L system - 2-d case.- 1.4.1 The system.- 1.4.2 Mathematical scaling.- 1.4.3 The G-L functional as a Lyapunov functional.- 1.4.4 Gauge invariance.- 1.4.5 A uniform bound on |?|.- 1.5 Exterior evolutionary Maxwell system.- 1.5.1 Review of the Maxwell system.- 1.5.2 The G-L superconductivity model.- 1.5.3 The setting of the problem.- 1.6 Exterior steady-state Maxwell system.- 1.7 Surface energy, superconductor classification.- 1.7.1 The sign of ?ns when ? ? 1.- 1.7.2 The sign of ?ns when ? ? 1.- 1.7.3 The case $$\mathcal{K} = 1/\sqrt 2 $$.- 1.7.4 Conclusion.- 1.8 Difference between 2-d and 3-d models.- 1.9 Bibliographical remarks.- 2 Mathematical Foundation.- 2.1 Co-dimension one phase transition problems.- 2.1.1 Steady state problems.- 2.1.2 Evolutionary problems.- 2.1.3 Long time behaviour.- 2.2 Co-dimension two phase transition problems.- 2.2.1 Steady state problems on bounded domains.- 2.2.2 Steady state problems on ?2.- 2.2.3 Evolutionary problems.- 2.2.4 Long time behaviour.- 2.3 Mathematical description of vortices in ?2.- 2.4 Asymptotic methods for describing vortices in ?2.- 2.4.1 Steady state case in ?2.- 2.4.2 Evolutionary case in ?2 - Introduction..- 2.4.3 Evolutionary case in ?2 - far field expansion:.- 2.4.4 Evolutionary case in ?2 - local structure of the far fieldsolution near a vortex.- 2.4.5 Evolutionary case in ?2 - Core expansion.- 2.4.6 Evolutionary case in ?2 - Matching of the core and far fieldexpansions.- 2.4.7 Vortex motion equation.- 2.5 Asymptotic methods for describing vortices in ?3.- 2.5.1 Steady state case in ?3.- 2.5.2 Evolutionary case in ?3.- 2.6 Bibliographical remarks.- 3 Asymptotics Involving Magnetic Potential.- 3.1 Basic facts concerning fluid vortices.- 3.2 Asymptotic analysis.- 3.2.1 2-D steady state case.- 3.2.2 Evolutionary case.- 3.2.3 Far field.- 3.2.4 Core region.- 3.3 Asymptotic analysis of densely packed vortices.- 3.3.1 Outer region - a mean field model.- 3.3.2 Intermediate region.- 3.3.3 Core region.- 3.4 Bibliographical remarks.- 4 Steady State Solutions.- 4.1 Existence of steady state solutions.- 4.1.1 The outside field is a given function, 2-d case.- 4.1.2 The outside field is governed by the Maxwell system, 3-d case.- 4.2 Stability and mapping properties of solutions.- 4.2.1 Non-existence of local maxima.- 4.2.2 Boundedness of the order parameter.- 4.2.3 Constant solutions and mixed state solutions.- 4.3 Co-dimension two vortex domain.- 4.4 Breakdown of superconductivity.- 4.5 A linearized problem.- 4.6 Bibliographical remarks.- 5 Evolutionary Solutions.- 5.1 2-d solutions with given external field.- 5.1.1 Mathematical setting.- 5.1.2 Existence and uniqueness of solutions.- 5.1.3 Proof of Theorem 1.2.- 5.1.4 Proof of Theorem 1.1.- 5.2 Existence of 3-d evolutionary solutions.- 5.3 The existence of an ?-limit set as t ? ?.- 5.4 An abstract theorem on global attractors.- 5.5 Global atractor for the G-L sstem.- 5.6 Physical bounds on the global attractor.- 5.7 The uniqueness of the long time limit of the evolutionary G-L so-lutions.- 5.8 Bibliographical remarks.- 6 Complex G-L Type Phase Transition Theory.- 6.1 Existence and basic properties of solutions.- 6.2 BBH type upper bound for energy of minimizers.- 6.3 Global estimates.- 6.4 Local estimates.- 6.5 The behaviour of solutions near vortices.- 6.6 Global ?-independent estimates.- 6.7 Convergence of the solutions as ? ? 0.- 6.8 Main results on the limit functions.- 6.9 Renormalized energies.- 6.10 Bibliographical remarks.- 7 The Slow Motion of Vortices.- 7.1 Introduction.- 7.2 Preliminaries.- 7.3 Estimates from below for the mobilities.- 7.4 Estimates from above for the mobilities.- 7.5 Bibliographical remarks.- 8 Thin Plate/Film G-L Models.- 8.1 The outside Maxwell system - steady state case.- 8.1.1 The energy bound.- 8.1.2 Convergence properties of the resealed variables.- 8.1.3 Passing to the limit.- 8.2 The outside field is given - evolutionary case.- 8.2.1 Existence and uniqueness of solutions.- 8.2.2 The limit when ? ? 0.- 8.2.3 Some estimates.- 8.2.4 The convergence.- 8.3 The outside field is given - formal analysis.- 8.3.1 Variational formulation.- 8.3.2 Formal asymptotic analysis when ? ? 0.- 8.4 Bibliographical remarks.- 9 Pinning Theory.- 9.1 Local Pohozaev-type identity.- 9.2 Estimate the energy of minimizers.- 9.3 Local estimates.- 9.4 Global Estimates.- 9.5 Convergence of solutions and the term $$ \frac{1} {{\varepsilon ^2 }}\int\Omega {(\left| {\psi \varepsilon } \right|^2 - 1)^2 } $$.- 9.6 Properties of ?, A.- 9.7 Renormalized energy.- 9.8 Pinning of vortices in other circumstances.- 9.8.1 G-L model subject to thermo-perturbation or large horizon-tal field.- 9.8.2 An anisotropic G-L model.- 9.8.3 A thin film G-L model.- 9.9 Bibliographical remarks.- 10 Numerical Analysis.- 10.1 Introduction.- 10.2 Discretization.- 10.2.1 Weak formulation.- 10.2.2 Discretization.- 10.3 Stability estimates.- 10.4 Error estimates.- 10.5 A numerical example.- 10.6 Discretization using variable step length.- 10.7 A dual problem.- 10.7.1 Stability estimates.- 10.7.2 Error representation formula.- 10.8 A posteriori error analysis.- 10.8.1 Residuals.- 10.8.2 Proof of Theorem 4.1.- 10.9 Numerical implementation.- 10.9.1 Comparison of the schemes.- 10.10 Bibliographical remarks.- References.