Author | Hoffmann, Karl-Heinz. author |
---|---|
Title | Ginzburg-Landau Phase Transition Theory and Superconductivity [electronic resource] / by Karl-Heinz Hoffmann, Qi Tang |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2001 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8274-3 |
Descript | XII, 384 p. online resource |
1 Introduction -- 1.1 Brief history -- 1.2 The G-L phenomenological theory -- 1.3 Some considerations arising from scaling -- 1.4 The evolutionary G-L system โ 2-d case -- 1.5 Exterior evolutionary Maxwell system -- 1.6 Exterior steady-state Maxwell system -- 1.7 Surface energy, superconductor classification -- 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 -- 2.2 Co-dimension two phase transition problems -- 2.3 Mathematical description of vortices in ?2 -- 2.4 Asymptotic methods for describing vortices in ?2 -- 2.5 Asymptotic methods for describing vortices in ?3 -- 2.6 Bibliographical remarks -- 3 Asymptotics Involving Magnetic Potential -- 3.1 Basic facts concerning fluid vortices -- 3.2 Asymptotic analysis -- 3.3 Asymptotic analysis of densely packed vortices -- 3.4 Bibliographical remarks -- 4 Steady State Solutions -- 4.1 Existence of steady state solutions -- 4.2 Stability and mapping properties of 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 -- 5.2 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 โ steady state case -- 8.2 The outside field is given โ evolutionary case -- 8.3 The outside field is given โ formal analysis -- 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 -- 9.9 Bibliographical remarks -- 10 Numerical Analysis -- 10.1 Introduction -- 10.2 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 -- 10.8 A posteriori error analysis -- 10.9 Numerical implementation -- 10.10 Bibliographical remarks -- References