zbMATH — the first resource for mathematics

Static and moving vortices in Ginzburg-Landau theories. (English) Zbl 0867.35039
Baker, Garth (ed.) et al., Nonlinear partial differential equations in geometry and physics. The 1995 Barrett lectures, Knoxville, TN, USA, March 22–24, 1995. Basel: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 29, 71-111 (1997).
The paper consists of three lectures delivered by the author in March, 1995 at the University of Tennessee, Knoxville for the Barrett Lectures series. The main purpose is to give a brief description of some mathematical work concerning the static and moving vortices of solutions to Ginzburg-Landau equations arising in the theory of superconductivity.
The first lecture describes the main mathematical results due to Bethuel-Brézis-Hélein and recent improvements and simplifications, as well as the asymptotic behaviour of distributions of vortices for static solutions or approximate solutions.
The aim of the second lecture is to describe the important role of the so-called renormalized energy in the study of vortices. The connection between the critical points renormalized energy and solutions to Ginzburg-Landau equations is explained. Further, certain hysteresis phenomena are shown for the phase transition near the lower critical magnetic field.
The third lecture starts with the Gor’kov-Eliashberg equation for the evolution of Ginzburg-Landau. Global existence of classical solutions is proved. Also, the asymptotic behaviour of these solutions as well as dynamic properties of vortices are studied.
For the entire collection see [Zbl 0856.00026].

35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35Q60 PDEs in connection with optics and electromagnetic theory
82D55 Statistical mechanics of superconductors