Global existence and uniqueness of solutions on the time-dependent Ginzburg- Landau model for superconductivity. (English) Zbl 0843.35019

Du, Qiang

Appl. Anal. 53, No. 1-2, 1-17 (1994).

Summary: We consider the initial-boundary value problems of the time-dependent nonlinear Ginzburg-Landau equations in superconductivity. It is assumed that the material sample occupies a bounded domain in two- and three-dimensional spaces. We illustrate that the original equations are not well-posed. In order to fix the lack of uniqueness of the solutions, possible choices of the gauge are identified. Global existence and uniqueness of solutions are proved in a proper gauge. A by-product is the convergence of finite-dimensional Galerkin approximations which may be used in the numerical study of superconductivity phenomena.

MSC:

35G25	Initial value problems for nonlinear higher-order PDEs
82D55	Statistical mechanics of superconductors
65M99	Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99	Numerical methods for partial differential equations, boundary value problems
35R25	Ill-posed problems for PDEs

Keywords:

initial-boundary value problems; time-dependent nonlinear Ginzburg-Landau equations; Galerkin approximations