Time dependent Ginzburg-Landau equations of superconductivity

Abstract We study in this article the existence, uniqueness and long time behavior of the solutions of a nonstationary Ginzburg-Landau superconductivity model. We first prove the existence and uniqueness of solutions with H 1 initial data, which are crucial for the study of the global attractor. We also obtain, for the first time, the existence of global weak solutions of the model with L 2 initial data. It is then proved that the Ginzburg-Landau system admits a global attractor, which represents exactly all the long time dynamics of the system. The global attractor obtained consists of exactly the set of steady state solutions and its unstable manifold. Its Hausdorff and fractal dimensions are estimated in terms of the physically relevant Ginzburg-Landau parameter, diffusion parameter and applied magnetic field. We construct explicitly absorbing sets for some abstract semigroups having a Lyapunov functional and consequently prove the existence of global attractors. This abstract result is applied to the Ginzburg-Landau system, for which the existence of global attractor does not seem to be the direct consequence of some a priori estimates of solutions.

[1]  Anneliese Defranceschi,et al.  Limits of minimum problems with convex obstacles for vector valued functions , 1994 .

[2]  Sam D. Howison,et al.  Macroscopic Models for Superconductivity , 1992, SIAM Rev..

[3]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[4]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[5]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[6]  Qiang Du,et al.  Global existence and uniqueness of solutions of the time-dependent ginzburg-landau model for superconductivity , 1994 .

[7]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[8]  Zhiming Chen,et al.  On a non-stationary Ginzburg–Landau superconductivity model , 1993 .

[9]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[10]  G. M. Éliashberg,et al.  Generalization of the Ginzburg-Landau Equations for Non-Stationary Problems in the Case of Alloys with Paramagnetic Impurities - JETP 27, 328 (1968) , 1968 .

[11]  J. Preskill.  VORTICES AND MONOPOLES , 1986 .

[12]  R. Temam,et al.  Attractors Representing Turbulent Flows , 1985 .

[13]  F. D. Rosi,et al.  SUPERCONDUCTIVITY IN METALS AND ALLOYS , 1963 .

[14]  Tang Qi,et al.  On an evolutionary system of ginzburg-landau equations with fixed total magnetic flux , 1995 .

[15]  L. Cooper,et al.  Theory of superconductivity , 1957 .

[16]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[17]  R. Temam.  Navier-Stokes Equations , 1977 .

[18]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 1972 .

[19]  Radu Purice,et al.  A boundary value problem related to the Ginzburg-Landau model , 1991 .

[20]  T. Qi.  Remark on a relation between magnetic field and topological degree of order parameter in a type-II superconductivity model , 1993 .

[21]  Long Time Behavior of the Ginzburg-Landau Superconductivity Equations , 1995 .

[22]  Qiang Du,et al.  Analysis and Approximation of the Ginzburg-Landau Model of Superconductivity , 1992, SIAM Rev..

[23]  R. Temam,et al.  Inertial Forms of Navier-Stokes Equations on the Sphere , 1993, chao-dyn/9304004.

[24]  Zeros of a complex Ginzburg–Landau order parameter with applications to superconductivity , 1994, European Journal of Applied Mathematics.

[25]  L. Landau,et al.  On the theory of superconductivity , 1955 .