Time dependent Ginzburg-Landau equations of superconductivity. (English) Zbl 0900.35371

Tang, Qi; Wang, S.

Physica D 88, No. 3-4, 139-166 (1995).

Summary: 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.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
82D55	Statistical mechanics of superconductors
37C70	Attractors and repellers of smooth dynamical systems and their topological structure

Keywords:

existence; uniqueness; nonstationary Ginzburg-Landau superconductivity model; global attractor; Hausdorff and fractal dimensions

References:

[1]	Bardeen, J.; Cooper, L. N.; Schreiffer, J. R., Theory of Superconductivity, Phys. Rev., 108, 1175 (1957) · Zbl 0090.45401
[2]	Chen, Y. Y., Vortices for the Ginzburg-Landau equations—the nonsymmetric case in unbounded domain, Contemporary Mathematics, 108, 19-32 (1990) · Zbl 0705.35113
[3]	Z.M. Chen, K.H. Hoffmann and J. Liang, On a non-stationary Ginzburg-Landau superconductivity model, to appear in Math. Meth. Appl. Sci.; Z.M. Chen, K.H. Hoffmann and J. Liang, On a non-stationary Ginzburg-Landau superconductivity model, to appear in Math. Meth. Appl. Sci. · Zbl 0817.35111
[4]	Chapman, S. J.; Howison, S. D.; Ockendon, J. R., Macroscopic Models for Superconductivity, SIAM Review, 34, No. 4, 529-560 (1990) · Zbl 0769.73068
[5]	Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Memoirs of AMS, 53, No. 314 (1985) · Zbl 0567.35070
[6]	Du, Q., Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, Appl. Anal., 52, Nos. 1-2, 1-17 (1994) · Zbl 0843.35019
[7]	Du, Q.; Gunzburger, M. D.; Peterson, J. S., Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Review, 34, 54-81 (1992) · Zbl 0787.65091
[8]	Elliott, C.; Matano, H.; Tang, Q., Zeros of a complex Ginzburg-Landau order parameter with applications to superconductivity, Eur. J. Appl. Math., 5, No. 7, 437-448 (1994) · Zbl 0817.35112
[9]	C. Elliot and Q. Tang, Existence theorems for a evolutionary superconductivity model, preprint.; C. Elliot and Q. Tang, Existence theorems for a evolutionary superconductivity model, preprint.
[10]	de Gennes, P., Superconductivity in Metals and Alloys (1966), Benjamin: Benjamin New York · Zbl 0138.22801
[11]	Gor’kov, L. P.; Éliashberg, G. M., Generalization of the Ginzburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities, Sov. Phys. JETP, 27, 328-334 (1968)
[12]	Ginzburg, V.; Landau, L., (Landau, L. D.; ter Haar, I. D., Man of Physics (1965), Pergamon: Pergamon Oxford), 138-167, (in English)
[13]	Girault, V.; Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations (1986), Springer-Verlag: Springer-Verlag Berlin · Zbl 0413.65081
[14]	Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0456.35001
[15]	Jaffe, A.; Taubes, C., Vortices and Monopoles (1980), Birkhauser · Zbl 0457.53034
[16]	Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow (1969), Gordon and Breach, Science Publishers: Gordon and Breach, Science Publishers New York, London, Paris · Zbl 0184.52603
[17]	Ladyzhenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and quasi-linear equations of parabolic type (1967), Moscow · Zbl 0164.12302
[18]	J. Liang and Q. Tang, Asymptotic behavior of an evolutionary Ginzburg-Landau superconductivity model, to appear in Math. Anal. Appl.; J. Liang and Q. Tang, Asymptotic behavior of an evolutionary Ginzburg-Landau superconductivity model, to appear in Math. Anal. Appl. · Zbl 0845.35118
[19]	Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603
[20]	Monvel-Berthier, A. B.; Georgescu, V.; Purice, R., A boundary value problem related to the Ginzburg-Landau Model, Comm. Math. Phys., 142, 1-23 (1991) · Zbl 0742.35045
[21]	Tang, Q., Remark on a relation between the magnetic field and the topological degree of the order parameter in superconductivity, Physica D, 69, 320-326 (1993) · Zbl 0791.58032
[22]	Tang, Q., On an evolutionary system of Ginzburg-Landau equations with fixed total magnetic flux, Comm. in PDE, 20, Nos. 1-2, 1-36 (1995) · Zbl 0833.35132
[23]	Tang, Q.; Wang, S., Long time behavior of the Ginzburg-Landau superconductivity equations, Appl. Math. Lett., 8, No. 2, 31-34 (1995) · Zbl 0829.35122
[24]	Temam, R., Navier-Stokes Equations (1979), North-Holland · Zbl 0454.35073
[25]	Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1988), Springer-Verlag · Zbl 0662.35001
[26]	Temam, R., Navier-Stokes equations and nonlinear functional analysis, (CBMS-NSF Regional Conference Series in Applied Mathematics (1983), SIAM: SIAM Philadelphia) · Zbl 0833.35110
[27]	Temam, R.; Wang, S., Inertial forms of Navier-Stokes equations on the sphere, J. Funct. Anal., 117, No. 1, 215-242 (1993) · Zbl 0801.35109