Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations
Authors:
Valeria Berti and Stefania Gatti
Journal:
Quart. Appl. Math. 64 (2006), 617-639
MSC (2000):
Primary 35B41
DOI:
https://doi.org/10.1090/S0033-569X-06-01044-9
Published electronically:
October 16, 2006
MathSciNet review:
2284463
Full-text PDF Free Access
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Abstract: This article is devoted to the long-term dynamics of a parabolic-hyperbolic system arising in superconductivity. In the literature, the existence and uniqueness of the solution have been investigated but, to our knowledge, no asymptotic result is available. For the bidimensional model we prove that the system generates a dissipative semigroup in a proper phase-space where it possesses a (regular) global attractor. Then, we show the existence of an exponential attractor whose basin of attraction coincides with the whole phase-space. Thus, in particular, this exponential attractor contains the global attractor which, as a consequence, is of finite fractal dimension.
BFG V. Berti, M. Fabrizio, C. Giorgi, Gauge invariance and asymptotic behavior for the Ginzburg-Landau equations of superconductivity, J. Math. Anal. Appl. (to appear)
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BFG V. Berti, M. Fabrizio, C. Giorgi, Gauge invariance and asymptotic behavior for the Ginzburg-Landau equations of superconductivity, J. Math. Anal. Appl. (to appear)
CPS M. Conti, V. Pata, M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J. 55 (2006), 169–215.
D Q. Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, Appl. Anal. 53 (1994), 1–17.
EMZ M. Efendiev, A. Miranville, S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb R^3$, C.R. Acad. Sci. Paris Sér. I Math. 330 (2000), 713–718.
FGMZ P. Fabrie, C. Galusinski, A. Miranville, S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems 10 (2004), 211–238.
GGMP S. Gatti, M. Grasselli, A. Miranville, V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc. 134 (2006), 117-127.
GE L. Gor′kov and G. Èliashberg, Generalization of the Ginzburg-Landau equations for nonstationary problems in the case of alloys with paramagnetic impurities, Soviet Phys. JETP 27 (1968), 328–334.
PPV V. Pata, G. Prouse, M. I. Vishik, Traveling waves of dissipative non-autonomous hyperbolic equations in a strip, Adv. Differential Equations 3 (1998), 249–270.
RWW A. Rodriguez-Bernal, B. Wang, R. Willie, Asymptotic behaviour of the time-dependent Ginzburg-Landau equations of superconductivity, Math. Meth. Appl. Sci. 22 (1999), 1647–1669.
TW Q. Tang, S. Wang, Time dependent Ginzburg-Landau equations of superconductivity, Physica D 88 (1995), 139–166.
T R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer-Verlag, New York, 1988.
TK M. Tsutsumi, H. Kasai, The time-dependent Ginzburg-Landau-Maxwell equations, Nonlinear Analysis 37 (1999), 187–216.
WZ S. Wang, M. Q. Zhan, $L^p$ solutions to the time-dependent Ginzburg-Landau equations of superconductivity, Nonlinear Analysis 36 (1999), 661–677.
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Additional Information
Valeria Berti
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I-40126 Bologna, Italy
Email:
berti@dm.unibo.it
Stefania Gatti
Affiliation:
Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, I-44100 Ferrara, Italy
Email:
s.gatti@economia.unife.it
Keywords:
Ginzburg-Landau-Maxwell equations,
superconductivity,
strongly continuous semigroup,
global attractor,
exponential attractor
Received by editor(s):
April 11, 2005
Published electronically:
October 16, 2006
Additional Notes:
Research performed under the auspices of G.N.F.M. - I.N.D.A.M. and partially supported by the Italian MIUR PRIN Research Project “Dinamica a lungo termine e problemi di regolarità per modelli di cambiamento di fase" and by University of Bologna - Funds for selected topics.
Article copyright:
© Copyright 2006
Brown University
