Global and exponential attractors for a Ginzburg-Landau model of superfluidity

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April 2011, 4(2): 247-271. doi: 10.3934/dcdss.2011.4.247

Global and exponential attractors for a Ginzburg-Landau model of superfluidity

Alessia Berti ^1, , Valeria Berti ^2, and Ivana Bochicchio ^3,

1.	Facoltà di Ingegneria, Università e-Campus, 22060 Novedrate (CO), Italy
2.	Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy
3.	Dipartimento di Matematica e Informatica, Università di Salerno, 84084 Fisciano (SA), Italy

Received October 2008 Revised June 2009 Published November 2010

Abstract
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The long-time behavior of the solutions for a non-isothermal model in superfluidity is investigated. The model describes the transition between the normal and the superfluid phase in liquid ⁴He by means of a non-linear differential system, where the concentration of the superfluid phase satisfies a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techinques of semigroups theory, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.

Keywords: Superfluids, Lyapunov functional, Ginzburg-Landau equations, exponential attractors..

Mathematics Subject Classification: 35B41, 37B25, 82D5.

Citation: Alessia Berti, Valeria Berti, Ivana Bochicchio. Global and exponential attractors for a Ginzburg-Landau model of superfluidity. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 247-271. doi: 10.3934/dcdss.2011.4.247

References:

[1]	R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.
[2]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.
[3]	V. Berti and S. Gatti, Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, Quart. Appl. Math., 64 (2006), 617-639.
[4]	V. Berti and M. Fabrizio, Existence and uniqueness for a mathematical model in superfluidity, Math. Meth. Appl. Sci., 31 (2008), 1441-1459. doi: 10.1002/mma.981.
[5]	V. Berti, M. Fabrizio and C. Giorgi, Gauge invariance and asymptotic behavior for the Ginzburg-Landau equations of superconductivity, J. Math. Anal. Appl., 329 (2007), 357-375. doi: 10.1016/j.jmaa.2006.06.031.
[6]	M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, New York, 1996.
[7]	M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720. doi: 10.3934/cpaa.2005.4.705.
[8]	Q. Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, Appl. Anal., 53 (1994), 1-17. doi: 10.1080/00036819408840240.
[9]	A. Eden, C. Foias, B. Nicoalenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," John-Wiley, New York, 1994.
[10]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
[11]	M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C.R. Acad.Sci. Paris Ser. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.
[12]	M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions, Internat. J. Engrg. Sci., 44 (2006), 529-539. doi: 10.1016/j.ijengsci.2006.02.006.
[13]	M. Fabrizio, A Ginzburg-Landau model for the phase transition in Helium II, Z. Angew. Math. Phys., 61 (2010), 329-340. doi: 10.1007/s00033-009-0011-5.
[14]	P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211.
[15]	J. Fleckinger-Pellé, H. Kaper and P. Takac, Dynamics of the Ginzburg-Landau equations of superconductivity, Nonlinear Anal., 32 (1998), 647-665. doi: 10.1016/S0362-546X(97)00508-7.
[16]	S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127. doi: 10.1090/S0002-9939-05-08340-1.
[17]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Math. Soc., Providence, 1988.
[18]	H. G. Kaper and P. Takac, An equivalence relation for the Ginzburg-Landau equations of superconductivity, Z. Angew. Math. Phys., 48 (1997), 665-675. doi: 10.1007/s000330050054.
[19]	K. Mendelssohn, Liquid Helium, in "Handbuch Physik" (ed. S. Flugge), Vol. XV, Springer, Berlin (1956), 370-461.
[20]	R. Nibbi, Some generalized Poincaré inequalities and applications to problems arising in electromagnetism, J. Inequal. Appl., 4 (1999), 283-299. doi: 10.1155/S1025583499000405.
[21]	A. Rodriguez-Bernal, B. Wang and R. Willie, Asymptotic behaviour of the time-dependent Ginzburg-Landau equations of superconductivity, Math. Meth. Appl. Sci., 22 (1999), 1647-1669. doi: 10.1002/(SICI)1099-1476(199912)22:18<1647::AID-MMA97>3.0.CO;2-W.
[22]	Q. Tang Q and S. Wang, Time dependent Ginzburg-Landau superconductivity equations, Physica D, 88 (1995), 130-166.
[23]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1988.
[24]	D. R. Tilley and J. Tilley, "Superfluidity and Superconductivity," Graduate student series in physics 13, Bristol, 1990.
[25]	M. Tinkham, "Introduction to Superconductivity," McGraw-Hill, New York, 1975.

show all references

References:

[1]	R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.
[2]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.
[3]	V. Berti and S. Gatti, Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, Quart. Appl. Math., 64 (2006), 617-639.
[4]	V. Berti and M. Fabrizio, Existence and uniqueness for a mathematical model in superfluidity, Math. Meth. Appl. Sci., 31 (2008), 1441-1459. doi: 10.1002/mma.981.
[5]	V. Berti, M. Fabrizio and C. Giorgi, Gauge invariance and asymptotic behavior for the Ginzburg-Landau equations of superconductivity, J. Math. Anal. Appl., 329 (2007), 357-375. doi: 10.1016/j.jmaa.2006.06.031.
[6]	M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, New York, 1996.
[7]	M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720. doi: 10.3934/cpaa.2005.4.705.
[8]	Q. Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, Appl. Anal., 53 (1994), 1-17. doi: 10.1080/00036819408840240.
[9]	A. Eden, C. Foias, B. Nicoalenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," John-Wiley, New York, 1994.
[10]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
[11]	M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C.R. Acad.Sci. Paris Ser. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.
[12]	M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions, Internat. J. Engrg. Sci., 44 (2006), 529-539. doi: 10.1016/j.ijengsci.2006.02.006.
[13]	M. Fabrizio, A Ginzburg-Landau model for the phase transition in Helium II, Z. Angew. Math. Phys., 61 (2010), 329-340. doi: 10.1007/s00033-009-0011-5.
[14]	P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211.
[15]	J. Fleckinger-Pellé, H. Kaper and P. Takac, Dynamics of the Ginzburg-Landau equations of superconductivity, Nonlinear Anal., 32 (1998), 647-665. doi: 10.1016/S0362-546X(97)00508-7.
[16]	S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127. doi: 10.1090/S0002-9939-05-08340-1.
[17]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Math. Soc., Providence, 1988.
[18]	H. G. Kaper and P. Takac, An equivalence relation for the Ginzburg-Landau equations of superconductivity, Z. Angew. Math. Phys., 48 (1997), 665-675. doi: 10.1007/s000330050054.
[19]	K. Mendelssohn, Liquid Helium, in "Handbuch Physik" (ed. S. Flugge), Vol. XV, Springer, Berlin (1956), 370-461.
[20]	R. Nibbi, Some generalized Poincaré inequalities and applications to problems arising in electromagnetism, J. Inequal. Appl., 4 (1999), 283-299. doi: 10.1155/S1025583499000405.
[21]	A. Rodriguez-Bernal, B. Wang and R. Willie, Asymptotic behaviour of the time-dependent Ginzburg-Landau equations of superconductivity, Math. Meth. Appl. Sci., 22 (1999), 1647-1669. doi: 10.1002/(SICI)1099-1476(199912)22:18<1647::AID-MMA97>3.0.CO;2-W.
[22]	Q. Tang Q and S. Wang, Time dependent Ginzburg-Landau superconductivity equations, Physica D, 88 (1995), 130-166.
[23]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1988.
[24]	D. R. Tilley and J. Tilley, "Superfluidity and Superconductivity," Graduate student series in physics 13, Bristol, 1990.
[25]	M. Tinkham, "Introduction to Superconductivity," McGraw-Hill, New York, 1975.

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