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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

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 & 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. Google Scholar

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. Google Scholar

[3]

V. Berti and S. Gatti, Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, Quart. Appl. Math., 64 (2006), 617-639.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, New York, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Eden, C. Foias, B. Nicoalenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," John-Wiley, New York, 1994. Google Scholar

[10]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. Google Scholar

[11]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$, C.R. Acad.Sci. Paris Ser. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Math. Soc., Providence, 1988. Google Scholar

[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.  Google Scholar

[19]

K. Mendelssohn, Liquid Helium, in "Handbuch Physik" (ed. S. Flugge), Vol. XV, Springer, Berlin (1956), 370-461. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Q. Tang Q and S. Wang, Time dependent Ginzburg-Landau superconductivity equations, Physica D, 88 (1995), 130-166.  Google Scholar

[23]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1988. Google Scholar

[24]

D. R. Tilley and J. Tilley, "Superfluidity and Superconductivity," Graduate student series in physics 13, Bristol, 1990. Google Scholar

[25]

M. Tinkham, "Introduction to Superconductivity," McGraw-Hill, New York, 1975. Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975. Google Scholar

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. Google Scholar

[3]

V. Berti and S. Gatti, Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, Quart. Appl. Math., 64 (2006), 617-639.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, New York, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Eden, C. Foias, B. Nicoalenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," John-Wiley, New York, 1994. Google Scholar

[10]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. Google Scholar

[11]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$, C.R. Acad.Sci. Paris Ser. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Math. Soc., Providence, 1988. Google Scholar

[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.  Google Scholar

[19]

K. Mendelssohn, Liquid Helium, in "Handbuch Physik" (ed. S. Flugge), Vol. XV, Springer, Berlin (1956), 370-461. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Q. Tang Q and S. Wang, Time dependent Ginzburg-Landau superconductivity equations, Physica D, 88 (1995), 130-166.  Google Scholar

[23]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1988. Google Scholar

[24]

D. R. Tilley and J. Tilley, "Superfluidity and Superconductivity," Graduate student series in physics 13, Bristol, 1990. Google Scholar

[25]

M. Tinkham, "Introduction to Superconductivity," McGraw-Hill, New York, 1975. Google Scholar

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