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