American Institute of Mathematical Sciences

Advanced
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, (1975).   Google Scholar

[2]

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

[3]

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

[4]

V. Berti and M. Fabrizio, Existence and uniqueness for a mathematical model in superfluidity,, Math. Meth. Appl. Sci., 31 (2008), 1441.  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.  doi: 10.1016/j.jmaa.2006.06.031.  Google Scholar

[6]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996).   Google Scholar

[7]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity,, Commun. Pure Appl. Anal., 4 (2005), 705.  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.  doi: 10.1080/00036819408840240.  Google Scholar

[9]

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

[10]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19,, American Mathematical Society, (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.  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.  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.  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.  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.  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.  doi: 10.1090/S0002-9939-05-08340-1.  Google Scholar

[17]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Math. Soc., (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).  doi: 10.1007/s000330050054.  Google Scholar

[19]

K. Mendelssohn, Liquid Helium,, in, XV (1956), 370.   Google Scholar

[20]

R. Nibbi, Some generalized Poincaré inequalities and applications to problems arising in electromagnetism,, J. Inequal. Appl., 4 (1999), 283.  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.  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.   Google Scholar

[23]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer-Verlag, (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, (1975).   Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces,", Academic Press, (1975).   Google Scholar

[2]

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

[3]

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

[4]

V. Berti and M. Fabrizio, Existence and uniqueness for a mathematical model in superfluidity,, Math. Meth. Appl. Sci., 31 (2008), 1441.  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.  doi: 10.1016/j.jmaa.2006.06.031.  Google Scholar

[6]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996).   Google Scholar

[7]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity,, Commun. Pure Appl. Anal., 4 (2005), 705.  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.  doi: 10.1080/00036819408840240.  Google Scholar

[9]

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

[10]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19,, American Mathematical Society, (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.  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.  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.  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.  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.  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.  doi: 10.1090/S0002-9939-05-08340-1.  Google Scholar

[17]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Math. Soc., (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).  doi: 10.1007/s000330050054.  Google Scholar

[19]

K. Mendelssohn, Liquid Helium,, in, XV (1956), 370.   Google Scholar

[20]

R. Nibbi, Some generalized Poincaré inequalities and applications to problems arising in electromagnetism,, J. Inequal. Appl., 4 (1999), 283.  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.  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.   Google Scholar

[23]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer-Verlag, (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, (1975).   Google Scholar

[1]

Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145
[2]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205
[3]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[4]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[5]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280
[6]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476
[7]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754
[8]

Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 507-522. doi: 10.3934/dcds.1998.4.507
[9]

Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 835-851. doi: 10.3934/dcdss.2020048
[10]

Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075
[11]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801
[12]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715
[13]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121
[14]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[15]

Tianlong Shen, Jianhua Huang. Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 605-625. doi: 10.3934/dcdsb.2017029
[16]

Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095
[17]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181
[18]

Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109
[19]

Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046
[20]

Leonid Berlyand, Volodymyr Rybalko, Nung Kwan Yip. Renormalized Ginzburg-Landau energy and location of near boundary vortices. Networks & Heterogeneous Media, 2012, 7 (1) : 179-196. doi: 10.3934/nhm.2012.7.179

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences