×

zbMATH — the first resource for mathematics

Finite dimensional global and exponential attractors for a class of coupled time-dependent Ginzburg-Landau equations. (English) Zbl 1239.35026
Summary: We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover. First, we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phasespace which possesses a global attractor. Then we establish the existence of an exponential attractor. As a consequence, we show that the global attractor is of finite fractal dimension.

MSC:
35B41 Attractors
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
81V45 Atomic physics
35Q56 Ginzburg-Landau equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Baranov M A, Petrov D S. Low-energy collecive excitations in a superfluid trapped Fermi gas. Phys Rev A, 2000, 62: 041601(R) · doi:10.1103/PhysRevA.62.041601
[2] Berti V, Gatti S. Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations. Quart Appl Math, 2006, 64: 617–639 · Zbl 1120.35026
[3] Chen S H, Guo B L. Solution theory of the coupled time-dependent Ginzburg-Landau equations. Int J Dyn Syst Diff Equ, 2009, 2: 1–20 · Zbl 1187.35250
[4] Chen S H, Guo B L. Existence of the weak solution of coupled time-dependent Ginzburg-Landau equations. J Math Phys, 2010, 51: 033507 · Zbl 1309.35150 · doi:10.1063/1.3293968
[5] Chen S H, Guo B L. Classical solutions of time-dependent Ginzburg-Landau theory for atomic Fermi gases near the BCS-BEC crossover. Preprint, 2009 · Zbl 1228.35236
[6] Drechsler M, Zwerger W. Crossover from BCS-superconductivity to Bose-condensation. Ann Phys, 1992, 1: 15–23 · doi:10.1002/andp.19925040105
[7] Eden A, Foias C, Nicolaenko B, et al. Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics. Providence, RI: Masson, 1994 · Zbl 0842.58056
[8] Efendiev M, Miranville A, Zelik S. Exponential attractors for a nonlinear reaction-diffusion system in R3. C R Acad Sci Paris, 2000, 330: 713–718 · Zbl 1151.35315 · doi:10.1016/S0764-4442(00)00259-7
[9] Efendiev M, Miranville A, Zelik S. Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems. Proc Roy Soc Edinburgh Sect A, 2005, 135: 703–730 · Zbl 1088.37005 · doi:10.1017/S030821050000408X
[10] Fabrie P, Galusinski C, Miranville A, et al. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete Contin Dyn Syst, 2004, 10: 211–238 · Zbl 1060.35011 · doi:10.3934/dcds.2004.10.211
[11] Fang S M, Jin L Y, Guo B L. Global attractor for the initial boundary value problems for Ginzburg-Landau equations for atomic Fermi gases near the BCS-BEC crossover. Nonlinear Anal, 2010, 72: 4063–4070 · Zbl 1188.35027 · doi:10.1016/j.na.2010.01.037
[12] Gatti S, Grasselli M, Pata V. Exponential attractors for a conserved phase-field system with memory. Phys D, 2004, 189: 31–48 · Zbl 1051.37039 · doi:10.1016/j.physd.2003.10.005
[13] Gatti S, Grasselli M, Miranville A, et al. A construction of a robust family of exponential attractors. Proc Amer Math Soc, 2006, 134: 117–127 · Zbl 1078.37047 · doi:10.1090/S0002-9939-05-08340-1
[14] Lions J L, Magenes E. Problèmes aux Limites Non Homogènes et Applications. Paris: Dunod, 1968 · Zbl 0165.10801
[15] Machida M, Koyama T. Time-dependent Ginzburg-Landau theory for atomic Fermi gases near the BCS-BEC crossover. Phys Rev A, 2006, 74: 033603 · doi:10.1103/PhysRevA.74.033603
[16] Marshall R J, New G H C, Burnett K, et al. Exciting, cooling and vortex trapping in a Bose-condensed gas. Phys Rev A, 1999, 59: 2085–2093 · doi:10.1103/PhysRevA.59.2085
[17] Miranville A, Zelik S. Attractors for dissipative partial differential equations in bounded and unbounded domains. In: Handbook of Differential Equations: Evolutionary Equations, vol. IV. Amsterdam: Elsevier/North-Holland, 2008, 103–200 · Zbl 1221.37158
[18] Sa de Melo C A R, Randeria M, Engelbrecht J R. Crossover from BCS to Bose superconductivity: transition temperature and time-dependent Ginzburg-Landau theory. Phys Rev Lett, 1993, 71: 3202–3205 · doi:10.1103/PhysRevLett.71.3202
[19] Ohashi Y, Griffin A. BCS-BEC crossover in a gas of Fermi atoms with a Feshbach resonance. Phys Rev Lett, 2002, 89: 130402 · doi:10.1103/PhysRevLett.89.130402
[20] Temam R. Infinite-Dimensional Dynamical System in Mechanics and Physics. Applied Mathematical Sciences. New York: Springer, 1988 · Zbl 0662.35001
[21] Tempere J, Wouters M, Devereese J T. Path-intergral mean-field description of the vortex state in the BEC-to-BCS crossover. Phys Rev A, 2005, 71: 033631 · doi:10.1103/PhysRevA.71.033631
[22] Zheng S M. Nonlinear Evolution Equations. Boca Raton, FL: Chapman & Hall/CRC, 2004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.