Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain

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September 2015, 14(5): 1781-1801. doi: 10.3934/cpaa.2015.14.1781

Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain

1.	Unité de recherche: Ondelettes et Fractals, Faculté des Sciences de Monastir, Av. de l'environnement, 5000 Monastir

Received September 2014 Revised March 2015 Published June 2015

Abstract
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We study the long-time behavior of the solutions to a nonlinear damped driven Schrödinger type equation with quadratic potential on a strip. We prove that this behavior is described by a regular compact global attractor with finite fractal dimension.

Keywords: fractal dimension., global attractor, Bose-Einstein equation.

Mathematics Subject Classification: Primary: 35L05, 35Q55; Secondary: 76B0.

Citation: Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781

References:

[1]	B. Alouini, Long-time behavior of a Bose-Einstein equation in a two dimensional thin domain,, \emph{Communications in Pure and Applied Analysis}, 10 (2011), 1629. doi: 10.3934/cpaa.2011.10.1629. Google Scholar
[2]	B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain,, \emph{Discrete Continuous Dynam. Systems - B}, 19 (2014), 651. doi: 10.3934/dcdsb.2014.19.651. Google Scholar
[3]	J. M. Ball, Global attractors for damped semilinear wave equations,, \emph{Discrete Continuous Dynam. Systems - A}, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31. Google Scholar
[4]	B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator,, \emph{Proc. Indian. Acad. Sci. (Math. Sci.)}, 116 (2006), 337. doi: 10.1007/BF02829750. Google Scholar
[5]	C. C. Bradlay, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: observation of limited condensate number,, \emph{Phys. Rev. Lett.}, 78 (1997), 985. doi: 10.1103/PhysRevLett.78.985. Google Scholar
[6]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011). Google Scholar
[7]	R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential,, \emph{Annales Henri Poincare}, 3 (2002), 757. doi: 10.1007/s00023-002-8635-4. Google Scholar
[8]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, 10 (2003). Google Scholar
[9]	I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems,, University Lectures in Contemporary Mathematics, (2002). Google Scholar
[10]	I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping,, Memoirs of the American Mathematical Society, 195 (2008). Google Scholar
[11]	G. B. Folland, Fourier Analysis and Its Applications,, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, (1992). Google Scholar
[12]	O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$,, \emph{Advances in Differential Equations}, 3 (1998), 337. Google Scholar
[13]	O. Goubet and L. Legry, Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential,, \emph{Nonlinear Analysis}, 72 (2010), 4397. doi: 10.1016/j.na.2010.02.013. Google Scholar
[14]	A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anormalous dispersion,, \emph{Applied Physics Lettres}, 23 (1973), 14. doi: 10.1063/1.1654836. Google Scholar
[15]	P. Lauren\ccot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $R^N,N\leq 3$,, \emph{NoDEA}, 2 (1995), 357. doi: 10.1007/BF01261181. Google Scholar
[16]	Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential,, \emph{Appl. Math. Comput.}, 177 (2006), 482. Google Scholar
[17]	K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation,, \emph{Physica D: Nonlinear Phenomena}, 21 (1986), 381. doi: 10.1016/0167-2789(86)90012-6. Google Scholar
[18]	K. Promislow and J. N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator,, \emph{Nonlinearity}, 13 (2000), 675. doi: 10.1088/0951-7715/13/3/310. Google Scholar
[19]	J.C. Robinson, Infinite-Dimensionel Dynamical Systems, An Introduction To Dissipative Parabolic PDEs And The Theorie Of Global Attractors,, Cambridge Texts in Applied Mathematics. Cambridge University Press, (2001). Google Scholar
[20]	B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$,, \emph{J. Funct. Anal.}, 219 (2005), 340. doi: 10.1016/j.jfa.2004.06.013. Google Scholar
[21]	R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics,, Springer applied mathmatical sciences, 68 (1997). Google Scholar
[22]	X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors,, \emph{Physica D: Nonlinear Phenomena}, 88 (1995), 167. doi: 10.1016/0167-2789(95)00196-B. Google Scholar

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References:

[1]	B. Alouini, Long-time behavior of a Bose-Einstein equation in a two dimensional thin domain,, \emph{Communications in Pure and Applied Analysis}, 10 (2011), 1629. doi: 10.3934/cpaa.2011.10.1629. Google Scholar
[2]	B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain,, \emph{Discrete Continuous Dynam. Systems - B}, 19 (2014), 651. doi: 10.3934/dcdsb.2014.19.651. Google Scholar
[3]	J. M. Ball, Global attractors for damped semilinear wave equations,, \emph{Discrete Continuous Dynam. Systems - A}, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31. Google Scholar
[4]	B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator,, \emph{Proc. Indian. Acad. Sci. (Math. Sci.)}, 116 (2006), 337. doi: 10.1007/BF02829750. Google Scholar
[5]	C. C. Bradlay, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: observation of limited condensate number,, \emph{Phys. Rev. Lett.}, 78 (1997), 985. doi: 10.1103/PhysRevLett.78.985. Google Scholar
[6]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011). Google Scholar
[7]	R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential,, \emph{Annales Henri Poincare}, 3 (2002), 757. doi: 10.1007/s00023-002-8635-4. Google Scholar
[8]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, 10 (2003). Google Scholar
[9]	I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems,, University Lectures in Contemporary Mathematics, (2002). Google Scholar
[10]	I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping,, Memoirs of the American Mathematical Society, 195 (2008). Google Scholar
[11]	G. B. Folland, Fourier Analysis and Its Applications,, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, (1992). Google Scholar
[12]	O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$,, \emph{Advances in Differential Equations}, 3 (1998), 337. Google Scholar
[13]	O. Goubet and L. Legry, Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential,, \emph{Nonlinear Analysis}, 72 (2010), 4397. doi: 10.1016/j.na.2010.02.013. Google Scholar
[14]	A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anormalous dispersion,, \emph{Applied Physics Lettres}, 23 (1973), 14. doi: 10.1063/1.1654836. Google Scholar
[15]	P. Lauren\ccot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $R^N,N\leq 3$,, \emph{NoDEA}, 2 (1995), 357. doi: 10.1007/BF01261181. Google Scholar
[16]	Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential,, \emph{Appl. Math. Comput.}, 177 (2006), 482. Google Scholar
[17]	K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation,, \emph{Physica D: Nonlinear Phenomena}, 21 (1986), 381. doi: 10.1016/0167-2789(86)90012-6. Google Scholar
[18]	K. Promislow and J. N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator,, \emph{Nonlinearity}, 13 (2000), 675. doi: 10.1088/0951-7715/13/3/310. Google Scholar
[19]	J.C. Robinson, Infinite-Dimensionel Dynamical Systems, An Introduction To Dissipative Parabolic PDEs And The Theorie Of Global Attractors,, Cambridge Texts in Applied Mathematics. Cambridge University Press, (2001). Google Scholar
[20]	B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$,, \emph{J. Funct. Anal.}, 219 (2005), 340. doi: 10.1016/j.jfa.2004.06.013. Google Scholar
[21]	R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics,, Springer applied mathmatical sciences, 68 (1997). Google Scholar
[22]	X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors,, \emph{Physica D: Nonlinear Phenomena}, 88 (1995), 167. doi: 10.1016/0167-2789(95)00196-B. Google Scholar

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