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

Brahim Alouini ^1,

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 and 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, Communications in Pure and Applied Analysis, 10 (2011), 1629-1643. doi: 10.3934/cpaa.2011.10.1629.
[2]	B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Discrete Continuous Dynam. Systems - B, 19 (2014), 651-677. doi: 10.3934/dcdsb.2014.19.651.
[3]	J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynam. Systems - A, 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[4]	B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian. Acad. Sci. (Math. Sci.), 116 (2006), 337-360. doi: 10.1007/BF02829750.
[5]	C. C. Bradlay, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: observation of limited condensate number, Phys. Rev. Lett., 78 (1997), 985-989. doi: 10.1103/PhysRevLett.78.985.
[6]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[7]	R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Annales Henri Poincare, 3 (2002), 757-772. doi: 10.1007/s00023-002-8635-4.
[8]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
[9]	I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, ACTA 2002, Available from: http://www.emis.de/monographs/Chueshov/book.pdf
[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, $n^{\circ}$ 912, August 2008. viii+183 pp.
[11]	G. B. Folland, Fourier Analysis and Its Applications, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.
[12]	O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$, Advances in Differential Equations, 3 (1998), 337-360.
[13]	O. Goubet and L. Legry, Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential, Nonlinear Analysis, 72 (2010), 4397-4406. doi: 10.1016/j.na.2010.02.013.
[14]	A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anormalous dispersion, Applied Physics Lettres, 23 (1973), 14-24. doi: 10.1063/1.1654836.
[15]	P. Lauren\ccot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $R^N,N\leq 3$, NoDEA, 2 (1995), 357-369. doi: 10.1007/BF01261181.
[16]	Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput., 177 (2006), 482-487.
[17]	K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation, Physica D: Nonlinear Phenomena, 21 (1986), 381-393. doi: 10.1016/0167-2789(86)90012-6.
[18]	K. Promislow and J. N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator, Nonlinearity, 13 (2000), 675-698. doi: 10.1088/0951-7715/13/3/310.
[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, Cambridge, 2001. xviii+461 pp.
[20]	B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.
[21]	R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics, Springer applied mathmatical sciences, 68, Springer-Verlag, New York, 1997.
[22]	X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D: Nonlinear Phenomena, 88 (1995), 167-175. doi: 10.1016/0167-2789(95)00196-B.

show all references

References:

[1]	B. Alouini, Long-time behavior of a Bose-Einstein equation in a two dimensional thin domain, Communications in Pure and Applied Analysis, 10 (2011), 1629-1643. doi: 10.3934/cpaa.2011.10.1629.
[2]	B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Discrete Continuous Dynam. Systems - B, 19 (2014), 651-677. doi: 10.3934/dcdsb.2014.19.651.
[3]	J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynam. Systems - A, 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[4]	B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian. Acad. Sci. (Math. Sci.), 116 (2006), 337-360. doi: 10.1007/BF02829750.
[5]	C. C. Bradlay, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: observation of limited condensate number, Phys. Rev. Lett., 78 (1997), 985-989. doi: 10.1103/PhysRevLett.78.985.
[6]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[7]	R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Annales Henri Poincare, 3 (2002), 757-772. doi: 10.1007/s00023-002-8635-4.
[8]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
[9]	I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, ACTA 2002, Available from: http://www.emis.de/monographs/Chueshov/book.pdf
[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, $n^{\circ}$ 912, August 2008. viii+183 pp.
[11]	G. B. Folland, Fourier Analysis and Its Applications, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.
[12]	O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$, Advances in Differential Equations, 3 (1998), 337-360.
[13]	O. Goubet and L. Legry, Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential, Nonlinear Analysis, 72 (2010), 4397-4406. doi: 10.1016/j.na.2010.02.013.
[14]	A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anormalous dispersion, Applied Physics Lettres, 23 (1973), 14-24. doi: 10.1063/1.1654836.
[15]	P. Lauren\ccot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $R^N,N\leq 3$, NoDEA, 2 (1995), 357-369. doi: 10.1007/BF01261181.
[16]	Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput., 177 (2006), 482-487.
[17]	K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation, Physica D: Nonlinear Phenomena, 21 (1986), 381-393. doi: 10.1016/0167-2789(86)90012-6.
[18]	K. Promislow and J. N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator, Nonlinearity, 13 (2000), 675-698. doi: 10.1088/0951-7715/13/3/310.
[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, Cambridge, 2001. xviii+461 pp.
[20]	B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.
[21]	R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics, Springer applied mathmatical sciences, 68, Springer-Verlag, New York, 1997.
[22]	X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D: Nonlinear Phenomena, 88 (1995), 167-175. doi: 10.1016/0167-2789(95)00196-B.

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