Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain

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May 2014, 19(3): 651-677. doi: 10.3934/dcdsb.2014.19.651

Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain

Brahim Alouini ^1, and Olivier Goubet ^2,

1.	Unité de recherche: Ondelettes et Fractals, Faculté des Sciences de Monastir, Av. de l'environnement, 5000 Monastir
2.	LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue St Leu, 80039, Amiens Cedex

Received October 2013 Revised November 2013 Published February 2014

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

Keywords: Schrödinger equation, Bose-Einstein equation, global attractor..

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

Citation: Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651

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, Étude De L'équation De Bose-Einstein Dans Un Canal, Ph.D thesis, Monastir University in Monastir, 2013.
[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]	R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Annales Henri Poincare, 3 (2002), 757-772. doi: 10.1007/s00023-002-8635-4.
[7]	C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187, 2001.
[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]	G. Chen and J. Zhang, Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential, J. Math. Anal. Appl., 320 (2006), 591-598. doi: 10.1016/j.jmaa.2005.07.008.
[10]	G. B. Folland, Fourier Analysis and Its Applications, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.
[11]	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.
[12]	O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbbR^2$, Advances in Differential Equations, 3 (1998), 337-360.
[13]	M. Haase, The Functional Calculus For Sectoriel Operators, Operator Theory, Advances and Applications, Birkhäuser Verlag, Basel-Boston-Berlin, 169, 2006. doi: 10.1007/3-7643-7698-8.
[14]	E. Harboure, L. de Rosa, C. Segovia et J. L. Torrea, $\mathbfL^p$-Dimension free boundedness for Riesz transforms associated to Hermite functions, Math. Ann., 328 (2004), 653-682. doi: 10.1007/s00208-003-0501-2.
[15]	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.
[16]	P. Laurençot, Long-time behavior for weakly damped driven nonlinear schrödinger equations in $\mathbbR^N, N\leq 3$, NoDEA, 2 (1995), 357-369. doi: 10.1007/BF01261181.
[17]	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. doi: 10.1016/j.amc.2005.11.024.
[18]	Y. Meyer and R. Coifman, Wavelets: Calderòn-Zygmund and Multilinear Operators, Cambridge Studies in Advanced Mathematics, 48, Cambridge University Press, Cambridge, 1997.
[19]	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.
[20]	H. Pollard, The mean convergence of orthogonal series II, Trans. Amer. Math. Soc., 63 (1948), 355-367. Available from: http://www.ams.org/journals/tran/1948-063-02/S0002-9947-1948-0023941-X/S0002-9947-1948-0023941-X.pdf doi: 10.1090/S0002-9947-1948-0023941-X.
[21]	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.
[22]	J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452. doi: 10.1007/BF02570748.
[23]	J. Prüss and G. Simonett, $H^{\infty}-$calculus for the sum of non-commuting operators, Transactions Of The American Mathematical Society, 359 (2007), 3549-3565. Available from: http://www.math.vanderbilt.edu/ simonett/preprints/non-commuting.pdf. doi: 10.1090/S0002-9947-07-04291-2.
[24]	B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbbR^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.
[25]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 43, Princeton, New Jersey, 1993.
[26]	K. Stempak and J. L. Torrea, Poisson integrals and Riesz transforms for Hermite function expensions with weigths. Journal of Functional Analysis, 202 (2003), 443-472. doi: 10.1016/S0022-1236(03)00083-1.
[27]	R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics, Springer applied mathmatical sciences, 68, Springer-Verlag, New York, 1997.
[28]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, 1978.
[29]	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, Étude De L'équation De Bose-Einstein Dans Un Canal, Ph.D thesis, Monastir University in Monastir, 2013.
[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]	R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Annales Henri Poincare, 3 (2002), 757-772. doi: 10.1007/s00023-002-8635-4.
[7]	C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187, 2001.
[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]	G. Chen and J. Zhang, Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential, J. Math. Anal. Appl., 320 (2006), 591-598. doi: 10.1016/j.jmaa.2005.07.008.
[10]	G. B. Folland, Fourier Analysis and Its Applications, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.
[11]	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.
[12]	O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbbR^2$, Advances in Differential Equations, 3 (1998), 337-360.
[13]	M. Haase, The Functional Calculus For Sectoriel Operators, Operator Theory, Advances and Applications, Birkhäuser Verlag, Basel-Boston-Berlin, 169, 2006. doi: 10.1007/3-7643-7698-8.
[14]	E. Harboure, L. de Rosa, C. Segovia et J. L. Torrea, $\mathbfL^p$-Dimension free boundedness for Riesz transforms associated to Hermite functions, Math. Ann., 328 (2004), 653-682. doi: 10.1007/s00208-003-0501-2.
[15]	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.
[16]	P. Laurençot, Long-time behavior for weakly damped driven nonlinear schrödinger equations in $\mathbbR^N, N\leq 3$, NoDEA, 2 (1995), 357-369. doi: 10.1007/BF01261181.
[17]	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. doi: 10.1016/j.amc.2005.11.024.
[18]	Y. Meyer and R. Coifman, Wavelets: Calderòn-Zygmund and Multilinear Operators, Cambridge Studies in Advanced Mathematics, 48, Cambridge University Press, Cambridge, 1997.
[19]	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.
[20]	H. Pollard, The mean convergence of orthogonal series II, Trans. Amer. Math. Soc., 63 (1948), 355-367. Available from: http://www.ams.org/journals/tran/1948-063-02/S0002-9947-1948-0023941-X/S0002-9947-1948-0023941-X.pdf doi: 10.1090/S0002-9947-1948-0023941-X.
[21]	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.
[22]	J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452. doi: 10.1007/BF02570748.
[23]	J. Prüss and G. Simonett, $H^{\infty}-$calculus for the sum of non-commuting operators, Transactions Of The American Mathematical Society, 359 (2007), 3549-3565. Available from: http://www.math.vanderbilt.edu/ simonett/preprints/non-commuting.pdf. doi: 10.1090/S0002-9947-07-04291-2.
[24]	B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbbR^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.
[25]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 43, Princeton, New Jersey, 1993.
[26]	K. Stempak and J. L. Torrea, Poisson integrals and Riesz transforms for Hermite function expensions with weigths. Journal of Functional Analysis, 202 (2003), 443-472. doi: 10.1016/S0022-1236(03)00083-1.
[27]	R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics, Springer applied mathmatical sciences, 68, Springer-Verlag, New York, 1997.
[28]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, 1978.
[29]	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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