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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

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 & 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.  Google Scholar

[2]

B. Alouini, Étude De L'équation De Bose-Einstein Dans Un Canal, Ph.D thesis, Monastir University in Monastir, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Y. Meyer and R. Coifman, Wavelets: Calderòn-Zygmund and Multilinear Operators, Cambridge Studies in Advanced Mathematics, 48, Cambridge University Press, Cambridge, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 43, Princeton, New Jersey, 1993.  Google Scholar

[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.  Google Scholar

[27]

R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics, Springer applied mathmatical sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar

[28]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, 1978.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

B. Alouini, Étude De L'équation De Bose-Einstein Dans Un Canal, Ph.D thesis, Monastir University in Monastir, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Y. Meyer and R. Coifman, Wavelets: Calderòn-Zygmund and Multilinear Operators, Cambridge Studies in Advanced Mathematics, 48, Cambridge University Press, Cambridge, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 43, Princeton, New Jersey, 1993.  Google Scholar

[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.  Google Scholar

[27]

R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics, Springer applied mathmatical sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar

[28]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, 1978.  Google Scholar

[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.  Google Scholar

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