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Regularity of the 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 |
2. | LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue St Leu, 80039, Amiens Cedex |
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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