no. 7
Invariant measures for the defocusing Nonlinear Schrödinger equation
[Mesures invariantes pour l’équation de Schrödinger non linéaire]
Tzvetkov, Nikolay1
1 Université Lille I Département de Mathématiques 59 655 Villeneuve d’Ascq Cedex (France)
Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2543-2604.

On démontre l’existence et l’invariance d’une mesure de Gibbs par le flot de l’équation de Schrödinger non linéaire posée sur le disque du plan ℝ 2 . On démontre également une estimée qui donne une idée de ce qui pourrait arriver en dimension 3.

We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane ℝ 2 . We also prove an estimate giving some intuition to what may happen in 3 dimensions.

DOI : 10.5802/aif.2422
Classification : 35Q55, 35BXX, 37K05, 37L50, 81Q20
Keywords: Nonlinear Schrödinger, eigenfunctions, dispersive equations, invariant measures
Mot clés : Equation de Schrödinger non linéaire, fonctions propres, équations dispersives, mesures invariantes
Tzvetkov, Nikolay 1

1 Université Lille I Département de Mathématiques 59 655 Villeneuve d’Ascq Cedex (France) 
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     title = {Invariant measures for the defocusing {Nonlinear} {Schr\"odinger} equation},
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     volume = {58},
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Tzvetkov, Nikolay. Invariant measures for the defocusing Nonlinear Schrödinger equation. Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2543-2604. doi : 10.5802/aif.2422. http://archive.numdam.org/articles/10.5802/aif.2422/

[1] Anton, R. Cubic nonlinear Schrödinger equation on three dimensional balls with radial data (2006) (Preprint)

[2] Ayache, A.; Tzvetkov, N. L p properties of Gaussian random series (to appear in Trans. AMS) | Zbl

[3] Bourgain, J. Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., Volume 166 (1994), pp. 1-26 | DOI | MR | Zbl

[4] Bourgain, J. Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., Volume 176 (1996), pp. 421-445 | DOI | MR | Zbl

[5] Burq, N.; Gérard, P.; Tzvetkov, N. Zonal low regularity solutions of the nonlinear Schrödinger equation on S d (2002) (Unpublished manuscript)

[6] Burq, N.; Gérard, P.; Tzvetkov, N. Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. ENS, Volume 38 (2005), pp. 255-301 | Numdam | MR | Zbl

[7] Christ, M.; Colliander, J.; Tao, T. Ill-posedness for nonlinear Schrödinger and wave equations (2003) (Preprint)

[8] Ginibre, J. Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain), Séminaire Bourbaki, Exp. 796, Astérisque, Volume 237 (1996), pp. 163-187 | Numdam | Zbl

[9] Kuksin, S.; Shirikyan, A. Randomly forced CGL equation : stationary measures and the inviscid limit, J. Phys A, Volume 37 (2004), pp. 1-18 | DOI | MR | Zbl

[10] Lebowitz, J.; Rose, R.; Speer, E. Statistical dynamics of the nonlinear Schrödinger equation, J. Stat. Physics V, Volume 50 (1988), pp. 657-687 | DOI | MR | Zbl

[11] Stein, E.; Weiss, G.; Princeton University Press, Princeton N.J. Introduction to Fourier analysis on euclidean spaces (Princeton Mathematical Series), Volume 32 (1971) | MR | Zbl

[12] Tzvetkov, N. Invariant measures for the nonlinear Schrödinger equation on the disc, Dynamics of PDE, Volume 3 (2006), pp. 111-160 | MR | Zbl

[13] Zhidkov, P. Korteweg de Vries and nonlinear Schrödinger equations : qualitative theory, Lecture Notes in Mathematics, 1756, Springer-Verlag, Berlin, 2001 | MR | Zbl

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