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

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.

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.

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) 
@article{AIF_2008__58_7_2543_0,
     author = {Tzvetkov, Nikolay},
     title = {Invariant measures for the defocusing {Nonlinear} {Schr\"odinger} equation},
     journal = {Annales de l'Institut Fourier},
     pages = {2543--2604},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {7},
     year = {2008},
     doi = {10.5802/aif.2422},
     zbl = {1171.35116},
     mrnumber = {2498359},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2422/}
}
TY  - JOUR
AU  - Tzvetkov, Nikolay
TI  - Invariant measures for the defocusing Nonlinear Schrödinger equation
JO  - Annales de l'Institut Fourier
PY  - 2008
SP  - 2543
EP  - 2604
VL  - 58
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2422/
DO  - 10.5802/aif.2422
LA  - en
ID  - AIF_2008__58_7_2543_0
ER  - 
%0 Journal Article
%A Tzvetkov, Nikolay
%T Invariant measures for the defocusing Nonlinear Schrödinger equation
%J Annales de l'Institut Fourier
%D 2008
%P 2543-2604
%V 58
%N 7
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2422/
%R 10.5802/aif.2422
%G en
%F AIF_2008__58_7_2543_0
Tzvetkov, Nikolay. Invariant measures for the defocusing Nonlinear Schrödinger equation. Annales de l'Institut Fourier, Volume 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

Cited by Sources: