Invariant measures for the defocusing Nonlinear Schrödinger equation  [ Mesures invariantes pour l’équation de Schrödinger non linéaire ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 7, p. 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 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 2 . We also prove an estimate giving some intuition to what may happen in 3 dimensions.
DOI : https://doi.org/10.5802/aif.2422
Classification:  35Q55,  35BXX,  37K05,  37L50,  81Q20
Mots clés: Equation de Schrödinger non linéaire, fonctions propres, équations dispersives, mesures invariantes
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {7},
     year = {2008},
     pages = {2543-2604},
     doi = {10.5802/aif.2422},
     mrnumber = {2498359},
     zbl = {1171.35116},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_7_2543_0}
}
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://www.numdam.org/item/AIF_2008__58_7_2543_0/

[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 1145.60019

[3] Bourgain, J. Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., Tome 166 (1994), pp. 1-26 | Article | MR 1309539 | Zbl 0822.35126

[4] Bourgain, J. Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., Tome 176 (1996), pp. 421-445 | Article | MR 1374420 | Zbl 0852.35131

[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Burq, N.; Gérard, P.; Tzvetkov, N. Zonal low regularity solutions of the nonlinear Schrödinger equation on S d (2002) (Unpublished manuscript)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, Tome 38 (2005), pp. 255-301 | Numdam | MR 2144988 | Zbl 1116.35109

[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, Tome 237 (1996), pp. 163-187 | Numdam | Zbl 0870.35096

[9] Kuksin, S.; Shirikyan, A. Randomly forced CGL equation : stationary measures and the inviscid limit, J. Phys A, Tome 37 (2004), pp. 1-18 | Article | MR 2039838 | Zbl 1047.35061

[10] Lebowitz, J.; Rose, R.; Speer, E. Statistical dynamics of the nonlinear Schrödinger equation, J. Stat. Physics V, Tome 50 (1988), pp. 657-687 | Article | MR 939505 | Zbl 1084.82506

[11] Stein, E.; Weiss, G.; Princeton University Press, Princeton N.J. Introduction to Fourier analysis on euclidean spaces (Princeton Mathematical Series) Tome 32 (1971) | MR 304972 | Zbl 0232.42007

[12] Tzvetkov, N. Invariant measures for the nonlinear Schrödinger equation on the disc, Dynamics of PDE, Tome 3 (2006), pp. 111-160 | MR 2227040 | Zbl 1142.35090

[13] Zhidkov, P. Korteweg de Vries and nonlinear Schrödinger equations : qualitative theory, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1756 (2001) | MR 1831831 | Zbl 0987.35001