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 . On démontre également une estimée qui donne une idée de ce qui pourrait arriver en dimension .
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 . We also prove an estimate giving some intuition to what may happen in dimensions.
Keywords: Nonlinear Schrödinger, eigenfunctions, dispersive equations, invariant measures
Mot 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}, 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, Tome 58 (2008) no. 7, pp. 2543-2604. doi : 10.5802/aif.2422. http://archive.numdam.org/articles/10.5802/aif.2422/
[1] Cubic nonlinear Schrödinger equation on three dimensional balls with radial data (2006) (Preprint)
[2] properties of Gaussian random series (to appear in Trans. AMS) | Zbl
[3] Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., Volume 166 (1994), pp. 1-26 | DOI | MR | Zbl
[4] Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., Volume 176 (1996), pp. 421-445 | DOI | MR | Zbl
[5] Zonal low regularity solutions of the nonlinear Schrödinger equation on (2002) (Unpublished manuscript)
[6] 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] Ill-posedness for nonlinear Schrödinger and wave equations (2003) (Preprint)
[8] 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] Randomly forced CGL equation : stationary measures and the inviscid limit, J. Phys A, Volume 37 (2004), pp. 1-18 | DOI | MR | Zbl
[10] Statistical dynamics of the nonlinear Schrödinger equation, J. Stat. Physics V, Volume 50 (1988), pp. 657-687 | DOI | MR | Zbl
[11] Introduction to Fourier analysis on euclidean spaces (Princeton Mathematical Series), Volume 32 (1971) | MR | Zbl
[12] Invariant measures for the nonlinear Schrödinger equation on the disc, Dynamics of PDE, Volume 3 (2006), pp. 111-160 | MR | Zbl
[13] Korteweg de Vries and nonlinear Schrödinger equations : qualitative theory, Lecture Notes in Mathematics, 1756, Springer-Verlag, Berlin, 2001 | MR | Zbl
Cité par Sources :