Long time dynamics for the one dimensional non linear Schrödinger equation
Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2137-2198.

In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the L 2 critical and super-critical NLS (without harmonic potential).

Nous présentons d’abord dans cet article la construction de mesures de Gibbs pour l’équation de Schrödinger non linéaire associée à un potentiel harmonique. Nous démontrons ensuite que le problème de Cauchy correspondant est globalement bien posé pour des données initiales très peu régulières (sur le support de cette mesure). Finalement, nous démontrons aussi que ces mesures de Gibbs sont invariantes par le flot ainsi défini. Nous obtenons comme conséquence de cette approche que l’équation de Schrödinger non linéaire L 2 -critique et surcritique sur (sans potentiel harmonique) est globalement bien posée et diffuse pour ces données initiales.

DOI: 10.5802/aif.2825
Classification: 35BXX,  37K05,  37L50,  35Q55
Keywords: Nonlinear Schrödinger equation, potential, random data, Gibbs measure, invariant measure, global solutions
Burq, Nicolas 1; Thomann, Laurent 2; Tzvetkov, Nikolay 3

1 Laboratoire de Mathématiques, Bât. 425, Université Paris Sud, 91405 Orsay Cedex, France.
2 Laboratoire de Mathématiques J. Leray, Université de Nantes, UMR CNRS 6629 2, rue de la Houssinière, 44322 Nantes Cedex 03, France.
3 University of Cergy-Pontoise, UMR CNRS 8088, F-95000, Cergy-Pontoise, France.
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     title = {Long time dynamics for the one dimensional non linear {Schr\"odinger} equation},
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Burq, Nicolas; Thomann, Laurent; Tzvetkov, Nikolay. Long time dynamics for the one dimensional non linear Schrödinger equation. Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2137-2198. doi : 10.5802/aif.2825. http://www.numdam.org/articles/10.5802/aif.2825/

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