Long time dynamics for the one dimensional non linear Schrödinger equation

Burq, Nicolas; Thomann, Laurent; Tzvetkov, Nikolay

Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2137-2198.

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 $ℝ$ (sans potentiel harmonique) est globalement bien posée et diffuse pour ces données initiales.

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).

Classification : 35BXX, 37K05, 37L50, 35Q55
Keywords: Nonlinear Schrödinger equation, potential, random data, Gibbs measure, invariant measure, global solutions
Mot clés : Equation de Schrödinger non linéaire, données aléatoires, mesure de Gibbs, mesures invariants, solutions globales

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