Long time dynamics for the one dimensional non linear Schrödinger equation

2010/02/22 by Burq, Nicolas, Thomann, Laurent, Tzvetkov, Nikolay
#35BXX #35Q55 #37K05 #37L50 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1002.4054

Abstract

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 L² critical and super-critical NLS (without harmonic potential).

Related

• Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two
• Invariant Gibbs measures for 1D NLS in a trap
• Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise
• Probabilistic preservation of regularity for periodic nonlinear Schrödinger equations
• Almost sure global well-posedness for the energy supercritical Schrödinger equations