Remarks on the Gibbs measures for nonlinear dispersive equations

Nicolas Burq (1) , Laurent Thomann (2) , Nikolay Tzvetkov (3)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 IECL - Institut Élie Cartan de Lorraine
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

We show, by the means of several examples, how we can use Gibbs measures to construct global solutions to dispersive equations at low regularity. The construction relies on the Prokhorov compactness theorem combined with the Skorokhod convergence theorem. To begin with, we consider the non linear Schrödinger equation (NLS) on the tri-dimensional sphere. Then we focus on the Benjamin-Ono equation and on the derivative nonlinear Schrödinger equation on the circle. Next, we construct a Gibbs measure and global solutions to the so-called periodic half-wave equation. Finally, we consider the cubic 2d defocusing NLS on an arbitrary spatial domain and we construct global solutions on the support of the associated Gibbs measure.

Mots clés: global solutions; half-wave equation; derivative nonlinear Schrödinger equation; Benjamin-Ono equation; weak solutions; Nonlinear Schrödinger equation; random data; Gibbs measure

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques [math]

Nicolas Burq, Laurent Thomann, Nikolay Tzvetkov. Remarks on the Gibbs measures for nonlinear dispersive equations. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2018, 27 (3), pp.527--597. ⟨10.5802/afst.1578⟩. ⟨hal-01097696⟩