Remarks on the Gibbs measures for nonlinear dispersive equationsReport as inadecuate




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1 LM-Orsay - Laboratoire de Mathématiques d-Orsay 2 LMJL - Laboratoire de Mathématiques Jean Leray 3 AGM - Laboratoire d-Analyse, Géométrie et Modélisation

Abstract : 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.

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





Author: Nicolas Burq - Laurent Thomann - Nikolay Tzvetkov -

Source: https://hal.archives-ouvertes.fr/



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