Remarks on the Gibbs measures for nonlinear dispersive equations

Nicolas Burq; Laurent Thomann; Nikolay Tzvetkov

doi:10.5802/afst.1578

Nicolas Burq ¹; Laurent Thomann ²; Nikolay Tzvetkov ³

¹ Laboratoire de Mathématiques, Bât. 307, Université Paris Sud, 91405 Orsay Cedex, France
² Université de Lorraine, CNRS, IECL, 54000 Nancy, France
³ Université de Cergy-Pontoise, UMR CNRS 8088, Cergy-Pontoise, 95000

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 3, pp. 527-597.

Abstract
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 nonlinear 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 and of the Szegő equation. Finally, we consider the cubic $2 d$ defocusing NLS on an arbitrary spatial domain and we construct global solutions on the support of the associated Gibbs measure.

On montre, grâce à différents exemples, comment on peut utiliser des mesures de Gibbs pour construire des solutions globales, à basse régularité, pour des équations dispersives. La construction repose sur le théorème de compacité de Prokhorov, combiné avec le théorème de convergence de Skorokhod. D’abord, on considère l’équation de Schrödinger non-linéaire (NLS) sur la sphère de dimension 3. Ensuite, on étudie l’équation de Benjamin–Ono et l’équation de Schrödinger avec dérivée sur le cercle. Puis, on construit une mesure de Gibbs et une solution globales aux équations des demi-ondes et de Szegő avec conditions périodiques. Enfin, on considère NLS cubique défocalisante, en dimension deux, sur un domaine quelconque et on construit des solutions globales sur le support de la mesure de Gibbs correspondante.

Received: 2016-05-09
Accepted: 2016-10-03
Published online: 2018-09-19

MR Zbl

DOI: 10.5802/afst.1578

Classification: 35BXX, 37K05, 37L50, 35Q55
Keywords: nonlinear Schrödinger equation, Benjamin–Ono equation, derivative nonlinear Schrödinger equation, half-wave equation, Szegő equation, random data, Gibbs measure, weak solutions, global solutions

Author's affiliations:

Nicolas Burq ¹; Laurent Thomann ²; Nikolay Tzvetkov ³

¹ Laboratoire de Mathématiques, Bât. 307, Université Paris Sud, 91405 Orsay Cedex, France
² Université de Lorraine, CNRS, IECL, 54000 Nancy, France
³ Université de Cergy-Pontoise, UMR CNRS 8088, Cergy-Pontoise, 95000

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Nicolas Burq and Laurent Thomann and Nikolay Tzvetkov},
     title = {Remarks on the {Gibbs} measures for nonlinear dispersive equations},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {527--597},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {3},
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Nicolas Burq; Laurent Thomann; Nikolay Tzvetkov. Remarks on the Gibbs measures for nonlinear dispersive equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 3, pp. 527-597. doi : 10.5802/afst.1578. https://afst.centre-mersenne.org/articles/10.5802/afst.1578/

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