Vol. 11, No. 8, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 8, 1841–2148
Issue 7, 1587–1839
Issue 6, 1343–1586
Issue 5, 1083–1342
Issue 4, 813–1081
Issue 3, 555–812
Issue 2, 263–553
Issue 1, 1–261

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Invariant measure and long time behavior of regular solutions of the Benjamin–Ono equation

Mouhamadou Sy

Vol. 11 (2018), No. 8, 1841–1879
DOI: 10.2140/apde.2018.11.1841
Abstract

The Benjamin–Ono equation describes the propagation of internal waves in a stratified fluid. In the present work, we study large time dynamics of its regular solutions via some probabilistic point of view. We prove the existence of an invariant measure concentrated on C(T) and establish some qualitative properties of this measure. We then deduce a recurrence property of regular solutions and other corollaries using ergodic theorems. The approach used in this paper applies to other equations with infinitely many conservation laws, such as the KdV and cubic Schrödinger equations in one dimension. It uses the fluctuation-dissipation-limit approach and relies on a uniform smoothing lemma for stationary solutions to the damped-driven Benjamin–Ono equation.

Keywords
Benjamin–Ono equation, invariant measure, long time behavior, regular solutions, inviscid limit
Mathematical Subject Classification 2010
Primary: 35A09, 35B40, 35Q51, 35R60, 37K10
Milestones
Received: 14 November 2016
Revised: 10 January 2018
Accepted: 14 February 2018
Published: 6 June 2018
Authors
Mouhamadou Sy
Laboratoire AGM UMR 8088 CNRS
Université de Cergy-Pontoise
Cergy-Pontoise
France