Invariant measure and long time behavior of regular solutions of the Benjamin–Ono equation

Sy, Mouhamadou

doi:10.2140/apde.2018.11.1841

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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 $C^{\infty} (T)$ 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

Vol. 11, No. 8, 2018