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

TL;DR

This paper investigates the long-term behavior of regular solutions to the Benjamin-Ono equation by establishing an invariant measure and analyzing its properties using probabilistic methods, ergodic theory, and smoothing techniques.

Contribution

It introduces the existence of an invariant measure for the Benjamin-Ono equation and applies probabilistic and ergodic methods to study the equation's long-time dynamics.

Findings

Existence of an invariant measure on smooth solutions

Recurrence property of solutions established

Method applicable to other integrable equations with conservation laws

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 Schr\"odinger equations in 1D. It uses the fluctuation-dissipation-limit approach and relies on a \textit{uniform} smoothing lemma for stationary solutions to the damped-driven Benjamin-Ono equation.