## Analysis & PDE

• Anal. PDE
• Volume 11, Number 8 (2018), 1841-1879.

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

#### 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.

#### Article information

Source
Anal. PDE, Volume 11, Number 8 (2018), 1841-1879.

Dates
Revised: 10 January 2018
Accepted: 14 February 2018
First available in Project Euclid: 15 January 2019

https://projecteuclid.org/euclid.apde/1547521440

Digital Object Identifier
doi:10.2140/apde.2018.11.1841

Mathematical Reviews number (MathSciNet)
MR3812859

Zentralblatt MATH identifier
1388.35175

#### Citation

Sy, Mouhamadou. Invariant measure and long time behavior of regular solutions of the Benjamin–Ono equation. Anal. PDE 11 (2018), no. 8, 1841--1879. doi:10.2140/apde.2018.11.1841. https://projecteuclid.org/euclid.apde/1547521440

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