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

Mouhamadou Sy

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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

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

Received: 14 November 2016
Revised: 10 January 2018
Accepted: 14 February 2018
First available in Project Euclid: 15 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35A09: Classical solutions 35B40: Asymptotic behavior of solutions 35Q51: Soliton-like equations [See also 37K40] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)

Benjamin–Ono equation invariant measure long time behavior regular solutions inviscid limit


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.

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