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Invariant measure and
long time behavior of regular solutions of the Benjamin–Ono
equation
Mouhamadou Sy |
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Vol. 11 (2018), No. 8, 1841–1879
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DOI: 10.2140/apde.2018.11.1841
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Abstract |
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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 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. |
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Keywords
Benjamin–Ono equation, invariant measure, long time
behavior, regular solutions, inviscid limit
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Mathematical Subject Classification 2010
Primary: 35A09, 35B40, 35Q51, 35R60, 37K10
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Milestones
Received: 14 November 2016
Revised: 10 January 2018
Accepted: 14 February 2018
Published: 6 June 2018
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Authors |
| Mouhamadou Sy | |
| Laboratoire AGM UMR 8088 CNRS Université de Cergy-Pontoise Cergy-Pontoise France |
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