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