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arXiv 1601.05055math.AP
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Benjamin-Ono方程正则解的不变测度和长时间行为

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

Mouhamadou Sy

摘要:

Benjamin-Ono方程描述了分层流体中内波的传播。本文从概率的角度研究了它的正则解的大时间动力学性质。证明了集中在$C^\infty(\T)$上的不变测度的存在性,并建立了该测度的一些定性性质.然后,我们推导出的正则解和其他推论使用遍历定理的递归性质。本文所用的方法也适用于其他具有无穷多个守恒律的方程,如一维KdV方程和三维薛定谔方程。它使用的波动耗散限制的方法,并依赖于一个\textit{uniform}平滑引理固定的阻尼驱动的Benjamin-Ono方程的解决方案。

英文摘要:

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 Schrödinger 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.

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