On the pathwise solutions to the CAMASSA–HOLM equation with multiplicative noise

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Original languageEnglish
Pages (from-to)1322-1366
Journal / PublicationSIAM Journal on Mathematical Analysis
Issue number1
Online published20 Feb 2018
Publication statusPublished - 2018



In this paper we consider the Camassa–Holm (CH) equation with multiplicative noise, which can be obtained when the nonhydrostatic pressure in the deterministic equation is subject to a turbulent velocity field. For the periodic boundary value problem for this SPDE, we establish the local existence and pathwise uniqueness of the pathwise solution in Sobolev spaces s with s > 3/2. For the linear noise case, conditions that lead to the global existence and the blow-up in finite time of the solution, and their associated probabilities, are also acquired. Finally, we study the pathwise dissipative effect of the linear noise on the periodic peakons to the deterministic CH equation.

Research Area(s)

  • Blow up, Global existence, Martingale solutions, Pathwise solutions, Periodic peakons, Stochastic Camassa–Holm equation

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