Communications in Mathematical Sciences

Volume 19 (2021)

Number 3

Global well-posedness of the stochastic Camassa–Holm equation

Pages: 607 – 627



Yong Chen (School of Science, Zhejiang Sci-Tech University, Hangzhou, China)

Jinqiao Duan (Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Il., U.S.A.)

Hongjun Gao (School of Mathematics, Southeast University, Nanjing, Jiangsu, China)


We establish the existence of global martingale solutions of the stochastic Camassa–Holm equation in $H^1(\mathbb{R})$. The construction of the solution is based on the regularization method and the stochastic compactness method. Furthermore, we use Borel–Cantelli Lemma to prove the global existence of mild solution of the stochastic Camassa–Holm equation with small noise in $L^2(\mathbb{R})$.


stochastic Camassa–Holm equation, martingale solutions, regularization, tightness

2010 Mathematics Subject Classification

35L05, 35R60, 60H15

Received 23 September 2019

Accepted 29 September 2020

Published 5 May 2021