Well-posedness for stochastic Camassa–Holm equation

The Camassa–Holm equation describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity. Due to uncertainty in the modelling and external environment, this modelling could be subject to random fluctuations. In this article, the stochastic Camassa–Holm equation with additive noise is considered. Using regularization, a local existence and uniqueness result in the Sobolev space Hs(R) with s>3/2 of stochastic Camassa–Holm equation is obtained. With the help of priori estimates, the local solution will blow up in Hq(R) in finite time for any q>3/2 when initial value satisfies some conditions.