Noise effect in a stochastic generalized Camassa-Holm equation

Abstract

<p style='text-indent:20px;'>In this paper we consider a stochastic variant of the generalized Camassa-Holm equation. We first establish some local results, including local existence, uniqueness and a blow-up criterion characterizing the possible blow-up of the solutions, for the corresponding stochastic partial differential equation. Then we study the effect of noise. For the relatively small noise, in terms of the dependence on initial data, we construct an example to show that the SPDE is weakly instable in the sense that either the exiting time is not strongly stable, or the dependence on initial data is not uniformly continuous. Finally, for the large noise, we prove that singularities can be prevented, i.e., large noise has regularization effect.</p>