Global existence and wave breaking for a stochastic two-component Camassa–Holm system

Abstract

In this paper, we study the stochastic two-component Camassa–Holm shallow water system on R and T≔R/2πZ. We first establish the existence, uniqueness, and blow-up criterion of the pathwise strong solution to the initial value problem with nonlinear noise. Then, we consider the impact of noise on preventing blow-up. In both nonlinear and linear noise cases, we establish global existence. In the nonlinear noise case, the global existence holds true with probability 1 if a Lyapunov-type condition is satisfied. In the linear noise case, we provide a lower bound for the probability that the solution exists globally. Furthermore, in the linear noise and the periodic case, we formulate a precise condition on initial data that leads to blow-up of strong solutions with a positive probability, and the lower bound for this probability is also estimated.