On the Pathwise Solutions to the Camassa--Holm Equation with Multiplicative Noise

Abstract

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 $H^s$ 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.