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

On a Stochastic Camassa–Holm Type Equation with Higher Order Nonlinearities

“…Remark 2.1 We remark here that F 3 (u) in (1.6) will disappear when k = 1. The proof for Theorem 2.1 combines the techniques as employed in the papers [3][4][5]18,20,29,58]. However, the Faedo-Galerkin method used e.g.…”

“…Here we notice that if k = 1, then F 3 (u) in (1.6) will disappear. Motivated by the recent papers [29,57,58], where the linear noise βudW with β ∈ R \ {0} is considered, we focus on the non-autonomous linear multiplicative noise case, namely (1.8). We transform (1.8) to a non-autonomous random system (5.2).…”

“…For the stochastic modified CH equation with linear multiplicative noise, we refer to [10]. When the multiplicative noise is given by a one-dimensional Wiener process with H s diffusion coefficient, the stochastic CH equation was considered in second author's work [58]. In this paper, we consider the local existence and uniqueness of pathwise solutions to (1.5) as well as blow-up criteria for more general noise given by a cylindrical Wiener process with nonlinear coefficient.…”

“…For example, we refer to [21,23,53,54] for linear stochastic transport equations and to what concerns nonlinear stochastic conservation laws, we refer to [27,50,51]. In [29,46,57,58] the dissipation of energy caused by linear multiplicative noise has been analyzed.…”

“…For general noise terms, it is difficult to determine whether a local pathwise solution to (1.5) is globally defined. As shown in [29,46,48,57,58], the linear noise h(t, u)dW = βudW , with β ∈ R \ {0} and W being a standard 1-D Brownian motion, acts dissipatively for many SPDEs. Motivated by these works, we prove some results in Theorems 2.3 and 2.4 for the global dynamics of (1.5) with linear multiplicative noise , that is…”