Well-posedness for stochastic Camassa–Holm equation

Abstract

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 H s (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 H q (R) in finite time for any q > 3/2 when initial value satisfies some conditions.

Rare Events in the Stochastic Camassa–holm Equation

“…< q < min{1, s − 1}, similar to the estimates in[5, Proposition 3.1], E sup t∈[0,T ∧τ R ] w η q ≤ C( + 1)η (s−q)/2 e1+ √ R . (3.10) By using the estimates of Chen et al [5, (4.48)-(4.50)], the inequality (3.10), Gronwall's inequality and Itô's formula, E sup…”

“…Lemma 3.1 [2,5]. Under the Assumptions 2.4, the following estimates hold for any η satisfying 0 < η < 1 and s > 0.…”

“…The CH equation was derived by Camassa and Holm [4,12] as a model of water waves. The well-posedness of (1.1) in H s (R) for s > 3/2 was established by Chen et al [5]. In this paper, we consider the rare events as described by a large deviation principle (LDP) for the stochastic CH equation (1.1).…”

On the stochastic Dullin–Gottwald–Holm equation: global existence and wave-breaking phenomena

“…Under generic assumptions on h(t, u), we will show that (1.8) has a local unique pathwise solution (see Theorem 2.1 below). Here we remark that Chen et al in [9] have considered the stochastic CH equation with additive noise. For the linear multiplicative noise case, we refer to [48] for the stochastic CH equation, and to [8] for a stochastic modified CH equation.…”

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

“…The first objective of this paper is to analyze the local existence and uniqueness of pathwise solutions as well as blow-up criteria for problem (1.5) with nonlinear multiplicative noise (see Theorem 2.1). We note that for the CH equation with additive noise, existence and uniqueness has been obtained in [11]. For the stochastic modified CH equation with linear multiplicative noise, we refer to [10].…”