Abstract
We consider a class of stochastic evolution equations that include in particular the stochastic Camassa–Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces H^s with s>3/2. Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the initial data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.
Highlights
The Dullin–Gottwald–Holm (DGH) equation is a third-order dispersive evolution equation given by ut − α2uxxt + c0ux + 3uux + γuxxx = α2 (2uxuxx + uuxxx) in (0, ∞) × R
We notice that the CH equation exhibits two interesting phenomenon, namely soliton interaction and wave breaking, while the KdV equation does not model breaking waves [35] (when c0 = 0, (1.2) admits a smooth soliton)
Bressan&Constantin [5,6] developed a new approach to the analysis of the CH equation, and proved the existence of a global conservative and dissipative solutions
Summary
The Dullin–Gottwald–Holm (DGH) equation is a third-order dispersive evolution equation given by ut − α2uxxt + c0ux + 3uux + γuxxx = α2 (2uxuxx + uuxxx) in (0, ∞) × R. It was derived by Dullin et al in [20] as a model governing planar solutions to Euler’s equations in the shallow–water regime. While (1.1) equals to the following Camassa–Holm (CH) equation for the choices γ = 0 and α = 1, ut − uxxt + c0ux + 3uux = 2uxuxx + uuxxx Both (1.2) and (1.3) have been studied widely in the literature. The first goal of the present paper is to analyze the existence and uniqueness of pathwise solutions and to determine possible
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