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

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 $$H^s$$ H s with $$s>3/2$$ 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.

Well-posedness for a stochastic Camassa-Holm type equation with higher order nonlinearities

“…For the stochastic modified CH equation with linear multiplicative noise, we refer to [9]. In [47], global existence and noise effect for (1.4) are studied in the periodic case, i.e., x ∈ T, and we refer to [48] for the stochastic Dullin-Gottwald-Holm equation.…”

“…After taking limits to obtain a solution, one can improve the regularity to H s again, and the technical difficulty in this step is to prove the time continuity of the solution because the classical Itô formula is not applicable (see in Remark 1.1). To overcome this difficulty, as in [41,48], we apply a mollifier J ε to equation and estimate E J ε u 2 H s first (see (2.19) and (5.1)). We also remark that the techniques in removing the cut-off have been used in [3,23,41].…”

Global existence and blow-up for a stochastic transport equation with non-local velocity

“…Here we use the terminology "fast growing" condition, described by (1.11) in Assumption (B), to cancel (notice that G ′′ < 0 in (1.11)) the growth of the non-local transport term (Hu)u such that EG( u 2 H s ) can be controlled with a Lyapunov type function G. The idea of using a Lyapunov type function is motivated by the works [7,36,49,51].…”

“…• Motivated by previous works [24,50,51], to study (1.4), we make use of Girsanov type transform to obtain a PDE with random coefficient instead of a SPDE to study blow-up. In this linear noise case, the blow-up criterion (1.15) also holds true.…”

“…Remark 4.1. We remark that the using of Lyapunov is motivated by the non-explosion text [36], see also [7,49,51]. In the above proof, (2.8) is used to obtain (4.3), but we remark that (2.9) is also used implicitly.…”