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

Abstract

The subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces H s with s > 3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.

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

“…We conclude this preparatory section with some results from [47], which are needed to establish the theorems on global existence. (ii) Let a(t) = λb 2 (t) with λ < 1 2 and τ R = inf{t ≥ 0 : X(t) > R} with R > 1, then…”

“…We consider the initial value problem (1.8). The proof of existence and uniqueness of pathwise solutions can be carried out by standard procedures used in many works, see [2,3,28,29,[47][48][49] for more details. Therefore we only give a sketch.…”

“…With such cut-off, both the drift and diffusion coefficients in the problem become locally Lipschitz continuous and grow linearly in u (cf. [47][48][49]). Thus the approximation solutions exist globally.…”

“…To finish the proof of Theorem 2.1, we only need to verify the blow-up criterion (2.5). Motivated by [16,47], we first consider, in next lemma, the relationship between the explosion time of u(t) H s and the explosion time of u(t) W 1,∞ for (1.8). The results of the lemma will not only immediately imply the blow-up criterion (2.5) but also be used in the next sections.…”

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

“…As in [47], the additional noise term can be used to account for the randomness arising from the energy exchange mechanisms. Indeed, in [40,58,59], the weakly dissipative term (1 − ∂ 2 xx )(λu) with λ > 0 was added to the governing equations.…”