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

• Published in: Journal of dynamics and differential equations Vol. 33; no. 4; pp. 1823 - 1852
• Main Authors: Rohde, Christian, Tang, Hao
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Existence theorems; Mathematics; Mathematics and Statistics; Noise; Ordinary Differential Equations; Partial Differential Equations; Perturbation; Regularization; Sobolev space; Stability; Stochastic generalized Camassa–Holm equation; Exiting time; Secondary: 35A01; Pathwise solution; Global existence; 35B30; Noise effect; 35Q51; Primary: 60H15; Dependence on initial data
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-020-09872-1

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.