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

TL;DR

This paper investigates a stochastic Camassa-Holm equation with higher nonlinearities, establishing local well-posedness, blow-up criteria, and showing that strong noise can prevent blow-up, highlighting the regularization effect of stochastic perturbations.

Contribution

It provides the first detailed analysis of continuous dependence on initial data and blow-up criteria for stochastic Camassa-Holm equations with higher nonlinearities.

Findings

Solution map is continuous in Sobolev spaces

Stability of exiting time is characterized

Strong noise can ensure global existence almost surely

Abstract

This paper aims at studying a generalized Camassa--Holm equation under random perturbation. We establish a local well-posedness result in the sense of Hadamard, i.e., existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces $H^s$ with $s>3/2$ for $x\in\mathbb{R}$. The analysis on continuous dependence on initial data for nonlinear stochastic partial differential equations has gained less attention in the literature so far. In this work, we first show that the solution map is continuous. Then we introduce a notion of stability of exiting time. We provide an example showing that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data. Finally, we analyze the regularization effect of nonlinear noise in preventing blow-up. Precisely, we demonstrate that global existence holds true almost surely provided that the noise is strong enough.