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 with . 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 inital 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.
Keywords
Cite
@article{arxiv.2003.07206,
title = {On the stochastic Dullin-Gottwald-Holm equation: Global existence and wave-breaking phenomena},
author = {Christian Rohde and Hao Tang},
journal= {arXiv preprint arXiv:2003.07206},
year = {2020}
}