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 Hs with 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.
%0 Journal Article
%1 rohde2021stochastic
%A Rohde, Christian
%A Tang, Hao
%D 2021
%I Springer
%J Nonlinear Differential Equations and Applications NoDEA
%K EXC2075 pn1
%N 5
%P 1--34
%R 10.1007/s00030-020-00661-9
%T On the stochastic Dullin--Gottwald--Holm equation: global existence and wave-breaking phenomena
%U https://doi.org/10.1007/s00030-020-00661-9
%V 28
%X 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 Hs with 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.
@article{rohde2021stochastic,
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 Hs with 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.},
added-at = {2022-01-24T10:40:14.000+0100},
author = {Rohde, Christian and Tang, Hao},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/23995c6831ba43fd76694ec4ee2f2388a/simtechpuma},
doi = {10.1007/s00030-020-00661-9},
interhash = {d5624940bad31d972363b26744e8ac31},
intrahash = {3995c6831ba43fd76694ec4ee2f2388a},
journal = {Nonlinear Differential Equations and Applications NoDEA},
keywords = {EXC2075 pn1},
number = 5,
pages = {1--34},
publisher = {Springer},
timestamp = {2022-01-27T10:50:56.000+0100},
title = {On the stochastic Dullin--Gottwald--Holm equation: global existence and wave-breaking phenomena},
url = {https://doi.org/10.1007/s00030-020-00661-9},
volume = 28,
year = 2021
}
