dc.contributor.author | Holden, Helge | |
dc.contributor.author | Karlsen, Kenneth Aksel Hvistendahl | |
dc.contributor.author | Pang, Ho Cheung | |
dc.date.accessioned | 2023-02-27T10:21:01Z | |
dc.date.available | 2023-02-27T10:21:01Z | |
dc.date.created | 2022-11-15T08:21:11Z | |
dc.date.issued | 2022 | |
dc.identifier.citation | Discrete and Continuous Dynamical Systems. Series A. 2022, 43 (2), 568-618. | en_US |
dc.identifier.issn | 1078-0947 | |
dc.identifier.uri | https://hdl.handle.net/11250/3054142 | |
dc.description.abstract | We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa–Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in () using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod–Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-Itô correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional Itô formula and a DiPerna–Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators. | en_US |
dc.language.iso | eng | en_US |
dc.publisher | American Institute of Mathematical Sciences | en_US |
dc.rights | Navngivelse 4.0 Internasjonal | * |
dc.rights.uri | http://creativecommons.org/licenses/by/4.0/deed.no | * |
dc.title | Global well-posedness of the viscous Camassa–Holm equation with gradient noise | en_US |
dc.title.alternative | Global well-posedness of the viscous Camassa–Holm equation with gradient noise | en_US |
dc.type | Peer reviewed | en_US |
dc.type | Journal article | en_US |
dc.description.version | publishedVersion | en_US |
dc.source.pagenumber | 568-618 | en_US |
dc.source.volume | 43 | en_US |
dc.source.journal | Discrete and Continuous Dynamical Systems. Series A | en_US |
dc.source.issue | 2 | en_US |
dc.identifier.doi | 10.3934/dcds.2022163 | |
dc.identifier.cristin | 2073918 | |
dc.relation.project | Norges forskningsråd: 250070 | en_US |
dc.relation.project | Norges forskningsråd: 301538 | en_US |
cristin.ispublished | true | |
cristin.fulltext | original | |
cristin.qualitycode | 2 | |