Global existence of dissipative solutions to the Camassa--Holm equation with transport noise
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913186176827392 |
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| author | Galimberti, Luca Holden, Helge Karlsen, Kenneth H. Pang, Peter H. C. |
| author_facet | Galimberti, Luca Holden, Helge Karlsen, Kenneth H. Pang, Peter H. C. |
| contents | We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich--Itô correction term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_07046 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Global existence of dissipative solutions to the Camassa--Holm equation with transport noise Galimberti, Luca Holden, Helge Karlsen, Kenneth H. Pang, Peter H. C. Analysis of PDEs Primary: 35R60, 35G25, Secondary: 35A01, 35D30 We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich--Itô correction term. |
| title | Global existence of dissipative solutions to the Camassa--Holm equation with transport noise |
| topic | Analysis of PDEs Primary: 35R60, 35G25, Secondary: 35A01, 35D30 |
| url | https://arxiv.org/abs/2211.07046 |