Regularization by Nonlinear Noise for PDEs: Well-posedness and Finite Time Extinction
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915408667213824 |
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| author | Hong, Wei Li, Shihu Liu, Wei |
| author_facet | Hong, Wei Li, Shihu Liu, Wei |
| contents | This work focuses on the regularization by nonlinear noise for a class of partial differential equations that may only have local solutions. In particular, we obtain the global existence, uniqueness and the Feller property for stochastic 3D Navier-Stokes equations, which provide positive answers to a longstanding open problem in this field.
Moreover, we discover a new phenomenon that for a potentially explosive deterministic system, an appropriate intervention of nonlinear noise can not only prevent blow-up but also lead to the finite time extinction of the associated stochastic system. Our main results have broad applications, including stochastic $p$-Laplace equations with heat sources, stochastic surface growth models and stochastic quasi-geostrophic equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06840 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularization by Nonlinear Noise for PDEs: Well-posedness and Finite Time Extinction Hong, Wei Li, Shihu Liu, Wei Probability Analysis of PDEs 60H15, 60H50, 35Q30, 35R60 This work focuses on the regularization by nonlinear noise for a class of partial differential equations that may only have local solutions. In particular, we obtain the global existence, uniqueness and the Feller property for stochastic 3D Navier-Stokes equations, which provide positive answers to a longstanding open problem in this field. Moreover, we discover a new phenomenon that for a potentially explosive deterministic system, an appropriate intervention of nonlinear noise can not only prevent blow-up but also lead to the finite time extinction of the associated stochastic system. Our main results have broad applications, including stochastic $p$-Laplace equations with heat sources, stochastic surface growth models and stochastic quasi-geostrophic equations. |
| title | Regularization by Nonlinear Noise for PDEs: Well-posedness and Finite Time Extinction |
| topic | Probability Analysis of PDEs 60H15, 60H50, 35Q30, 35R60 |
| url | https://arxiv.org/abs/2407.06840 |