On the two-dimensional singular stochastic viscous nonlinear wave equations
We study the stochastic viscous nonlinear wave equations (SvNLW) on $\mathbb{T}^2$, forced by a fractional derivative of the space-time white noise $\xi $. In particular, we consider SvNLW with the singular additive forcing $D^\frac{1}{2}\xi $ such that solutions are expected to be merely distributi...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2022-12-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.377/ |
| _version_ | 1827785257257009152 |
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| author | Liu, Ruoyuan Oh, Tadahiro |
| author_facet | Liu, Ruoyuan Oh, Tadahiro |
| author_sort | Liu, Ruoyuan |
| collection | DOAJ |
| description | We study the stochastic viscous nonlinear wave equations (SvNLW) on $\mathbb{T}^2$, forced by a fractional derivative of the space-time white noise $\xi $. In particular, we consider SvNLW with the singular additive forcing $D^\frac{1}{2}\xi $ such that solutions are expected to be merely distributions. By introducing an appropriate renormalization, we prove local well-posedness of SvNLW. By establishing an energy bound via a Yudovich-type argument, we also prove pathwise global well-posedness of the defocusing cubic SvNLW. Lastly, in the defocusing case, we prove almost sure global well-posedness of SvNLW with respect to certain Gaussian random initial data. |
| first_indexed | 2024-03-11T16:16:37Z |
| format | Article |
| id | doaj.art-c3cdef78b70b4e19aec545ed697e649a |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:37Z |
| publishDate | 2022-12-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-c3cdef78b70b4e19aec545ed697e649a2023-10-24T14:20:22ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-12-01360G111227124810.5802/crmath.37710.5802/crmath.377On the two-dimensional singular stochastic viscous nonlinear wave equationsLiu, Ruoyuan0Oh, Tadahiro1School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United KingdomSchool of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United KingdomWe study the stochastic viscous nonlinear wave equations (SvNLW) on $\mathbb{T}^2$, forced by a fractional derivative of the space-time white noise $\xi $. In particular, we consider SvNLW with the singular additive forcing $D^\frac{1}{2}\xi $ such that solutions are expected to be merely distributions. By introducing an appropriate renormalization, we prove local well-posedness of SvNLW. By establishing an energy bound via a Yudovich-type argument, we also prove pathwise global well-posedness of the defocusing cubic SvNLW. Lastly, in the defocusing case, we prove almost sure global well-posedness of SvNLW with respect to certain Gaussian random initial data.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.377/ |
| spellingShingle | Liu, Ruoyuan Oh, Tadahiro On the two-dimensional singular stochastic viscous nonlinear wave equations Comptes Rendus. Mathématique |
| title | On the two-dimensional singular stochastic viscous nonlinear wave equations |
| title_full | On the two-dimensional singular stochastic viscous nonlinear wave equations |
| title_fullStr | On the two-dimensional singular stochastic viscous nonlinear wave equations |
| title_full_unstemmed | On the two-dimensional singular stochastic viscous nonlinear wave equations |
| title_short | On the two-dimensional singular stochastic viscous nonlinear wave equations |
| title_sort | on the two dimensional singular stochastic viscous nonlinear wave equations |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.377/ |
| work_keys_str_mv | AT liuruoyuan onthetwodimensionalsingularstochasticviscousnonlinearwaveequations AT ohtadahiro onthetwodimensionalsingularstochasticviscousnonlinearwaveequations |