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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2410.05034 |
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| _version_ | 1866916425374892032 |
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| author | Herr, Sebastian Röckner, Michael Spitz, Martin Zhang, Deng |
| author_facet | Herr, Sebastian Röckner, Michael Spitz, Martin Zhang, Deng |
| contents | This work is devoted to the stochastic Zakharov system in dimension four, which is the energy-critical dimension. First, we prove local well-posedness in the energy space $H^1\times L^2$ up to the maximal existence time and a blow-up alternative. Second, we prove that for large data solutions exist globally as long as energy and wave mass are below the ground state threshold. Third, we prove a regularization by noise phenomenon: the probability of global existence and scattering goes to one if the strength of the (non-conservative) noise goes to infinity. The proof is based on the refined rescaling approach and a new functional framework, where both Fourier restriction and local smoothing norms are used as well as a (uniform) double endpoint Strichartz and local smoothing inequality for the Schrödinger equation with certain rough and time dependent lower order perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05034 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The energy-critical stochastic Zakharov system Herr, Sebastian Röckner, Michael Spitz, Martin Zhang, Deng Analysis of PDEs This work is devoted to the stochastic Zakharov system in dimension four, which is the energy-critical dimension. First, we prove local well-posedness in the energy space $H^1\times L^2$ up to the maximal existence time and a blow-up alternative. Second, we prove that for large data solutions exist globally as long as energy and wave mass are below the ground state threshold. Third, we prove a regularization by noise phenomenon: the probability of global existence and scattering goes to one if the strength of the (non-conservative) noise goes to infinity. The proof is based on the refined rescaling approach and a new functional framework, where both Fourier restriction and local smoothing norms are used as well as a (uniform) double endpoint Strichartz and local smoothing inequality for the Schrödinger equation with certain rough and time dependent lower order perturbations. |
| title | The energy-critical stochastic Zakharov system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.05034 |