The energy-critical stochastic nonlinear Schrödinger equation: well-posedness and blow-up
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910135196057600 |
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| author | Millet, Annie Roudenko, Svetlana |
| author_facet | Millet, Annie Roudenko, Svetlana |
| contents | We investigate the focusing and defocusing energy-critical stochastic nonlinear Schrödinger equation, subject to random perturbations in the form of either additive or multiplicative (Stratonovich) noise. We establish local well-posedness for random or deterministic initial data $u_0$ in $\dot{H}^1(\mathbb{R}^n)$ or $H^1(\mathbb{R}^n)$, depending on the noise type. In the focusing case we provide quantitative estimates regarding the existence time and probability. Moreover, we derive blow-up criteria for solutions with positive energy in both cases of noise, provided that the noise intensity is sufficiently small, showing that blow-up occurs before a certain given positive time with positive probability, thus, extending deterministic results of Kenig-Merle [24] for the energy-critical NLS equation to the stochastic setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14852 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The energy-critical stochastic nonlinear Schrödinger equation: well-posedness and blow-up Millet, Annie Roudenko, Svetlana Analysis of PDEs Probability 60H15, 35R60, 35Q55 We investigate the focusing and defocusing energy-critical stochastic nonlinear Schrödinger equation, subject to random perturbations in the form of either additive or multiplicative (Stratonovich) noise. We establish local well-posedness for random or deterministic initial data $u_0$ in $\dot{H}^1(\mathbb{R}^n)$ or $H^1(\mathbb{R}^n)$, depending on the noise type. In the focusing case we provide quantitative estimates regarding the existence time and probability. Moreover, we derive blow-up criteria for solutions with positive energy in both cases of noise, provided that the noise intensity is sufficiently small, showing that blow-up occurs before a certain given positive time with positive probability, thus, extending deterministic results of Kenig-Merle [24] for the energy-critical NLS equation to the stochastic setting. |
| title | The energy-critical stochastic nonlinear Schrödinger equation: well-posedness and blow-up |
| topic | Analysis of PDEs Probability 60H15, 35R60, 35Q55 |
| url | https://arxiv.org/abs/2604.14852 |