Global well posedness and ergodic results in regular Sobolev spaces for the nonlinear Schrödinger equation with multiplicative noise and arbitrary power of the nonlinearity
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916304022142976 |
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| author | Brzeźniak, Zdzisław Ferrario, Benedetta Maurelli, Mario Zanella, Margherita |
| author_facet | Brzeźniak, Zdzisław Ferrario, Benedetta Maurelli, Mario Zanella, Margherita |
| contents | We consider the nonlinear Schrödinger equation on the $d$-dimensional torus $\mathbb T^d$, with the nonlinearity of polynomial type $|u|^{2σ}u$. For any $σ\in \mathbb N$ and $s>\frac d2$ we prove that adding to this equation a suitable stochastic forcing term there exists a unique global solution for any initial data in $H^s(\mathbb T^d)$. The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting. Moreover we prove existence of invariant measures and their uniqueness under more restrictive assumptions on the noise term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well posedness and ergodic results in regular Sobolev spaces for the nonlinear Schrödinger equation with multiplicative noise and arbitrary power of the nonlinearity Brzeźniak, Zdzisław Ferrario, Benedetta Maurelli, Mario Zanella, Margherita Probability Analysis of PDEs 35Q55, 35R60, 60H30, 60G10, 60H15 We consider the nonlinear Schrödinger equation on the $d$-dimensional torus $\mathbb T^d$, with the nonlinearity of polynomial type $|u|^{2σ}u$. For any $σ\in \mathbb N$ and $s>\frac d2$ we prove that adding to this equation a suitable stochastic forcing term there exists a unique global solution for any initial data in $H^s(\mathbb T^d)$. The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting. Moreover we prove existence of invariant measures and their uniqueness under more restrictive assumptions on the noise term. |
| title | Global well posedness and ergodic results in regular Sobolev spaces for the nonlinear Schrödinger equation with multiplicative noise and arbitrary power of the nonlinearity |
| topic | Probability Analysis of PDEs 35Q55, 35R60, 60H30, 60G10, 60H15 |
| url | https://arxiv.org/abs/2406.19214 |