Global Well-posedness for the periodic fractional cubic NLS in 1D
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908574106517504 |
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| author | Megretski, Alexandre Skouloudis, Nikolaos |
| author_facet | Megretski, Alexandre Skouloudis, Nikolaos |
| contents | We consider the defocusing periodic fractional nonlinear Schrödinger equation
$$ i \partial_t u +\left(-Δ\right)^αu=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< α< 1$ and the operator $(-Δ)^α$ is the fractional Laplacian with symbol $\lvert k \rvert ^{2α}$. We establish global well-posedness in $H^s(\mathbb{T})$ for $s\geq \frac{1-α}{2}$ and we conjecture this threshold to be sharp as it corresponds to the pseudo-Galilean symmetry exponent. Our proof uses the $I$-method to control the $H^s(\mathbb{T})$-norm of solutions with infinite energy initial data. A key component of our approach is a set of improved long-time bilinear Strichartz estimates on the rescaled torus, which allow us to exploit the subcritical nature of the equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01204 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Well-posedness for the periodic fractional cubic NLS in 1D Megretski, Alexandre Skouloudis, Nikolaos Analysis of PDEs We consider the defocusing periodic fractional nonlinear Schrödinger equation $$ i \partial_t u +\left(-Δ\right)^αu=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< α< 1$ and the operator $(-Δ)^α$ is the fractional Laplacian with symbol $\lvert k \rvert ^{2α}$. We establish global well-posedness in $H^s(\mathbb{T})$ for $s\geq \frac{1-α}{2}$ and we conjecture this threshold to be sharp as it corresponds to the pseudo-Galilean symmetry exponent. Our proof uses the $I$-method to control the $H^s(\mathbb{T})$-norm of solutions with infinite energy initial data. A key component of our approach is a set of improved long-time bilinear Strichartz estimates on the rescaled torus, which allow us to exploit the subcritical nature of the equation. |
| title | Global Well-posedness for the periodic fractional cubic NLS in 1D |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.01204 |