Critical local well-posedness of the nonlinear Schrödinger equation on the torus
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915035436023808 |
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| author | Kwak, Beomjong Kwon, Soonsik |
| author_facet | Kwak, Beomjong Kwon, Soonsik |
| contents | In this paper, we study the local well-posedness of nonlinear Schrödinger equations on tori $\mathbb{T}^{d}$ at the critical regularity. We focus on cases where the nonlinearity $|u|^{a}u$ is non-algebraic with small $a>0$. We prove the local well-posedness for a wide range covering the mass-supercritical regime. Moreover, we supplementarily investigate the regularity of the solution map. In pursuit of lowering $a$, we prove a bilinear estimate for the Schrödinger operator on tori $\mathbb{T}^{d}$, which enhances previously known multilinear estimates. We design a function space adapted to the new bilinear estimate and a package of Strichartz estimates, which is not based on conventional atomic spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17147 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical local well-posedness of the nonlinear Schrödinger equation on the torus Kwak, Beomjong Kwon, Soonsik Analysis of PDEs 35Q55 In this paper, we study the local well-posedness of nonlinear Schrödinger equations on tori $\mathbb{T}^{d}$ at the critical regularity. We focus on cases where the nonlinearity $|u|^{a}u$ is non-algebraic with small $a>0$. We prove the local well-posedness for a wide range covering the mass-supercritical regime. Moreover, we supplementarily investigate the regularity of the solution map. In pursuit of lowering $a$, we prove a bilinear estimate for the Schrödinger operator on tori $\mathbb{T}^{d}$, which enhances previously known multilinear estimates. We design a function space adapted to the new bilinear estimate and a package of Strichartz estimates, which is not based on conventional atomic spaces. |
| title | Critical local well-posedness of the nonlinear Schrödinger equation on the torus |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/2411.17147 |