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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2411.16610 |
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| _version_ | 1866929603981869056 |
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| author | Himonas, A. Alexandrou Yan, Fangchi |
| author_facet | Himonas, A. Alexandrou Yan, Fangchi |
| contents | This work studies the initial-boundary value problem for both the linear Schrödinger equation and the cubic nonlinear Schrödinger equation on the half-space in higher dimensions ($n\ge 2$). First, the forced linear problem is solved on the half-space via the Fokas method and then using the obtained solution formula new and interesting linear estimates are derived with data and forcing in appropriate spaces. Second, the well-posedness of the nonlinear problem on the half-space is proved with initial data in Sobolev spaces $H^s(\mathbb{R}^n_+)$, with $s>\frac{n}{2}-1$, and boundary data in natural Bourgain spaces $\mathcal{B}^s$ that reflect the boundary regularity of the linear problem. The proof method consists of showing that the iteration map defined via the Fokas solution formula is a contraction by establishing sharper trilinear estimates. The presence of the boundary introduces solution spaces that involve temporal Bourgain spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16610 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The nonlinear Schrödinger equation on the half-space Himonas, A. Alexandrou Yan, Fangchi Analysis of PDEs 35Q55, 35G31, 35G16, 37K10 This work studies the initial-boundary value problem for both the linear Schrödinger equation and the cubic nonlinear Schrödinger equation on the half-space in higher dimensions ($n\ge 2$). First, the forced linear problem is solved on the half-space via the Fokas method and then using the obtained solution formula new and interesting linear estimates are derived with data and forcing in appropriate spaces. Second, the well-posedness of the nonlinear problem on the half-space is proved with initial data in Sobolev spaces $H^s(\mathbb{R}^n_+)$, with $s>\frac{n}{2}-1$, and boundary data in natural Bourgain spaces $\mathcal{B}^s$ that reflect the boundary regularity of the linear problem. The proof method consists of showing that the iteration map defined via the Fokas solution formula is a contraction by establishing sharper trilinear estimates. The presence of the boundary introduces solution spaces that involve temporal Bourgain spaces. |
| title | The nonlinear Schrödinger equation on the half-space |
| topic | Analysis of PDEs 35Q55, 35G31, 35G16, 37K10 |
| url | https://arxiv.org/abs/2411.16610 |