Local well-posedness for cubic fractional Schrödinger equations with derivatives on the right-hand side
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917968734060544 |
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| author | Dughayshim, Ahmed Farina, Silvino Reyes Schikorra, Armin |
| author_facet | Dughayshim, Ahmed Farina, Silvino Reyes Schikorra, Armin |
| contents | For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-Δ)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data $\|u(0)\|_{H^{\frac{n-2s}{2}}(\mathbb{R}^n)} \ll 1$.
This equation is a model equation for for $s$-Schrödinger map equation \[ \partial_t ψ= ψ\wedge (-Δ)^s ψ: \quad ψ: \mathbb{R}^n \times \mathbb{R} \to \mathbb{S}^{2}, \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20971 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local well-posedness for cubic fractional Schrödinger equations with derivatives on the right-hand side Dughayshim, Ahmed Farina, Silvino Reyes Schikorra, Armin Analysis of PDEs For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-Δ)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data $\|u(0)\|_{H^{\frac{n-2s}{2}}(\mathbb{R}^n)} \ll 1$. This equation is a model equation for for $s$-Schrödinger map equation \[ \partial_t ψ= ψ\wedge (-Δ)^s ψ: \quad ψ: \mathbb{R}^n \times \mathbb{R} \to \mathbb{S}^{2}, \] |
| title | Local well-posedness for cubic fractional Schrödinger equations with derivatives on the right-hand side |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.20971 |