Well-posedness for the NLS hierarchy
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913571437281280 |
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| author | Adams, Joseph |
| author_facet | Adams, Joseph |
| contents | We prove well-posedness for higher-order equations in the so-called NLS hierarchy (also known as part of the AKNS hierarchy) in almost critical Fourier-Lebesgue spaces and in modulation spaces. We show the $j$th equation in the hierarchy is locally well-posed for initial data in $\hat H^s_r(\mathbb{R})$ for $s \ge \frac{j-1}{r'}$ and $1 < r \le 2$ and also in $M^s_{2, p}(\mathbb{R})$ for $s = \frac{j-1}{2}$ and $2 \le p < \infty$. Supplementing our results with corresponding ill-posedness results in Fourier-Lebesgue spaces shows optimality. Using the conserved quantities derived in Koch-Tataru (2018) we argue that the hierarchy equations are globally well-posed for data in $H^s(\mathbb{R})$ for $s \ge \frac{j-1}{2}$.
Our arguments are based on the Fourier restriction norm method in Bourgain spaces adapted to our data spaces and bi- & trilinear refinements of Strichartz estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedness for the NLS hierarchy Adams, Joseph Analysis of PDEs 35Q55 (Primary) 35Q41 (Secondary) We prove well-posedness for higher-order equations in the so-called NLS hierarchy (also known as part of the AKNS hierarchy) in almost critical Fourier-Lebesgue spaces and in modulation spaces. We show the $j$th equation in the hierarchy is locally well-posed for initial data in $\hat H^s_r(\mathbb{R})$ for $s \ge \frac{j-1}{r'}$ and $1 < r \le 2$ and also in $M^s_{2, p}(\mathbb{R})$ for $s = \frac{j-1}{2}$ and $2 \le p < \infty$. Supplementing our results with corresponding ill-posedness results in Fourier-Lebesgue spaces shows optimality. Using the conserved quantities derived in Koch-Tataru (2018) we argue that the hierarchy equations are globally well-posed for data in $H^s(\mathbb{R})$ for $s \ge \frac{j-1}{2}$. Our arguments are based on the Fourier restriction norm method in Bourgain spaces adapted to our data spaces and bi- & trilinear refinements of Strichartz estimates. |
| title | Well-posedness for the NLS hierarchy |
| topic | Analysis of PDEs 35Q55 (Primary) 35Q41 (Secondary) |
| url | https://arxiv.org/abs/2402.07652 |