Well-posedness in the full scaling-subcritical range for a class of nonlocal NLS on the line
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917368604655616 |
|---|---|
| author | Hadama, Sonae |
| author_facet | Hadama, Sonae |
| contents | In this paper, we study a class of one-dimensional nonlocal nonlinear Schrödinger equations on the line with nonlinearity given by a Fourier multiplier whose symbol has subcritical high-frequency growth. In terms of symbol order, this class is intermediate between the cubic nonlinear Schrödinger equation and the Calogero--Moser derivative nonlinear Schrdöinger equation. We prove local well-posedness in $L^2(\mathbb{R})$ throughout the full scaling-subcritical range. Due to derivative loss, the standard Duhamel integral is not directly meaningful for rough data. To avoid this problem, we first construct the propagator $S_V$ for rough time-dependent potentials $V$, and then prove an Ozawa-Tsutsumi type bilinear Strichartz estimate for the perturbed flow $S_V$. These linear theories yield a concrete construction of rough solutions without using any equation-specific algebraic structure. For real-valued symbols, mass is conserved, and the local solutions are therefore global. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28055 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Well-posedness in the full scaling-subcritical range for a class of nonlocal NLS on the line Hadama, Sonae Analysis of PDEs Primary 35Q55, Secondary 35A01, 35A02 In this paper, we study a class of one-dimensional nonlocal nonlinear Schrödinger equations on the line with nonlinearity given by a Fourier multiplier whose symbol has subcritical high-frequency growth. In terms of symbol order, this class is intermediate between the cubic nonlinear Schrödinger equation and the Calogero--Moser derivative nonlinear Schrdöinger equation. We prove local well-posedness in $L^2(\mathbb{R})$ throughout the full scaling-subcritical range. Due to derivative loss, the standard Duhamel integral is not directly meaningful for rough data. To avoid this problem, we first construct the propagator $S_V$ for rough time-dependent potentials $V$, and then prove an Ozawa-Tsutsumi type bilinear Strichartz estimate for the perturbed flow $S_V$. These linear theories yield a concrete construction of rough solutions without using any equation-specific algebraic structure. For real-valued symbols, mass is conserved, and the local solutions are therefore global. |
| title | Well-posedness in the full scaling-subcritical range for a class of nonlocal NLS on the line |
| topic | Analysis of PDEs Primary 35Q55, Secondary 35A01, 35A02 |
| url | https://arxiv.org/abs/2603.28055 |