Local and global well-posedness for the kinetic derivative NLS on $\mathbb{R}$
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arXiv
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| Format: | Preprint |
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2025
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| author | Kishimoto, Nobu Lee, Kiyeon |
| author_facet | Kishimoto, Nobu Lee, Kiyeon |
| contents | We investigate the local and global well-posedness of the kinetic derivative nonlinear Schrödinger equation (KDNLS) on $\mathbb{R}$, described by \[ i\partial_t u + \partial_x^2 u = iα\partial_x (|u|^2 u) + iβ\partial_x (H(|u|^2) u), \] where $α, β\in \mathbb{R}$, and $H$ represents the Hilbert transformation. For KDNLS, the $L^2$ norm of a solution is decreasing (resp. increasing, conserved) when $β$ is negative (resp. positive, zero). Focusing on the Sobolev spaces $H^2$ and $H^2 \cap H^{1,1}$, we establish local well-posedness via the energy method combined with gauge transformations to address resonant interactions in both cases of negative and positive $β$. For the dissipative case $β< 0$, we further demonstrate global well-posedness by deriving an a priori bound in $H^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local and global well-posedness for the kinetic derivative NLS on $\mathbb{R}$ Kishimoto, Nobu Lee, Kiyeon Analysis of PDEs We investigate the local and global well-posedness of the kinetic derivative nonlinear Schrödinger equation (KDNLS) on $\mathbb{R}$, described by \[ i\partial_t u + \partial_x^2 u = iα\partial_x (|u|^2 u) + iβ\partial_x (H(|u|^2) u), \] where $α, β\in \mathbb{R}$, and $H$ represents the Hilbert transformation. For KDNLS, the $L^2$ norm of a solution is decreasing (resp. increasing, conserved) when $β$ is negative (resp. positive, zero). Focusing on the Sobolev spaces $H^2$ and $H^2 \cap H^{1,1}$, we establish local well-posedness via the energy method combined with gauge transformations to address resonant interactions in both cases of negative and positive $β$. For the dissipative case $β< 0$, we further demonstrate global well-posedness by deriving an a priori bound in $H^2$. |
| title | Local and global well-posedness for the kinetic derivative NLS on $\mathbb{R}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.20271 |