Local and global well-posedness for the kinetic derivative NLS on $\mathbb{R}$
Nobu Kishimoto, Kiyeon Lee
TL;DR
This work establishes local well-posedness for the kinetic derivative NLS (KDNLS) on the real line in $H^2(\mathbb{R})$ and $H^2\cap H^{1,1}(\mathbb{R})$ for all real parameters $\alpha,\beta$, addressing resonant interactions with a combination of energy methods and frequency-restricted gauge transformations. It develops a regularized problem and a normal-form (modified energy) framework, yielding robust a priori estimates in high-regularity and weighted spaces, and provides stability against perturbations by proving difference estimates in $L^2$, $H^2$, and weighted spaces. In the dissipative regime $\beta<0$, the authors obtain a global-in-time $H^2$ bound, hence forward-global well-posedness in $H^2$ (and in $H^2\cap H^{1,1}$); this marks a first global result for KDNLS on $\mathbb{R}$. The study also highlights how the real-line geometry permits local well-posedness for $\beta>0$, contrasting with torus ill-posedness, and sets the stage for future asymptotic analysis of small solutions.
Abstract
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 $H^2$.