On 1D Mass-subcritical Nonlinear Schrödinger equations in modulation spaces $M^{p, p'} (p<2)$
Divyang G. Bhimani, Diksha Dhingra, Vijay Kumar Sohani
TL;DR
This work proves global well-posedness for the 1D mass-subcritical NLS $iu_t + u_{xx} \pm |u|^{\alpha-1}u=0$ with $1<\alpha<5$ for large data in modulation spaces $M^{p,p'}$ with $p<2$ near 2. The authors adapt Bourgain's high-low decomposition to modulation spaces and leverage generalized Strichartz estimates in Fourier-Lebesgue spaces, together with the embedding $M^{p,p'} \hookrightarrow \widehat{L^{p}}$, to control the nonlinear evolution. They construct a local theory via data-splitting $u_0=\phi_0+\psi_0$ and then propagate the solution through an iterative scheme with carefully chosen time slices, yielding global solutions under explicit parameter regimes for $p_0$, $r$, and $\alpha$. This extends prior results for $p>2$ to the $p<2$ regime, broadening the understanding of low-regularity dynamics in modulation spaces and suggesting a robust approach for similar dispersive equations. The findings have implications for dispersive PDEs with rough data and highlight modulation spaces as a natural setting for global dynamics near scaling-critical regimes.
Abstract
We establish global well-posedness for the 1D mass-subcritical nonlinear Schrödinger equation $$iu_t +u_{xx} \pm |u|^{α-1}u=0 \quad (1< α<5)$$ for large Cauchy data in modulation spaces $M^{p,\frac{p}{p-1}}(\mathbb R)$ with $4/3<p<2$ and $p$ sufficiently close to $2$. This complements the work of Vargas-Vega (2001) and Chaichenets et al. (2017), where they established the result for $p$ larger than $2$. The proof adopts Bourgain's high-low decomposition method inspired by the work of Vargas-Vega (2001) and Hyakuna-Tsutsumi (2012) to the modulation space setting and exploits generalized Strichartz estimates in Fourier-Lebesgue spaces.