Low-regularity global solution of the inhomogeneous nonlinear Schr\"odinger equations in modulation spaces

TL;DR

This paper proves the first global well-posedness results for the inhomogeneous nonlinear Schrödinger equation in modulation spaces, extending understanding of low-regularity solutions and employing advanced decomposition techniques.

Contribution

It establishes the first global well-posedness results for INLS in modulation spaces, including cases with low regularity Sobolev spaces.

Findings

Global well-posedness in modulation spaces for INLS.

Local well-posedness in certain Lebesgue and Sobolev spaces.

Application of Bourgain's high-low decomposition method.

Abstract

The study of low regularity Cauchy data for nonlinear dispersive PDEs has successfully been achieved using modulation spaces $M^{p,q}$ in recent years. In this paper, we study the inhomogeneous nonlinear Schr\"odinger equation (INLS) $$iu_t + \Delta u\pm |x|^{-b}|u|^{\alpha}u=0,$$ where $\alpha, b>0,$ on whole space $\mathbb R^n$ in modulation spaces. In the subcritical regime $(0<\alpha< \frac{4-2b}{n}),$ we establish local well-posedness in $L^{2}+M^{\alpha+2,\frac{\alpha+2}{\alpha+1}}( \supset L^2 + H^s \ \text{for} \ s>\frac{n\alpha}{2(\alpha+2)}).$ By adapting Bourgain's high-low decomposition method, we establish global well-posedness in $M^{p,\frac{p}{p-1}}$ with $2<p$ and $p$ sufficiently close to 2. This is the first global well-posedness result for INLS on modulation spaces, which contains certain Sobolev $H^s$ $(0<s<1)$ and $L^p_s-$Sobolev spaces.