On the nonlinear Schr\"odinger equation in spaces of infinite mass and low regularity

TL;DR

This paper investigates the nonlinear Schrödinger equation with initial data of infinite mass and low regularity, establishing local well-posedness across a broad parameter range and global existence for a specific defocusing cubic case.

Contribution

It introduces a new well-posedness framework for NLS in spaces of infinite mass and low regularity, and proves global existence for the defocusing cubic in 3D.

Findings

Well-posedness in $ ext{Z}^s_p$ spaces for all $s,p$ in the specified range

Global existence for defocusing cubic NLS in 3D with infinite mass

Linear Schrödinger group well-defined in these low-regularity, infinite-mass spaces

Abstract

We study the nonlinear Schr\"odinger equation with initial data in $\mathcal{Z}^s_p(\mathbb{R}^d)=\dot{H}^s(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, where $0<s<\min\{d/2,1\}$ and $2<p<2d/(d-2s)$. After showing that the linear Schr\"odinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.