Large Data Well-posedness in the Energy Space of the Chern-Simons-Schr\"odinger System

TL;DR

This paper proves local and global well-posedness for the Chern-Simons-Schr"odinger system in the energy space, utilizing dispersive analysis and adapted function spaces to handle the system's long-range electromagnetic interactions.

Contribution

It establishes well-posedness results in the energy space for the Chern-Simons-Schr"odinger system, including a novel approach to handle derivative nonlinearities non-perturbatively.

Findings

Local well-posedness in $H^s$ for $s \,\geq 1$

Global regularity via energy conservation

Use of $U^p$ and $V^p$ spaces for dispersive estimates

Abstract

We consider the initial-value problem for the Chern-Simons-Schr\"odinger system, which is a gauge-covariant Schr\"{o}dinger system in $\mathbb{R}_t\times\mathbb{R}^2_x$ with a long-range electromagnetic field. We show that, in the Coulomb gauge, it is locally well-posed in $H^s$ for $s\ge 1$, and the solution map satisfies a local-in-time weak Lipschitz bound. By energy conservation, we also obtain a global regularity result. The key is to retain the non-perturbative part of the derivative nonlinearity in the principal operator, and exploit the dispersive properties of the resulting paradifferential-type principal operator using adapted $U^p$ and $V^p$ spaces.