The cubic Dirac equation: Small initial data in $H^{\frac12}(\mathbb{R}^2)$

TL;DR

This paper proves global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space $H^{1/2}(R^2)$, using sharp Strichartz estimates and adapted coordinate frames.

Contribution

It introduces a novel approach employing sharp endpoint Strichartz estimates and coordinate frame adaptations to analyze the cubic Dirac equation in two dimensions.

Findings

Established global well-posedness for small data in $H^{1/2}(R^2)$

Proved scattering results for the cubic Dirac equation

Developed a new method using adapted coordinate frames

Abstract

Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space $H^{\frac12}(\mathbb{R}^2)$ is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein-Gordon equation in dimension $n=2$, which is captured by constructing an adapted systems of coordinate frames.