Global solution to the cubic Dirac equation in two space dimensions

TL;DR

This paper proves the global existence, uniform decay, and scattering of solutions to the cubic Dirac equation in two dimensions for all mass values, including an improved decay when the mass is zero.

Contribution

It provides the first comprehensive analysis of the cubic Dirac equation in 2D, establishing uniform-in-mass global solutions and explicit scattering behavior.

Findings

Global existence of solutions for all masses in [0,1]

Uniform pointwise decay for solutions across all masses

Explicit scattering speed and improved decay at zero mass

Abstract

We are interested in the cubic Dirac equation with mass $m \in [0, 1]$ in two space dimensions, which is also known as the Soler model. We conduct a thorough study on this model with initial data sufficiently small in high regularity Sobolev spaces. First, we show the global existence of the model, which is uniform-in-mass. In addition, we derive a unified pointwise decay result valid for all $m \in [0, 1]$. Last but not least, we prove the cubic Dirac equations scatter linearly with an explicit scattering speed. When the mass $m=0$, we can show an improved pointwise decay result.