Global existence for an L^2 critical Nonlinear Dirac equation in one dimension

TL;DR

This paper establishes global existence for the L^2 critical nonlinear Dirac equation in one dimension by decomposing solutions and ruling out charge concentration, extending previous local results to a global scale.

Contribution

It introduces a novel approach using null coordinates and solution decomposition to prove global existence for all s>0 in the L^2 critical case.

Findings

Proves global existence for the nonlinear Dirac equation in 1D from L^2 data.

Develops a method to prevent charge concentration at points.

Extends local existence results to global for all s>0.

Abstract

We prove global existence from $L^2$ initial data for a nonlinear Dirac equation known as the Thirring model. Local existence in $H^s$ for $s>0$, and global existence for $s>1/2$, has recently been proven by Selberg and Tesfahun by using $X^{s, b}$ spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara, Nakanishi, and Tsugawa, we first prove local existence in $L^2$ by using null coordinates, where the time of existence depends on the profile of the initial data. To extend this to a global existence result we need to rule out concentration of $L^2$ norm, or charge, at a point. This is done by decomposing the solution into an approximately linear component and a component with improved integrability. We then prove global existence for all $s>0$.