Asymptotic stability for the Dirac--Klein-Gordon system in two space dimensions

TL;DR

This paper proves global existence, sharp decay, and linear scattering for the Dirac--Klein-Gordon system in 2D, establishing the first asymptotic stability result in this setting with novel methods.

Contribution

It provides the first asymptotic stability result for the 2D Dirac--Klein-Gordon system with a massive Klein-Gordon and massless Dirac field, using new weighted energy estimates.

Findings

Global solutions exist for the system in 2D

Solutions exhibit sharp time decay and linear scattering

The method relaxes previous assumptions on nonlinear structures

Abstract

We study the Dirac--Klein-Gordon system in $1+2$ spacetime dimensions. We show global existence of the solutions, as well as sharp time decay and linear scattering. One key advance is that we provide the first asymptotic stability result for the Dirac--Klein-Gordon system in $1+2$ spacetime dimensions in the case of a massive Klein-Gordon field and a massless Dirac field. The nonlinearities are below-critical in two spatial dimensions, and so our method requires the identification of special structures within the system and novel weighted energy estimates. Another key advance, is that our proof allows us to weaken certain conditions on the nonlinear structures that have been assumed in the literature.