Global Behavior of Small Data Solutions for The 2D Dirac-Klein-Gordon Equations

TL;DR

This paper analyzes the global behavior of small data solutions for the 2D Dirac-Klein-Gordon system, establishing sharp decay estimates and linear scattering results, thereby demonstrating asymptotic stability for general initial data.

Contribution

It provides the first sharp decay and scattering results for the 2D Dirac-Klein-Gordon system with general small initial data, without support restrictions.

Findings

Proved sharp time decay estimates for solutions.

Established linear scattering for the system.

Demonstrated asymptotic stability of solutions.

Abstract

In this paper, we are interested in the two-dimensional Dirac-Klein-Gordon system, which is a basic model in particle physics. We investigate the global behaviors of small data solutions to this system in the case of a massive scalar field and a massless Dirac field. More precisely, our main result is twofold: 1) we show sharp time decay for the pointwise estimates of the solutions which imply the asymptotic stability of this system; 2) we show the linear scattering result of this system which is a fundamental problem when it is viewed as dispersive equations. Our result is valid for general small, high-regular initial data, in particular, there is no restriction on the support of the initial data.