Global solution to the 3D Dirac--Klein-Gordon system with uniform energy bounds

TL;DR

This paper proves the uniform boundedness of the total energy for solutions to the 3D Dirac--Klein-Gordon system with small initial data, using advanced energy estimates and scattering analysis.

Contribution

It introduces a new weighted conformal energy estimate and demonstrates uniform energy bounds for the system, extending previous global existence results.

Findings

Established uniform energy bounds for the system

Developed a new weighted conformal energy estimate

Provided scattering results for the solutions

Abstract

On the (1+3) dimensional Minkowski spacetime, for small, regular initial data, it is well-known that the Dirac-Klein-Gordon system admits a global solution. In the present paper, we aim to establish the uniform boundedness of the total energy of the solution for this system. The proof relies on Klainerman's vector field and Alinhac's ghost weight methods. The main difficulty originates from the slow decay nature of the Dirac and wave components in three space dimensions. To overcome the difficulty, a sharp understanding of the structure for this system, and a new weighted conformal energy estimate are required. In addition, we also provide a few scattering results for the system.