Global Well-posedness of the 1D Dirac-Klein-Gordon system in Sobolev spaces of negative index

TL;DR

This paper establishes the global well-posedness of the 1D Dirac-Klein-Gordon system in Sobolev spaces with negative and positive indices, using advanced harmonic analysis techniques.

Contribution

It extends well-posedness results to negative Sobolev spaces for the Dirac component, employing the I-method and null structure analysis.

Findings

Proves global well-posedness in negative Sobolev spaces for the Dirac-Klein-Gordon system.

Utilizes the I-method and null structure to handle low regularity.

Establishes bilinear spacetime estimates for the system.

Abstract

We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of almost conservation law and I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. Our proof also relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type.