Bilinear Estimates and Applications to Global Well-posedness for the Dirac-Klein-Gordon Equation on R^{1+1}

TL;DR

This paper establishes optimal bilinear estimates in specific function spaces and applies them using the I-method to prove global well-posedness for the Dirac-Klein-Gordon equation on one-dimensional space-time.

Contribution

It introduces new bilinear estimates for X^{s, b}_\pm spaces and applies the I-method to extend global existence results for the Dirac-Klein-Gordon system.

Findings

Proved bilinear estimates optimal up to endpoints.

Extended global well-posedness below the charge class.

Applied dyadic decomposition techniques in the proof.

Abstract

We prove new bilinear estimates for the X^{s, b}_\pm(R^2) spaces which are optimal up to endpoints. These estimates are often used in the theory of nonlinear Dirac equations on R^{1+1}. The proof of the bilinear estimates follows from a dyadic decomposition in the spirit of Tao [21] and D'Ancona, Foschi, and Selberg [11]. As an application, by using the I-method of Colliander, Keel, Staffilani, Takaoka, and Tao, we extend the work of Tesfahun [23] on global existence below the charge class for the Dirac-Klein- Gordon equation on R^{1+1}.