Global solutions for the Dirac-Klein-Gordon system in two space dimensions

TL;DR

This paper proves global well-posedness for the Dirac-Klein-Gordon system in two dimensions using advanced function spaces and null structure analysis, extending previous methods to this coupled system.

Contribution

It establishes global solutions for low-regularity data in 2D, applying Bourgain-type spaces and null structure techniques to a coupled Dirac-Klein-Gordon system.

Findings

Global well-posedness for L^2 data in Dirac component.

Existence of global smooth solutions for smooth initial data.

Application of refined bilinear Strichartz estimates and null structure analysis.

Abstract

The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth (classical) solution. The proof uses function spaces of Bourgain type based on Besov spaces - previously applied by Colliander, Kenig and Staffilani for generalized Benjamin-Ono equations and also by Bejenaru, Herr, Holmer and Tataru for the 2D Zakharov system - and the null structure of the system detected by d'Ancona, Foschi and Selberg, and a refined bilinear Strichartz estimate due to Selberg. The global existence proof uses an idea of Colliander, Holmer and Tzirakis for the 1D Zakharov system.