Local well-posedness of the two-dimensional Dirac-Klein-Gordon equations in Fourier-Lebesgue spaces

TL;DR

This paper establishes local well-posedness for the 2D Dirac-Klein-Gordon system in Fourier-Lebesgue spaces, lowering regularity requirements and leveraging null conditions to improve previous results.

Contribution

It extends well-posedness results to Fourier-Lebesgue spaces with lower regularity, utilizing null conditions to enhance prior classical case findings.

Findings

Lowered regularity assumptions for well-posedness

Utilized null conditions in nonlinearities

Improved upon previous classical results

Abstract

The local well-posedness problem is considered for the Dirac-Klein-Gordon system in two space dimensions for data in Fourier-Lebesgue spaces $\hat{H}^{s,r}$ , where $\|f\|_{\hat{H}^{s,r}} = \| \langle \xi \rangle^s \hat{f}\|_{L^{r'}}$ and $r$ and $r'$ denote dual exponents. We lower the regularity assumptions on the data with respect to scaling improving the results of d'Ancona, Foschi and Selberg in the classical case $r=2$ . Crucial is the fact that the nonlinearities fulfill a null condition as detected by these authors.