Local well-posedness for the nonlinear Dirac equation in two space dimensions

TL;DR

This paper proves local well-posedness for the cubic nonlinear Dirac equation in two dimensions for initial data with regularity above a critical threshold, utilizing null structures and bilinear estimates.

Contribution

It establishes local well-posedness in H^s for s > 1/2 using Bourgain-Klainerman-Machedon type spaces, extending previous techniques to the Dirac equation.

Findings

Well-posedness for s > 1/2 in two dimensions

Utilization of null structure in the nonlinearity

Application of bilinear Strichartz estimates

Abstract

The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in H^s for s > 1/2. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon system before and bilinear Strichartz type estimates for the wave equation by Selberg and Foschi-Klainerman.