Local well-posedness of Dirac equations with nonlinearity derived from honeycomb structure in 2 dimensions

TL;DR

This paper establishes local well-posedness results for 2D Dirac equations with specific nonlinearities related to honeycomb structures, identifying regularity thresholds and demonstrating flow smoothness failure.

Contribution

It proves local well-posedness for Dirac equations with power and Hartree nonlinearities in 2D honeycomb structures, and analyzes flow smoothness limitations.

Findings

Well-posedness in $H^s$ for $s>rac78$ (power nonlinearity)

Well-posedness in $H^s$ for $s>rac38$ (Hartree nonlinearity)

Flow smoothness failure demonstrated

Abstract

The aim of this paper is to show the local well-posedness of 2 dimensional Dirac equations with power type and Hartree type nonlinearity derived from honeycomb structure in $H^s$ for $s>\frac78$ and $s>\frac38$, respectively. We also provide the smoothness failure of flows of Dirac equations.