Conditional large-data global well-posedness of Dirac equation with Hartree-type nonlinearity

TL;DR

This paper proves global well-posedness and scattering for large-data solutions of the nonlinear Dirac equation with Hartree-type nonlinearity in two and three dimensions, under certain conditions, advancing understanding of its long-term behavior.

Contribution

It extends previous results by establishing global solutions and scattering for large data using modified multilinear estimates and the Majorana condition.

Findings

Proved global well-posedness for large data

Established scattering results under specific conditions

Improved multilinear estimates for Dirac equations

Abstract

We study the Cauchy problems for the Hartree-type nonlinear Dirac equations with Yukawa-type potential in two and three spatial dimensions. This paper improves our previous results \cite{chohlee,cholee}; we establish global well-posedness and scattering for large data with a certain condition. Firstly we investigate the long-time behavior of solutions to the Dirac equation satisfies good control provided that a particular dispersive norm of solutions is bounded. The key of our proof relies on modifying multilinear estimates obtained in our previous papers. Secondly, we obtain large data scattering by exploiting the Majorana condition.