The massless and the non-relativistic limit for the cubic Dirac equation

TL;DR

This paper proves global well-posedness and scattering for cubic Dirac equations in 2D and 3D, and establishes optimal convergence results in massless and non-relativistic limits using novel bilinear estimates.

Contribution

It introduces a new approach employing bilinear Fourier restriction estimates and atomic spaces to analyze the Dirac equation's limits, providing first results on scattering state convergence.

Findings

Global well-posedness for small data in Sobolev spaces

Scattering of solutions in the considered regimes

Convergence of scattering states and wave operators in limits

Abstract

Massive and massless Dirac equations with Lorentz-covariant cubic nonlinearities are considered in spatial dimension $d=2,3$. Global well-posedness of the Cauchy problem for small initial data in scale-invariant Sobolev spaces and scattering of solutions is proved by a new approach which uses bilinear Fourier restriction estimates and atomic function spaces. Furthermore, global uniform convergence results, both in the massless and in the non-relativistic limit, are proved at optimal regularity. In both regimes, these are the first results which imply convergence of scattering states and wave operators.