Scattering and non-scattering of the Hartree-type nonlinear Dirac system at critical regularity

TL;DR

This paper proves global well-posedness and scattering for the Hartree-type nonlinear Dirac equation with exponential decay potentials, requiring minimal angular regularity, and also establishes non-scattering results for Coulomb potential cases.

Contribution

It introduces a new approach using angular regularity to achieve critical regularity results for the nonlinear Dirac system with Hartree potentials.

Findings

Global well-posedness and scattering for $b>0$ with small $L^2$ data and angular regularity.

Minimal angular regularity needed for global existence.

Non-scattering results for Coulomb potential ($b=0$) solutions.

Abstract

We consider Cauchy problem of the Hartree-type nonlinear Dirac equation with potentials given by $V_b(x) = \frac1{4\pi}\frac{e^{-b|x|}}{|x|}\, (b \ge 0)$. In previous works, a standard argument is to utilise null form estimates in order to prove global well-posedness for $H^s$-data, $s>0$. However, the null structure inside the equations is not enough to attain the critical regularity. We impose an extra regularity assumption with respect to the angular variable. Firstly, we prove global well-posedness and scattering of Dirac equations with Hartree-type nonlinearity for $b>0$ for small $L^2_x$-data with additional angular regularity. We also show that only small amount of angular regularity is required to obtain global existence of solutions. Secondly, we obtain non-scattering result for a certain class of solutions with the Coulomb potential $b=0$.