Small data scattering of Dirac equations with Yukawa type potentials in $L_x^2(\mathbb R^2)$

TL;DR

This paper proves small data scattering for the nonlinear massive Dirac equation with Yukawa potentials in two dimensions within the $L_x^2$ space, extending previous results from Sobolev spaces.

Contribution

It demonstrates small data scattering in $L_x^2$ for the Dirac equation with Yukawa potentials, using bilinear and modulation estimates, which was not previously established.

Findings

Small data scattering in $L_x^2( ^2)$ space.

Extension of scattering results from $H^s$ to $L_x^2$.

Utilization of bilinear and modulation estimates.

Abstract

We revisit the Cauchy problem of nonlinear massive Dirac equation with Yukawa type potentials $\mathcal F^{-1}\left[(b^2 + |\xi|^2)^{-1}\right]$ in 2 dimensions. The authors of \cite{tes2d, geosha} obtained small data scattering and large data global well-posedness in $H^s$ for $s > 0$, respectively. In this paper we show that the small data scattering occurs in $L_x^2(\mathbb R^2)$. This can be done by combining bilinear estimates and modulation estimates of \cite{yang, tes2d}.