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

2021/06/03 by Cho, Yonggeun, Hong, Seokchang, Lee, Kiyeon · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.01633

Abstract

We consider Cauchy problem of the Hartree-type nonlinear Dirac equation with potentials given by V_b(x) = \frac14π\frace^-b|x||x| (b ≥ 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 L²_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.

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