Small data scattering of 2d Hartree type Dirac equations

• Published in: Journal of mathematical analysis and applications Vol. 506; no. 1; p. 125549
• Main Authors: Cho, Yonggeun, Lee, Kiyeon, Ozawa, Tohru
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-02-2022
• Subjects: Coulomb type potential; Dirac equations; Global well-posedness; Nonexistence of scattering; Small data scattering; Dirac equations; Coulomb type potential; Small data scattering; Nonexistence of scattering; Global well-posedness
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2021.125549

In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity c(|⋅|−γ⁎〈ψ,βψ〉)βψ with c∈R∖{0}, 0<γ<2. Our aim is to show the small data global well-posedness and scattering in Hs for s>γ−1 and 1<γ<2. The difficulty stems from the singularity of the low-frequency part |ξ|−(2−γ)χ{|ξ|≤1} of potential. To overcome it we adapt Up−Vp space argument and bilinear estimates of [27,25] arising from the null structure. We also provide nonexistence result for scattering in the long-range case 0<γ≤1.