Decay estimates for the $ 3D $ relativistic and non-relativistic Vlasov-Poisson systems

<p style='text-indent:20px;'>We study the small data global regularity problem of <inline-formula><tex-math id="M2">\begin{document}$ 3D $\end{document}</tex-math></inline-formula> Vlasov-Poisson system for both relativistic case and non-relativistic case. The main goal this paper is twofold. (i) Based on a Fourier method, which works systematically case, we give short proof sharp decay estimate id="M3">\begin{document}$ system. Moreover, show that nonlinear solution scatters to linear in cases. result estimates not new, see Hwang-Rendall-Velázquez [<xref ref-type="bibr" rid="b9">9</xref>] Smulevici rid="b23">23</xref>]. (ii) method presented serves as good comparison more complicated id="M4">\begin{document}$ Vlasov-Nordström rid="b24">24</xref>] id="M5">\begin{document}$ Vlasov-Maxwell rid="b25">25</xref>].</p>