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

Abstract

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