Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system

In this paper, we obtain the uniform estimates with respect to the Knudsen number \begin{document}$ \varepsilon $\end{document} for the fluctuations \begin{document}$ g^{\pm}_{\varepsilon} $\end{document} to the two-species Vlasov-Poisson-Boltzmann (in briefly, VPB) system. Then, we prove the existence of the global-in-time classical solutions for two-species VPB with all \begin{document}$ \varepsilon \in (0,1] $\end{document} on the torus under small initial data and rigorously derive the convergence to the two-fluid incompressible Navier-Stokes-Fourier-Poisson (in briefly, NSFP) system as \begin{document}$ \varepsilon $\end{document} go to 0.