Time Evolution of an Infinitely Extended Vlasov System with Singular Mutual Interaction

We study the time evolution of an infinitely extended system in the mean field approximation, governed by the Vlasov equation. This system is confined in an unbounded cylinder by an external force singular on the border. The mutual interaction is assumed singular at short distance as 1/rα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/r^\alpha $$\end{document} with α<2/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha < 2/3$$\end{document} (or α<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <1$$\end{document} in case of an external Lorentz force) and with a short range. The initial density is assumed bounded. Differently from studies which assume initial data compact in space and/or in velocities, here we consider a system having infinite mass and an exponential bound on the velocities, according to the Maxwell–Boltzmann law.