Generalized Symplectization of Vlasov Dynamics and Application to the Vlasov–Poisson System

In this paper, we study a Hamiltonian structure of the Vlasov–Poisson system, first mentioned by Fröhlich et al. (Commun Math Phys 288:1023–1058, 2009). To begin with, we give a formal guideline to derive a Hamiltonian on a subspace of complex-valued L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}^{2}}$$\end{document} integrable functions α on the one particle phase space RZ2d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{2d}_{{\bf Z}}}$$\end{document}; s.t. f=α2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f={\left|{\alpha}\right|}^2}$$\end{document} is a solution of a collisionless Boltzmann equation. The only requirement is a sufficiently regular energy functional on a subspace of distribution functions f∈L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f \in \mathcal{L}^{1}}$$\end{document}. Secondly, we give a full well-posedness theory for the obtained system corresponding to Vlasov–Poisson in d≧3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d \geqq 3}$$\end{document} dimensions. Finally, we adapt the classical globality results (Lions and Perthame in Invent Math 105:415–430, 1991; Pfaffelmoser in J Differ Equ 95:281–303, 1992; Schaeffer in Commun Partial Differ Equ 16(8–9):1313–1335, 1991) for d = 3 to the generalized system.