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 integrable functions α on the one particle phase space RZ2d; s.t. f=α2 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. Secondly, we give a full well-posedness theory for the obtained system corresponding to Vlasov-Poisson in d≧3 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. [ABSTRACT FROM AUTHOR]