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

Abstract

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 ${\mathcal{L}^{2}}$ integrable functions α on the one particle phase space ${\mathbb{R}^{2d}_{{\bf Z}}}$; s.t. ${f={\left|{\alpha}\right|}^2}$ is a solution of a collisionless Boltzmann equation. The only requirement is a sufficiently regular energy functional on a subspace of distribution functions ${f \in \mathcal{L}^{1}}$. Secondly, we give a full well-posedness theory for the obtained system corresponding to Vlasov–Poisson in ${d \geqq 3}$ 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.