Generalized symplectization of Vlasov dynamics and application to the Vlasov-Poisson system

TL;DR

This paper develops a Hamiltonian framework for the Vlasov-Poisson system, providing a formal derivation, establishing well-posedness in higher dimensions, and extending global existence results.

Contribution

It introduces a generalized symplectization approach for Vlasov dynamics, including a rigorous well-posedness theory and global existence results for the Vlasov-Poisson system.

Findings

Hamiltonian structure derived for Vlasov-Poisson

Well-posedness established in dimensions d≥3

Global existence results extended to the generalized system

Abstract

In this paper, we study a Hamiltonian structure of the Vlasov-Poisson system, first mentioned by Fr\"ohlich, Knowles, and Schwarz. To begin with, we give a formal guideline to derive a Hamiltonian on a subspace of complex-valued $L^2$ integrable functions $\alpha$ on the one particle phase space $\mathbb{R}^{2d}$, 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 $f\in L^1$. Secondly, we give a full well-posedness theory for the obtained system corresponding to Vlasov-Poisson in $d\geq3$ dimensions. Finally, we adapt the classical globality results for $d=3$ to the generalized system.