The Vlasov-Poisson equation in $\mathbb{R}^3$ with infinite charge and velocities

TL;DR

This paper establishes existence and uniqueness results for the Vlasov-Poisson equation in three-dimensional space with initial data that are not integrable in space and have unbounded velocities, under specific decay conditions.

Contribution

It generalizes previous results by proving well-posedness for initial data with infinite charge and velocities, extending the class of admissible initial conditions.

Findings

Existence and uniqueness of solutions under new decay assumptions.

Extension of previous compact support results to unbounded velocities.

Broader applicability to physical models with infinite charge and velocities.

Abstract

We consider the Vlasov-Poisson equation in $\mathbb{R}^3$ with initial data which are not $L^1$ in space and have unbounded support in the velocities. Assuming for the density a slight decay in space and a strong decay in velocities, we prove existence and uniqueness of the solution, thus generalizing the analogous result given in [5] for data compactly supported in the velocities.