# Uniqueness and stability for the Vlasov-Poisson system with spatial density in Orlicz spaces

## TL;DR

This paper proves the uniqueness and stability of solutions to the Vlasov-Poisson system when the spatial density is in certain Orlicz spaces, extending previous results and providing quantitative estimates.

## Contribution

It extends the uniqueness results for the Vlasov-Poisson system to densities in Orlicz spaces and offers a stability estimate based on Wasserstein distance.

## Key findings

- Uniqueness of solutions in Orlicz spaces established
- Quantitative stability estimate derived
- Extension of Loeper's and previous results

## Abstract

In this paper, we establish uniqueness of the solution of the Vlasov-Poisson system with spatial density belonging to a certain class of Orlicz spaces. This extends the uniqueness result of Loeper (which holds for uniformly bounded density) and the uniqueness result of the second author. Uniqueness is a direct consequence of our main result, which provides a quantitative stability estimate for the Wasserstein distance between two weak solutions with spatial density in such Orlicz spaces, in the spirit of Dobrushin's proof of stability for mean-field PDEs. Our proofs are built on the second-order structure of the underlying characteristic system associated to the equation.