Communications in Mathematical Sciences

Volume 19 (2021)

Number 3

An improved small data theorem for the Vlasov–Poisson system

Pages: 721 – 736



Jack Schaeffer (Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)


A collisionless plasma is modeled by the Vlasov–Poisson system. Smooth solutions are considered in three spatial dimensions with compactly supported initial data. The main theorem of this work is a small data result that improves an earlier theorem of Bardos and Degond in that it does not require the derivatives of the initial data to be small. Another theorem is presented here that gives a sufficient condition that ensures that the charge density decays as $t^{-3}$, which is the rate which occurs when asymptotically all particles disperse freely.


partial differential equations, kinetic theory, Vlasov equation

2010 Mathematics Subject Classification

35L60, 35Q83, 82C22, 82D10

In memory of Robert Glassey.

Received 17 June 2020

Accepted 27 October 2020

Published 5 May 2021