Smoothing Estimates of the Vlasov-Poisson-Landau System

TL;DR

This paper proves that solutions to the Vlasov-Poisson-Landau system become instantly smooth and establishes global existence with optimal decay rates, enhancing understanding of the system's regularity and long-term behavior.

Contribution

It provides the first proof of immediate smoothing effects and global existence with decay for the Vlasov-Poisson-Landau system under both hard and soft potentials.

Findings

Solutions become instantly smooth in all variables.

Global existence with optimal large-time decay is established.

The results apply to both hard and soft potentials.

Abstract

In this work, we consider the smoothing effect of Vlasov-Poisson-Landau system for both hard and soft potential. In particular, we prove that any classical solutions becomes immediately smooth with respect to all variables. We also give a proof on the global existence to Vlasov-Poisson-Landau system with optimal large time decay. These results give the regularity to Vlasov-Poisson-Landau system. The proof is based on the time-weighted energy method building upon the pseudo-differential calculus.