The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian

TL;DR

This paper proves the global existence, uniqueness, and long-term stability of solutions near Maxwellian for the Vlasov-Poisson-Boltzmann and Landau systems with Coulomb interactions, using semigroup and energy methods.

Contribution

It establishes the first global well-posedness results for these systems with polynomial weights near Maxwellian, including Coulomb potential cases.

Findings

Global existence and uniqueness of solutions

Large time decay to equilibrium

Solutions in polynomial-weighted Sobolev spaces

Abstract

In this work, we consider the Vlasov-Poisson-Boltzmann system without angular cutoff and the Vlasov-Poisson-Landau system with Coulomb potential near a global Maxwellian $\mu$. We establish the global existence, uniqueness and large time behavior for solutions in a polynomial-weighted Sobolev space $H^2_{x, v}( \langle v \rangle^k)$ for some constant $k >0$. The proof is based on extra dissipation generated from semigroup method and energy estimates on electrostatic field.