Optimal Time Decay of the Vlasov-Poisson-Boltzmann System in ${\mathbb{R}}^3$

TL;DR

This paper investigates the decay rates of solutions to the Vlasov-Poisson-Boltzmann system in three-dimensional space, revealing that electric fields slow convergence to equilibrium compared to the Boltzmann equation without forces.

Contribution

It establishes the optimal algebraic decay rates for solutions, quantifying the electric field's impact on convergence speed in the Vlasov-Poisson-Boltzmann system.

Findings

Electric field reduces convergence speed by 1/4 in decay rate exponent.

Fourier analysis used to derive decay properties in linearized case.

Nonlinear energy estimates confirm optimal decay rates under initial data conditions.

Abstract

The Vlasov-Poisson-Boltzmann System governs the time evolution of the distribution function for the dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over ${\mathbb{R}}^3$. It is shown that the electric field which is indeed responsible for the lowest-order part in the energy space reduces the speed of convergence and hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces, where the exact difference between both power indices in the algebraic rates of convergence is 1/4. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.