Spectrum analysis and optimal decay rates of the bipolar Vlasov-Poisson-Boltzmann equations

TL;DR

This paper analyzes the spectral properties and decay rates of solutions to the bipolar Vlasov-Poisson-Boltzmann equations, establishing optimal convergence rates for the electric field and distribution function towards equilibrium.

Contribution

It provides the first spectral analysis and establishes the optimal decay rates for solutions of the bipolar Vlasov-Poisson-Boltzmann system and its modified version.

Findings

Electric field decays exponentially in the bVPB system.

Distribution function converges at rate (1+t)^{-3/4} to Maxwellian.

Both electric field and distribution function converge at rate (1+t)^{-3/4} in the mVPB system.

Abstract

In the present paper, we consider the initial value problem for the bipolar Vlasov-Poisson-Boltzmann (bVPB) system and its corresponding modified Vlasov-Poisson-Boltzmann (mVPB). We give the spectrum analysis on the linearized bVPB and mVPB systems around their equilibrium state and show the optimal convergence rate of global solutions. It was showed that the electric field decays exponentially and the distribution function tends to the absolute Maxwellian at the optimal convergence rate $(1+t)^{-3/4}$ for the bVPB system, yet both the electric field and the distribution function converge to equilibrium state at the optimal rate $(1+t)^{-3/4}$ for the mVPB system.