The Vlasov-Poisson-Boltzmann System without Angular Cutoff

Abstract

This paper is concerned with the Vlasov-Poisson-Boltzmann system for plasma particles of two species in three space dimensions. The Boltzmann collision kernel is assumed to be angular non-cutoff with 3 < γ < −2s and 1/2 ≤ s < 1, where γ , s are two parameters describing the kinetic and angular singularities, respectively. We establish the global existence and convergence rates of classical solutions to the Cauchy problem when initial data is near Maxwellians. This extends the results in Duan et al. (J Diff Eqs 252(12):6356–6386, 2012, Math Models Methods Appl Sci 23(6):927, 2013) for the cutoff kernel with −2 ≤ γ ≤ 1 to the case −3 < γ < −2 as long as the angular singularity exists instead and is strong enough, i.e., s is close to 1. The proof is based on the time-weighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in Gressman and Strain (J Amer Math Soc 24(3):771–847, 2011) and the Vlasov-Poisson-Landau system in Guo (J Amer Math Soc 25:759–812, 2012).