Global solutions to The Vlasov–Poisson–Boltzmann system with weak angular singularity

Abstract

We prove the global existence of smooth solutions near Maxwellians to the Cauchy problem of non-cutoff Vlasov–Poisson–Boltzmann equation for soft potentials, provided that the weak angular singularity assumption holds and the algebraic decay initial perturbation is sufficiently small. This extends the work of Duan and Liu (2013), in which the case of the strong angular singularity 12≤s<1 is considered, to the case of the weak angular singularity 0<s<12.Our analysis is based on the recent studies of the non-cutoff Boltzmann equation in Gressman and Strain (2011) and the Vlasov–Poisson–Landau system in Guo (2012), we introduce a time decay factor (1+t)−ϵ and two algebraic weights such that the strategy in Guo (2012) can be applied to the case of the non-cutoff soft Vlasov–Poisson–Boltzmann system with weak singularity.