The Vlasov-Poisson-Boltzmann System With Weak Angular Singularity

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 R.-J.and Liu S.-Q.,Comm.Math.Phys.324(2013),no.1,1-45),in which the case of the strong angular singularity 1/2?s<1 is considered,to the case of the weak angular singularity 0<s<1/2.Our analysis is based on the recent studies of the non-cutoff Boltzmann equation in(Gressman P.T.and Strain R.M.,J.Amer.Math.Soc.24(2011),no.3,771-847)and the Vlasov-Poisson-Landau system in(Guo Y.,J.Amer.Math.Soc.25(2012),no.3,759-812),we introduce a time decay factor(1+t)-? and two algebraic weights such that the strategy in(Guo Y.,J.Amer.Math.Soc.25(2012),no.3,759-812)can be applied to the case of the non-cutoff soft Vlasov-Poisson-Boltzmann system with weak singularity.