Regularity of the Vlasov–Poisson–Boltzmann System Without Angular Cutoff

In this paper we study the regularity of the non-cutoff Vlasov–Poisson–Boltzmann system for plasma particles of two species in the whole space ℝ³ with hard potential. The existence of global-in-time nearby Maxwellian solutions is known for soft potential from Duan and Liu (Commun Math Phys 324(1):1–45, 2013). However the smoothing effect of these solutions has been a challenging open problem. We establish the global existence and regularizing effect to the Cauchy problem for hard potential with large time decay. Hence, the solutions are smooth with respect to (t, x, v) for any positive time t > 0. This gives regularity to the Vlasov–Poisson–Boltzmann system, which enjoys a similar smoothing effect as the Boltzmann equation. The proof is based on the time-weighted energy method building also upon the pseudo-differential calculus.