On the trend to equilibrium for the Vlasov–Poisson–Boltzmann equation

The dynamics of dilute electrons and plasma can be modeled by Vlasov–Poisson–Boltzmann equation, for which the equilibrium state can be a global Maxwellian. In this paper, we show that the rate of convergence to equilibrium is O(t−∞), by using a method developed for the Boltzmann equation without external force in [L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math. 159 (2005) 245–316]. In particular, the idea of this method is to show that the solution f cannot stay near any local Maxwellians for long. The improvement in this paper is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro–micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components.