Exterior Problem of the Linear Vlasov–Poisson–Boltzmann System

We will present in this paper a linear theory on a flow under the effect of a self-induced electric field past an obstacle governed by the Vlasov-Poisson-Boltzmann equation. The bulk velocity at infinity acts like a driving force on the flow so that some nontrivial stationary solution profile exists. Similar to Ukai and Asano's work on the exterior problem for the Boltzmann equation, the spectrum of the linear operator is used through an approach in the spirit of the limiting absorption in scattering theory. However, to take care of the electric potential that satisfies the coupled Poisson equation, a new decomposition of the solution operator is needed to show the compactness.