The Linear Boltzmann Equation with Space Periodic Electric Field

We investigate the well posedness of the stationary linear Boltzmann equation with space periodic electric field. We discuss the different behaviors that occur depending if the average electric field vanishes or not. The existence follows by perturbation techniques and stability arguments under uniform a priori estimates. The uniqueness of the weak solution holds for space periodic electric fields with non vanishing average, under the constraint of given current. The main ingredients of the proof rely on the dissipation properties of the linear collision operator and the derivation of refined estimates. Introduction This paper is concerned with the free space linear Boltzmann equation (0.1) v(p)∂xf + F (x)∂pf = Q(f), (x, p) ∈ R. The unknown f = f(x, p) represents the number density of a population of charged particles, with x ∈ R the space variable and p ∈ R the momentum variable. The velocity p 7→ v(p) is defined by (0.2) v(p) = p m ( 1 + p mc0 )−1/2 where m is the mass of the particles and c0 is the light speed in vacuum. The kinetic energy associated to v(p) is then given by E(p) = mc0 (( 1 + p mc0 )1/2 − 1 ) so that E ′(p) = v(p), p ∈ R. The collision operator Q, which is an integral operator with respect to the variable p, is defined as follows: Q(g)(p) =Mθ(p)〈g〉s(p)− σ(p)g(p), p ∈ R with (0.3) 〈g〉s(p) = ∫ R s(p, p′)g(p′) dp′ and σ(p) = ∫ R s(p, p)Mθ(p) dp′ = 〈Mθ〉s(p) ; Date: January 1, 2001 and, in revised form, June 22, 2001. 2000 Mathematics Subject Classification. 82D10, 78A35, 35Q99.