Spectral properties of the space nonhomogeneous linearized Boltzmann operator

Abstract

Spectral properties of the Boltzmann operator linearized around a localMaxwellian have been investigated. We show that the spectrum is the same in all spaces Lp, 1 Q p < -, It consists of a half-plane Re h Q -uo and a countably many eigenvalues in a strip -vo < Re X Q 0. We analyse eigenvalues with Re h = 0 and show that when linearization is performed around a space nonhomogeneous Maxwellian all eigenvalues lie in the open strip -ha <Re h < 0. For a space homogeneous Maxwellian the fivefold degenerate eigenvalue h = 0 is the only one with Re h = 0.

Spectral Properties of the Linearized Boltzmann Operator in L p for 1≤p≤∞

“…In the following part of the paper we will need a version of compactness theorem for S(λ)K valid for bounded spatial region with periodic boundary condition [11]. To this end we define linearized Boltzmann operator acting in a bounded spatial domain.…”

“…Properties of the operators A p , A p , S p and S p were investigated in [11]. For the sake of completeness we introduce several lemmas (without proofs) that we will need later:…”

“…The operator is given by expression (2.8) with x ∈ Ω ⊂ R 3 and f a periodic function of x ∈ Ω. We can define an operator S p (λ) and Proof is given in [11].…”

“…The analysis of Fourier transformed operator B k in L 2 ξ was first done in [7] with an excellent extension in [11] for the classical operator and in [5] for the relativistic one. For the reader's convenience we cite the main results of these papers in the beginning of this section.…”

“…His approach required additional smoothing properties of the operator K which one can quite easily prove for classical hard spheres model but which are not always available for general form of interaction especially for relativistic interactions. Palczewski [11,12], using techniques developed in the neutron transport theory, described the spectrum of the linearized Boltzmann operator with periodic boundary conditions. These results were extended to the system with external potential in the work of Tabota and Eshima [13] and in [14] for time dependent forces.…”