Linear half-space problems in kinetic theory: Abstract formulation and regime transitions

Abstract

In this work, a general formulation, which is based on steady boundary layer problems for the Boltzmann equation, of a half-space problem is considered. The number of conditions on the indata at the interface needed to obtain well-posedness is investigated. The solutions will converge exponentially fast “far away” from the interface. For linearized kinetic half-space problems similar to the one of evaporation and condensation in kinetic theory, slowly varying modes might occur near regime transitions where the number of conditions needed to obtain well-posedness changes (corresponding to transition between evaporation and condensation, or subsonic and supersonic evaporation/condensation), preventing uniform exponential speed of convergence. However, those modes might be eliminated by imposing extra conditions on the indata at the interface. Flow velocities at the far end for which regime transitions occur are presented for Boltzmann equations: for monatomic and polyatomic single species and mixtures; as well as bosons and fermions.

Linearized Boltzmann collision operator: II. Polyatomic molecules modeled by a continuous internal energy variable

“…Remark 4 By Proposition 4 and Corollary 1−3 the linearized operator L fulfills the properties assumed on the linear operators in [3], and hence, the results therein can be applied to hard sphere like models.…”

“…For hard sphere like models the linearized collision operator satisfies all the properties of the general linear operator in the abstract half-space problem considered in [3], and, hence, the general existence results in [3] apply.…”