Asymptotic behavior of one-dimensional discrete-velocity models in a slab

We prove results on the asymptotic behavior of solutions to discrete-velocity models of the Boltzmann equation in the one-dimensional slab 0xM=(Mi) compatible with the boundary conditions, and under a technical assumption meaning “strong thermalization” at the boundaries, we prove three types of results:I.If no velocity has x-component 0, there are real-valued functions ß1(t) and ß2(t) such that in a measure-theoretic sense fi(0, t)?ß1(t)Mi, fi(1, t)?ß2(t)Mi as t?8. ß1 and ß2 are closely related and satisfy functional equations which suggest that ß1(t)?1 and ß2(t)?1 as t?8.II.Under the additional assumption that there is at least one non-trivial collision term containing a product fkfl with ?k=?l, where ?k denotes the x-component of the velocity associated with fk, we show that in a measure-theoretic sense ß1(t) and ß2(t) converge to 1 as t?8. This entails L1-convergence of the solution to the unique wall Maxwellian. For this result, ?k=?l=0 is admissible.III.In the absence of any collision terms, but under the assumption that there is an irrational quotient (?i+¦?j¦)/(?l+¦?k¦) (here ?i, ?l>0 and ?j,?kL8.