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 0x<1 with=" general=" stochastic=" boundary=" conditions=" at=" x="0" and=" x="1." assuming=" that=" there=" is=" a=" constant=">wall Maxwellian M=(M _i) 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 1(t) and 2(t) such that in a measure-theoretic sense f i(0, t) 1 (t)M i , f i(1, t) 2 (t)M i as t. 1 and 2 are closely related and satisfy functional equations which suggest that 1(t)1 and 2(t)1 as t.
II.	Under the additional assumption that there is at least one non-trivial collision term containing a product f k f l with k = l , where k denotes the x-component of the velocity associated with f k , we show that in a measure-theoretic sense 1(t) and 2(t) converge to 1 as t. This entails L 1-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 , k <0), renewal=" theory=" entails=" that=" the=" solution=" converges=" to=" the=" unique=" wall=" maxwellian=" in=">L .

Communicated by L. Arkeryd