ISSN:
1432-0673
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract We prove results on the asymptotic behavior of solutions to discrete-velocity models of the Boltzmann equation in the one-dimensional slab 0〈x〈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 β1(t) and β2(t) such that in a measure-theoretic sense f i(0, t)→β 1 (t)M i , f i(1, t)→β 2 (t)M i as t→∞. β 1 and β 2 are closely related and satisfy functional equations which suggest that β 1(t)→1 and β 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 ν k =ν l , where ν k denotes the x-component of the velocity associated with f k , we show that in a measure-theoretic sense β 1(t) and β 2(t) converge to 1 as t→∞. This entails L 1-convergence of the solution to the unique wall Maxwellian. For this result, ν k =ν l =0 is admissible. III. In the absence of any collision terms, but under the assumption that there is an irrational quotient (ν i +¦ν j ¦)/(ν l +¦ν k ¦) (here ν i , ν l 〉0 and ν j ,ν k 〈0), renewal theory entails that the solution converges to the unique wall Maxwellian in L ∞.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00375020
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