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 ∞.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Avoid common mistakes on your manuscript.
References
L. Arkeryd & R. Illner, The Broadwell Model in a Box: Strong Convergence to Equilibrium. SIAM J. Appl. Math., to appear.
C. Cercignani, Theory and Application of the Boltzmann Equation. Springer-Verlag (1988).
Çinlar, Introduction to Stochastic Processes. Prentice-Hall (1975).
W. Feller, An Introduction to Probability Theory and its Applications, II. Wiley (1966).
F. R. Gantmacher, Application of the Theory of Matrices. Interscience (1959).
R. Gatignol, Kinetic Theory Boundary Conditions for Discrete Velocity Gas. Phys. Fluids 20, 2022–2030 (1977).
R. Illner & T. Platkowski, Discrete Velocity Models of the Boltzmann Equation: A Survey on the Mathematical Aspects of the Theory. SIAM Review 30, 213–255 (1988).
S. Kawashima, Global Solutions to the Initial Boundary Value Problems for the Discrete Boltzmann Equation. Nonlin. Anal. Th. Meths. Appls. 17, 577–597 (1991).
Author information
Authors and Affiliations
Additional information
Communicated by L. Arkeryd
Rights and permissions
About this article
Cite this article
Bose, C., Grzegorczyk, P. & Illner, R. Asymptotic behavior of one-dimensional discrete-velocity models in a slab. Arch. Rational. Mech. Anal. 127, 337–360 (1994). https://doi.org/10.1007/BF00375020
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00375020