×

zbMATH — the first resource for mathematics

Large-time behavior of the Broadwell model of a discrete velocity gas. (English) Zbl 0626.76080
We study the behavior of solutions of the one-dimensional Broadwell model of a discrete velocity gas [J. E. Broadwell, Phys. Fluids 7, 1243- 1247 (1964; Zbl 0123.211)]. The particles have velocity \(\pm 1\) or 0; the total mass is assumed finite. We show that at large time the interaction is negligible and the solution tends to a free state in which all the mass travels outward at speed 1. The limiting behavior is stable with respect to the initial state. No smallness assumptions are made.

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
Citations:
Zbl 0123.211
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Broadwell, J. E.: Shock structure in a simple discrete velocity gas. Phys. Fluids7, 1243-47 (1964) · Zbl 0123.21102
[2] Cabannes, H.: Solution globale du problème de Cauchy en théorie cinétique discrète. J. Mec.17, 1-22 · Zbl 0439.76064
[3] Caflisch, R., Papanicolaou, G.: The fluid-dynamical limit of a nonlinear model Boltzmann equation. Commun. Pure Appl. Math.32, 589-619 (1979) · Zbl 0438.76059
[4] Courant, R., Hilbert, D.: Methods of mathematical physics. Vol. II, New York: Interscience 1962 · Zbl 0099.29504
[5] Godunov, S. K., Sultangazin, U. M.: On discrete models of the kinetic Boltzmann equation. Russ. Math. Surv.26, No. 3, 1-56 (1971) · Zbl 0228.35074
[6] Illner, R., Reed, M. C.: The decay of solutions of the Carleman model. Math. Methods Appl. Sci.3, 121-127 (1981) · Zbl 0563.76073
[7] Illner, R.: Global existence results for discrete velocity models of the Boltzmann equation in several dimensions. (Preprint) · Zbl 0514.76073
[8] Illner, R.: The Broadwell model for initial values inL + 1 (?). Commun. Math. Phys.93, 341-353 (1984) · Zbl 0592.35103
[9] Inoue, K., Nishida, T.: On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas. Appl. Math. Optim.3, 27-49 (1976) · Zbl 0361.76081
[10] Kaniel, S., Shinbrot, M.: The Boltzmann equation, II: Some discrete velocity models. (to appear in J. Mec.) · Zbl 0455.76071
[11] Kawashima, S.: Global solution of the initial value problem for a discrete velocity model. Proc. Jpn. Acad.57, 19-24 (1981) · Zbl 0476.76071
[12] Leguillon: Solution globale d’un problème avec conditions aux limites en theorie cinetique discrète. C.R. Acad. Sci. ParisA-B 285, 1125-28 (1977) · Zbl 0371.76053
[13] Nishida, T., Mimura, M.: On the Broadwell’s model for a simple discrete velocity gas. Proc. Jpn. Acad.50, 812-17 (1974) · Zbl 0326.35051
[14] Tartar, L.: Existence globale pom un system hyperbolique semilinéarie de la théorie cinetique des gaz. Sémin. Goulaouic-Schwarz, No. 1 (1975/76)
[15] Tartar, L.: Some existence theorems for semilinear hyperbolic systems in one space variable, MRC Technical Summary Report, University of Wisconsin (1980)
[16] Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience 1969 · Zbl 0186.40901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.