zbMATH — the first resource for mathematics

Large-time behavior of discrete velocity Boltzmann equations. (English) Zbl 0637.76070
The asymptotic behaviour of equations representing one-dimensional motions in a gas with a discrete set of velocities is studied. This fictitious gas is described by the following system of equations (discrete velocity Boltzmann equations): $u_{i,t}+c_ iu_{i,x}=\sum^{n}_{j,k=1}a_ i^{jk}u_ ju_ k\equiv F_ i(u_ 1,..,u_ n),\quad i=1,...,n,$ where $$u_ i(x,t)$$ is the density of particles with speeds $$c_ i$$, $$a_ i^{jk}$$ are the interaction coefficients (they are constant satisfying conditions for mass and momentum conservation and an entropy condition; $$a_ i^{jk}=a_ i^{kj})$$. The initial condition $$u_ i(x,0)=u_{0i}(x)$$, $$i=1,..,n$$, with $$u_{0i}\geq 0$$ in $$L^ 1\cap L^{\infty}$$ is prescribed. The solution $$u_ i(x,t)$$ is in class $$C(0,T;L^ p({\mathbb{R}}))$$ for $$1\leq p\leq \infty$$, with $$T>0$$ arbitrary. $$u_ i(x,t)$$ as $$t\to \infty$$ have the form $$u_ i^{\infty}(x-c_ it)$$ it corresponds to a state in which the interaction is negligible. 3 theorems are proved $$(L^ 1$$-asymptotics; boundedness and $$L^ p$$- asymptotics; uniform asymptotics).
Reviewer: I.Grosu

MSC:
 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82B40 Kinetic theory of gases in equilibrium statistical mechanics
Full Text:
References:
 [1] Beale, J.T.: Large-time behavior of the Broadwell model of a discrete velocity gas. Commun. Math. Phys.102, 217-235 (1985) · Zbl 0626.76080 [2] Cabannes, H.: Solution globale du probleme de Cauchy en th?orie cin?tique discr?te. J. M?c.17, 1-22 (1978) [3] Cabannes, H.: The discrete Boltzmann equation (theory and applications), lecture notes, University of California, Berkeley (1980) · Zbl 0462.76078 [4] Cabannes, H.: Comportement asymptotique des solutions de l’?quation de Boltzmann discr?te. C.R. Acad. Sci. Paris302, S. 1, 249-253 (1986) · Zbl 0585.76112 [5] Gatignol, R.: Th?orie cin?tique des gaz ? r?partition discr?te de vitesses. Lecture Notes in Physics, Vol. 36. Berlin, Heidelberg, New York: Springer 1975 [6] Godunov, S.K., Sultangazin, U.M.: On discrete models of the kinetic Boltzmann equation. Russ. Math. Surveys26, No. 3, 1-56 (1971) · Zbl 0228.35074 [7] Hamdache, K.: Existence globale et comportement asymptotique pour l’?quation de Boltzmann ? r?partition discr?te des vitesses. J. M?c. Th?or. Appl.3, 761-785 (1984) · Zbl 0605.76091 [8] Hamdache, K.: On the discrete velocity models of the Boltzmann equation. Preprint · Zbl 0605.76091 [9] Illner, R.: Global existence results for discrete velocity models of the Boltzmann equation in several dimensions. J. M?c. Th?or. Appl.1, 611-622 (1982) · Zbl 0514.76073 [10] Kaniel, S., Shinbrot, M.: The Boltzmann equation, II: Some discrete velocity models. J. M?c.19, 581-593 (1980) · 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] Nishida, T., Mimura, M.: On the Broadwell’s model for a simple discrete velocity gas. Proc. Jpn. Acad.50, 812-817 (1974) · Zbl 0326.35051 [13] Tartar, L.: Existence globale pom un syst?me hyperbolique semilin?arie de la th?orie cin?tique des gaz. S?minaire Goulaouic-Schwarz, No. 1 (1975/1976) [14] Tartar, L.: Some existence theorems for semilinear hyperbolic systems in one space variable, MRC Technical Summary Report, University of Wisconsin (1980)
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.