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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
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[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)
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