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Global existence to the BGK model of Boltzmann equation. (English) Zbl 0694.35134
The author has proved results on existence and stability for solutions to the BGK model of Boltzmann equation (1) \[ \partial_ tf+v\cdot \nabla_ xf+f=M[f],\quad t\geq 0,\quad x\in {\mathbb{R}}^ N,\quad v\in {\mathbb{R}}^ N, \] \[ M[f]=(\rho /(2\pi T)^{N/2})\exp (-| v-u|^ 2/(2T)), \] \[ (\rho,\rho u,\rho | u|^ 2+\rho T)(t,x)=\int_{{\mathbb{R}}^ N}(1,v,| v|^ 2)f(t,x,v)dv. \] The proof mainly relies on the strong compactness of \(\rho\), u, T and on a new estimate on the third moment of f: \(\int | v|^ 3f dv\). The entropy relation for (1) is also proved.
Reviewer: B.G.Pachpatte

35Q99 Partial differential equations of mathematical physics and other areas of application
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35B35 Stability in context of PDEs
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[1] Bhatnagar, P.L.; Gross, E.P.; Krook, M., A model for collision processes in gases, Phys. rev., 94, 511, (1954) · Zbl 0055.23609
[2] Bardos, C.; Degond, P., Global existence for the Vlasov-Poisson equation in espace variables with small initial data, Ann. inst. H. Poincaré anal. non linéaire, 2, No. 2, 101-118, (1985) · Zbl 0593.35076
[3] Bardos, C.; Golse, F.; Perthame, B.; Sentis, R., The nonaccretive radiative transfer equations. existence of solutions and rosseland approximation, J. func. anal., 77, 434, (1988) · Zbl 0655.35075
[4] Cercignani, C., Theory and applications of the Boltzmann equation, (1975), Scottish Academic Press San Diego
[5] \scR. Di Perna and P. L. Lions, On the Cauchy problem for Boltzmann equations, Ann. of Math., in press · Zbl 0662.35016
[6] \scR. Di Perna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, to Comm. Pure Appl. Math., in press.
[7] Dunford, N.; Schwartz, G.T., Linear operations, (1958), Interscience New York
[8] Golse, F.; Perthame, B.; Sentis, R., Un résultat de compacité pour LES équations du transport et application au calcul de la valeur propre principale d’un opérateur de transport, C.R. acad. sc. Paris, 301, I, 341-344, (1985) · Zbl 0591.45007
[9] Golse, F.; Lions, P.L.; Perthame, B.; Sentis, R., Regularity of the moments of the solution of a transport equation, J. funct. anal., 74, No. 1, 110-125, (1988) · Zbl 0652.47031
[10] \scB. Perthame, Relations between Boltzmann Equations and Boltzmann Schemes for Gas Dynamics, in preparation. · Zbl 0714.76078
[11] \scB. Perthame, Entropies and Entropy Boltzmann Schemes for Compressible Euler Equations, in preparation.
[12] Philippi, P.C.; Brun, R., Kinetic modeling of polyatomic gas mixture, Phys. A, 105, 147, (1981)
[13] Truesdell; Muncaster, Fundamentals of Maxwell’s kinetic of a simple monotonic gas, (1980), Academic Press New York
[14] Welander, P., Ark. phys., 7, 507, (1954)
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