Perthame, B. Global existence to the BGK model of Boltzmann equation. (English) Zbl 0694.35134 J. Differ. Equations 82, No. 1, 191-205 (1989). 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 Cited in 90 Documents MSC: 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 Keywords:estimate; third moment; entropy relation × Cite Format Result Cite Review PDF Full Text: DOI References: [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: Scottish Academic Press San Diego · Zbl 0403.76065 [5] Di Perna, R.; Lions, P. L., Solutions globales de l’Equation de Boltzmann, C.R. Acad. Sc. Paris, 306, I, 343 (1988), see also · Zbl 0662.35016 [7] Dunford, N.; Schwartz, G. T., Linear Operations (1958), Interscience: Interscience New York · Zbl 0084.10402 [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 [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: Academic Press New York · Zbl 1515.80001 [14] Welander, P., Ark. Phys., 7, 507 (1954) · Zbl 0057.23301 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.