Cercignani, Carlo A remarkable estimate for the solutions of the Boltzmann equation. (English) Zbl 0762.35090 Appl. Math. Lett. 5, No. 5, 59-62 (1992). The author proves a global (in time) a priori bound for a certain moment of classical solutions of the Boltzmann equation. This moment contains the collision kernel. The assumptions are that the solution is one- dimensional in space, and that the collision kernel satisfies certain cut off properties. The proof uses the collision invariants and the \(H\)- theorem. Reviewer: H.Lange (Köln) Cited in 1 ReviewCited in 13 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 45K05 Integro-partial differential equations 82C70 Transport processes in time-dependent statistical mechanics Keywords:moment of classical solutions; collision kernel PDFBibTeX XMLCite \textit{C. Cercignani}, Appl. Math. Lett. 5, No. 5, 59--62 (1992; Zbl 0762.35090) Full Text: DOI References: [1] Cercignani, C., The Boltzmann Equation and its Applications (1988), Springer: Springer New York · Zbl 0646.76001 [2] Cercignani, C., Mathematical Methods in Kinetic Theory (1990), Plenum Press: Plenum Press New York · Zbl 0726.76083 [3] Di Perna, R.; Lions, P. L., On the Cauchy problem for the Boltzmann equation. Global existence and stability, Ann. of Math., 130, 321-366 (1989) · Zbl 0698.45010 [4] Golse, F.; Lions, P. L.; Perthame, B.; Sentis, R., Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76, 110-125 (1988) · Zbl 0652.47031 [5] Nishida, T.; Mimura, M., On the Broadwell model for a simple discrete velocity gas, Proc. Japan. Acad., 50, 812-817 (1974) · Zbl 0326.35051 [6] Tartar, L., Existence globale pour un système hyperbolique semi-linéaire de la théorie cinétique des gaz, Séminaire Goulaouic-Schwartz, Ecole Polytechnique, No. 1 (1975-1976) · Zbl 0336.35069 [7] Cabannes, H., Solution globale du problème de Cauchy en théorie cinétique discrète, J. de Mécan., 17, 1-22 (1978) · Zbl 0439.76064 [8] Cabannes, H., Solution globale d’un problème de Cauchy en théorie cinétique discrète. Modèle plan, C.R. Acad. Sci. Paris, 284, 269-272 (1977), (sèrie A) · Zbl 0343.35073 [9] Cabannes, H., Solution globale de l’équation de Boltzmann discrète pour les modèles spatiaux reguliers à 12 ou 20 vitesses, Mech. Res. Comm., 10, 317-322 (1983) · Zbl 0578.76076 [10] Beale, T., Large-time behavior of discrete velocity Boltzmann equations, Comm. Math. Phys., 106, 659-678 (1986) · Zbl 0637.76070 [11] Bony, J. M., Solutions globales bornées pour les modèles discrete de l’équation de Boltzmann, en dimension 1 d’espace, (Journées Equations aux Derivées Partielles, \(n^0\) XVI (1987), Ecole Polytechnique: Ecole Polytechnique Palaiseau, France) [12] Kawashima, S.; Cabannes, H., Initial-value problem in discrete kinetic theory, (Muntz, E. P.; Weaver, D. P.; Campbell, D. H., Rarefied Gas Dynamics: Theoretical and Computational Techniques (1989), AIAA: AIAA Washington, D.C), 148-154 [13] Bony, J. M., Existence globale et diffusion en théorie cinétique discréte, (Gatignol, R.; Soubbaramayer, Advances in Kinetic Theory and Continuum Mechanics (1991), Springer: Springer Berlin), 81-90 [14] Cercignani, C., Global existence for a model Boltzmann equation, J. Stat. Phys., 49, 1083 (1987) · Zbl 0960.82513 [15] Arkeryd, L., Existence theorems for certain kinetic equations and large data, Arch. Rat. Mech. Anal., 103, 139-149 (1988) · Zbl 0654.76073 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.