## Global existence in $$L^ 1$$ for the generalized Enskog equation.(English)Zbl 1083.82531

Summary: Various existence theorems are given for the generalized Enskog equation in $$\mathbb{R}^3$$ and in a bounded spatial domain with periodic boundary conditions. A very general form of the geometric factor $$Y$$ is allowed, including an explicit space, velocity, and time dependence. The method is based on the existence of a Lyapunov functional, an analog of the $$H$$ function in the Boltzmann equation, and utilizes a weak compactness argument in $$L^1$$.

### MSC:

 82C40 Kinetic theory of gases in time-dependent statistical mechanics 35Q99 Partial differential equations of mathematical physics and other areas of application
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### References:

 [1] R. L. DiPerna and P. L. Lions, On the Cauchy problem for the Boltzmann equations: Global existence and weak stability,Ann. Math. 130:321-366 (1989). · Zbl 0698.45010 [2] D. Enskog, Kinetische Theorie,Kgl. Svenska Akad. Handl. 63(4) (1921) [English transl. in S. Brush,Kinetic Theory, Vol. 3 (Pergamon Press, New York, 1972)]. [3] H. Van Beijeren and M. H. Ernst, The modified Enskog equation,Physica 68:437-456 (1973). [4] P. Résibois,H-theorem for the (modified) nonlinear Enskog equation,J. Stat. Phys. 19:593-609 (1978). [5] J. Karkheck and G. Stell, Maximization of entropy, kinetic equations, and irreversible thermodynamics,Phys. Rev. A 25:3302-3327 (1982). [6] M. Mareschal, J. Blawzdziewicz, and J. Piasecki, Local entropy production from the revised Enskog equation: General formulation for inhomogeneous fluids,Phys. Rev. Lett. 52:1169-1172 (1984). [7] N. Bellomo and M. Lachowicz, Kinetic equations for dense gases. A review of physical and mathematical results,Int. J. Mod. Phys. 13:1193-1205 (1987). [8] M. Lachowicz, On the local existence and uniqueness of solution of initial-value problem for the Enskog equation,Bull. Polish Acad. Sci. 31:89-96 (1983). · Zbl 0551.76067 [9] G. Toscani and N. Bellomo, The Enskog-Boltzmann equation in the whole spaceR 3: Some global existence, uniqueness and stability results,Comput. Math. Appl. 13:851-859 (1987). · Zbl 0633.35072 [10] J. Polewczak, Global existence and asymptotic behavior for the nonlinear Enskog equation,SIAM J. Appl. Math. 49:952-959 (1989). · Zbl 0688.35094 [11] C. Cercignani, Existence of global solutions for the space inhomogeneous Enskog equation,Trans. Th. Stat. Phys. 16:213-221 (1987). · Zbl 0629.76083 [12] L. Arkeryd, On the Enskog equation in two space variables,Trans. Th. Stat. Phys. 15:673-691 (1986). · Zbl 0633.76076 [13] L. Arkeryd, On the Enskog equation with large initial data, Preprint, Department of Mathematics, University of Göteborg (1988). · Zbl 0654.76073 [14] J. Polewczak, Global existence inL 1 for the modified nonlinear Enskog equation in ?3,J. Stat. Phys. 56:159-173 (1989). · Zbl 0719.35071 [15] J. Karkheck and G. Stell, Kinetic mean-field theories,J. Chem. Phys. 75:1475-1487 (1981). [16] R. E. Edwards,Functional Analysis (Holt, Rinehart, and Winston, New York, 1965). · Zbl 0182.16101 [17] F. Golse, P. L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solution of a transport equation,J. Fund. Anal. 76:110-125 (1988). · Zbl 0652.47031 [18] R. H. Martin,Nonlinear Operators and Differential Equations in Banach Spaces (Wiley, New York, 1976). · Zbl 0333.47023 [19] T. Carleman,Problèmes mathématiques dans la théorie cinétique des gaz (Almiqvist & Wiksells Boktryckeri, Uppsala, Sweden, 1957). · Zbl 0077.23401 [20] C. Cercignani, Small data existence for the Enskog equation inL 1,J. Stat. Phys. 51:291-297 (1988). · Zbl 1086.82540
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