Abstract
The paper is concerned with the Enskog equation with a constant high density factor for large initial data in L 1(R n). The initial boundary value problem is investigated for bounded domains with irregular boundaries. The proof of an H-theorem for the case of general domains and boundary conditions is given. The main result guarantees the existence of global solutions of boundary value problems for large initial data with all v-moments initially finite and domains having boundary with finite Hausdorff measure and satisfying a cone condition. Existence and uniqueness are first proved for the case of bounded velocities. The solution has finite norm \(\int {(\sup _{0 \leqslant t \leqslant T} f(t_0 + t,x + vt))\sqrt {1 + v^2 } dq{\text{ }}dv,} \) where q = (t 0, x) is taken on all possible n-dimensional planes Q(v) in R n+l intersecting a fixed point and orthogonal to vectors (1, v), v ∈ R n.
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REFERENCES
L. Arkeryd, On the Enskog equation in two space variables, Transport Theory and Stat. Phys. 15:673-691 (1986).
L. Arkeryd, Existence theorems for certain kinetic equations and large data, Arch. Rat. Mech. Anal. 103:139-149 (1988).
L. Arkeryd, On the Enskog equation with large initial data, SIAM J. Math. Anal. 21:631-646 (1990).
L. Arkeryd and C. Cercignani, Global existence in L 1 for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation, J. of Stat. Phys. 59:845-867 (1990).
L. Arkeryd and N. B. Maslova, Boundary value problems and diffuse reflection, J. Stat. Phys. 77:1051-1077 (1994).
L. Arkeryd and A. Nouri, Asymptotics of the Boltzmann equation with diffuse reflection boundary conditions, to appear in Monatsheft für Mathematik (Nice, 1994).
T. Carleman, Théorie cinétique des gas (Almqvist & Wiksell, Uppsala, 1957).
C. Cercignani, The Boltzmann equation and its applications (Springer, New York, 1987).
R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equation: Global existence and weak stability, Ann. Math. 130:321-366 (1989).
D. Enskog, Kinetische Theorie der Wärmeleitung, Reibund und Selbstdiffusion in gevissen verdichteten Gasen und Flüssigkeiten, Kungl. Svenska Vetenskapsakad Handl. 63:3-44 (1922).
H. Federer, Colloqium in geometric measure theory, Bull. Am. Math. Soc. 84:291-338 (1978).
A. Heintz, Boundary value problems for nonlinear Boltzmann equation in domains with irregular boundaries, Ph.D. Thesis (1986), Leningrad State University.
A. Heintz, On solutions of boundary value problems for the Boltzmann equation in domains with irregular boundaries, in Statistical Mechanics, Numerical Method in Kinetic Theory of Gases, Novosibirsk, 1986, pp. 148-154.
A. Heintz, Initial boundary value problems in irregular domains for nonlinear kinetic equations of Boltzmann type. Preprint Chalmers University of Technology. NO 1996-20/ISSN 0347-2809, Göteborg. 1996.
P. Resibois, H-theorem for the (modified) nonlinear Enskog equation, J. of Stat. Phys. 19:593-609 (1978).
G. Toscani, On the Cauchy problem for the discrete Boltzmann equation with initial values in L 1 + (R), Tech. Report, (Universita di Ferrara, 1987).
S. Ukai, Solutions of the Boltzmann equation, in Patterns and Waves-Qualitative Analysis of Nonlinear Differential equations, H. Fujita, J. L. Lions, Papanicolau, and H. B. Keller, eds., Vol. 18, (Kinokuniya, 1986, pp. 37-96).
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Heintz, A. On the Initial Boundary Value Problems for the Enskog Equation in Irregular Domains. Journal of Statistical Physics 90, 663–695 (1998). https://doi.org/10.1023/A:1023268718526
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DOI: https://doi.org/10.1023/A:1023268718526