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On the solvability and asymptotics of the Boltzmann equation in irregular domains. (English) Zbl 0896.45007
The paper considers an initial boundary value problems for the Boltzmann equation in an irregular space domain and in an \(L^1\) setting. The authors generalize the existence results from L. Arkeryd and N. B. Maslova [J. Stat. Phys. 77, No. 5-6, 1051-1077 (1994; Zbl 0839.76073)], and the results on strong \(L^1\) asymptotics to the case of irregular domains with boundaries having finite Hausdorff measure and satisfying a cone condition. The temperature at the boundary is assumed to be constant. The boundary conditions have the regularity properties similar to Maxwellian diffuse reflection. The main results are the existence in the style of R. J. DiPerna and P. L. Lions [Ann. Math., II. Ser. 130, No. 2, 321-366 (1989; Zbl 0698.45010)], and the strong convergence to the equilibrium in \(L^1\) when the time tends to infinity.

MSC:
45K05 Integro-partial differential equations
45M05 Asymptotics of solutions to integral equations
82C40 Kinetic theory of gases in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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References:
[1] Adams, R. 1975. ”Sobolev spaces”. New York, London: Academic press. · Zbl 0314.46030
[2] DOI: 10.1007/BF00383222 · Zbl 0789.76075
[3] DOI: 10.1007/BF02183152 · Zbl 0839.76073
[4] Arkeyrd L., Nice (1994)
[5] DOI: 10.1007/BFb0090928
[6] DOI: 10.1016/0893-9659(91)90077-9 · Zbl 0744.45005
[7] Cercignani, C. 1987. ”The Boltzmann equation and its applications”. New York: Springer. · Zbl 0646.76001
[8] Cessenat M., C. R. Acad. Sci, Paris 300 pp 89– (1985)
[9] DOI: 10.1007/BF00375163 · Zbl 0705.76070
[10] DOI: 10.2307/1971423 · Zbl 0698.45010
[11] Federer H., Grundlehren math. Wiss., Band 153 (1969)
[12] DOI: 10.1090/S0002-9904-1978-14462-0 · Zbl 0392.49021
[13] DOI: 10.1016/0022-1236(88)90051-1 · Zbl 0652.47031
[14] DOI: 10.1007/BF01837113 · Zbl 0777.76084
[15] Heintz, H. 1986.Boundary value problems for nonlinear Boltzmann equation in domains with irregularboundaries. Ph.D.Thesis. 1986, Leningrad State Univesity.
[16] Heintz A., In Statistical Mechanics, Numerical Methods in Kinetic Theory of Gases pp 148– (1986)
[17] Heintz, A. 1996. ”Initial boundary value problems in irregular domains for nonlinear kinetic equationsof Boltzmann type”. Chalmers University of Technology, Goteborg Preprint No 1996–20/ISSN 0347–2809
[18] Heintz, A. 1996. ”Initial boundary value problems for the Enskog equations in irregular domains”. Chalmers University of Technology, Goreborg Preprint NO 1996–30/ISSN 0347–2809
[19] Lions P.L., J. Math. Kyoto Univ 34 pp 391– (1994)
[20] DOI: 10.1016/S0168-2024(08)70128-0
[21] Wennberg B., C.R.Acad.Sci. Paris 315 pp 1441– (1992)
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