Abstract
This paper studies the L p-behavior for 1≦p ≦ ∞ of solutions of the nonlinear, spatially homogeneous Boltzmann equation for a class of collision kernels including inverse k th-power forces with k>5 and angular cut-off. The following topics are treated: differentiability in L p together with global boundedness in time for L p-moments that exist initially, translation continuity in L p uniformly in time, and strong convergence to equilibrium.
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References
L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal. 45 (1972) 1–34.
L. Arkeryd, 37-01-estimates for the space homogeneous Boltzmann equation, J. Statist. Phys. 31 (1982), 347–361.
L. Arkeryd, On the Boltzmann equation in unbounded space far from equilibrium, and the limit of zero mean free path, Comm. Math. Phys. 105 (1986), 205–219.
L. Arkeryd, The non-linear Boltzmann equation far from equilibrium, to appear in Proceedings of the Hull conference on non-standard analysis 1986.
L. Arkeryd, R. Esposito, & M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data, Comm. Math. Phys. to appear.
J. Bergh & J. Löfström, Interpolation spaces, an introduction, Springer-Verlag, Berlin, 1976.
T. Carleman, Problèmes mathématiques dans la théorie cinétique des gaz, Almqvist & Wiksell, Uppsala, 1957.
C. Cercignani, Theory and application of the Boltzmann equation, Scottish Academic Press, 1975.
N. Dunford & J. T. Schwartz, Linear Operators Part I, Interscience Publishers, Inc., New York, 1958.
T. Elmroth, Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range, Arch. Rational Mech. Anal. 82 (1983), 1–12.
T. Elmroth, On the H-function and convergence towards equilibrium for a space-homogeneous molecular density, Siam J. Appl. Math. 44 (1984), 150–159.
T. Gustafsson, l p-estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal. 92 (1986), 23–57.
P. I. Lizorkin, Interpolation of weighted L p-spaces, Soviet Math. Dokl. 16 (1975).
R. H. Martin, Nonlinear operators and differential equations in Banach spaces, Wiley, New York, 1976.
N. B. Maslova & R. P. Tchubenko, Asymptotic properties of solutions of the Boltzmann equation, Dokl. Akad. Nauk SSSR, 202 (1972), 800–803. Vestnik, Leningrad Univ. n. 13, (1976), 90–97.
J. Mikusinski, The Bochner integral, Birkhäuser, 1978.
R. O'Neil, Convolution operators and L(p, q) spaces, Duke Math. J. 30 (1963), 129–142.
C. Truesdell & R. G. Muncaster, Fundamentals of Maxwell's kinetic theory of a simple monatomic gas, Academic Press, 1980.
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Communicated by L. Arkeryd
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Gustafsson, T. Global Lp-properties for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 103, 1–38 (1988). https://doi.org/10.1007/BF00292919
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DOI: https://doi.org/10.1007/BF00292919