Abstract
The nonlinear Bolzmann equation is discussed without cutoff approximations on potentials of infinite range. The Cauchy problem is solved locally in time, for both the spatially homogeneous and inhomogeneous cases. For the former case, this is done in function spaces of Gevrey classes in the velocity variables, and for the latter, in spaces of functions which are analytic in the space variables and of Gevrey classes in the velocity variables. The obtained existence theorem is of Cauchy-Kowalewski type. Also, the convergence of Grad’s angular cutoff approximations is established.
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References
L. Arkeryd: Intermolecular forces of infinite range and the Boltzmann equation. Arch Rational Mech. Anal.,77 (1981), 11–23.
K. Asano: Local solutions to the initial and initial boundary value problems for the Boltzmann equation with an external force. J. Math. Kyoto Univ. (to appear).
H. Grad: Asymptotic theory of the Boltzmann equation. Rarefied Gas Dynamics I. (Laurmann, J. A., Ed.), Academic Press, New York, 1963, 25–59.
M. Klaus: Boltzmann collision operator without cutoff. Helv. Phys. Acta,50 (1977), 893–903.
L. B. de Monvel and P. Krée: Pseudo-differential operators and Gevrey classes. Ann. Inst. Fourier,17 (1967), 295–323.
T. Nishida: A note on a theorem of Nirenberg. J. Differential Geometry,112 (1977), 629–633.
Y. P. Pao: Boltzmann collision operator with inverse power intermolecular potential, I, II. Comm. Pure Appl. Math.,27 (1974), 407–428, 559–581.
C. Truesdell and R. G. Muncuster: Fundamentals of Maxwell’s kinetic theory of a simple monoatomic gas. Pure Appl. Math. Vol. 83, Academic Press, New York, 1980.
S. Ukai and K. Asano: The Euler limit and initial layer of the nonlinear Boltzmann equation. Hokkaido Math. J.,12 (1983), 311–332.
—: Steady solutions of the Boltzmann equation for a gas flow past an obstacle, I. Existence. Arch. Rational Mech. Anal.,84 (1983), 249–291. II. Stability (preprint).
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Ukai, S. Local solutions in gevrey classes to the nonlinear Boltzmann equation without cutoff. Japan J. Appl. Math. 1, 141–156 (1984). https://doi.org/10.1007/BF03167864
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DOI: https://doi.org/10.1007/BF03167864