An abstract approach to nonlinear boltzmann-type equations |
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Authors: | G Toscani C V M van der Mee |
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Institution: | (1) Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, I-44100 Ferrara, Italy;(2) Dipartimento di Matematica, Università di Pavia, Strada Nuova 65, I-27100 Pavia, Italy |
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Abstract: | Summary Using a general existence and uniqueness theory for linear time dependent kinetic equations, for general inhomogeneous multidimensional
spatial and velocity domains and partially absorbing boundaries, we obtain local in time solutions of a class of nonlinear
Boltzmann type equations. For small initial-boundary data we obtain global in time solutions. The ideal norm on certain ideals
in the Banach space ofL
p-functions on phase space is used to measure the ?size? of initial-boundary data and solutions. Kaniel-Shinbrot type upper
and lower approximation arguments are applied. The combined length of the time interval of existence when applying the method
repeatedly is analyzed as a function of the size of the initial-boundary data. Specific applications to the nonlinear Boltzmann
equation itself and to the plane Broadwell model are given.
Research conducted under the auspices of C.N.R. (Consiglio Nazionale delle Ricerche), Gruppo Fisica-Matematica, and partially
supported by M.P.I. (Ministero della Pubblica Intruzione).
Research conducted as a visiting professor supported by C.N.R., Gruppo Fisica-Matematica. Permanente address: Dept. of Mathematics
and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. |
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Keywords: | |
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