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An abstract approach to nonlinear boltzmann-type equations

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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.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, I-44100, Ferrara, Italy

    G. Toscani

  2. Dipartimento di Matematica, Università di Pavia, Strada Nuova 65, I-27100, Pavia, Italy

    C. V. M. van der Mee

Authors
  1. G. Toscani
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  2. C. V. M. van der Mee
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Additional information

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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Toscani, G., van der Mee, C.V.M. An abstract approach to nonlinear boltzmann-type equations. Ann. Univ. Ferrara 34, 75–100 (1988). https://doi.org/10.1007/BF02824975

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  • DOI: https://doi.org/10.1007/BF02824975

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