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positivity in time dependent linear transport theory

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Abstract

We present methods using positive semigroups and perturbation theory in the application to the linear Boltzmann equation. Besides being a review, this paper also presents generalizations of known results and develops known methods in a more abstract setting.

In Section 1 we present spectral properties of the semigroup operatorsW a(t) of the absorption semigroup and its generatorT a. In Section 2 we treat the full semigroup (W(t);t≧0) as a perturbation of the absorption semigroup. We discuss part of the problems (perturbation arguments and existence of eigenvalues) which have to be solved in order to obtain statements about the large time behaviour ofW(·). In Section 3 we discuss irreducibility ofW(·).

In four appendices we present abstract methods used in Sections 1, 2 and 3.

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  1. Jürgen Voigt

    Present address: Mathematisches Institut der Universität München, Theresienstraße 39, D-8000, München 2, Federal Republic of Germany

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  1. Fach Mathematik, Universität Trier, F.R. Germany

    Jürgen Voigt

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Voigt, J. positivity in time dependent linear transport theory. Acta Appl Math 2, 311–331 (1984). https://doi.org/10.1007/BF02280857

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