positivity in time dependent linear transport theory |
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Authors: | Jürgen Voigt |
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Institution: | (1) Fach Mathematik, Universität Trier, F.R. Germany;(2) Present address: Mathematisches Institut der Universität München, Theresienstraße 39, D-8000 München 2, Federal Republic of Germany |
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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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Keywords: | 47D05 82A70 47B55 47G05 35K22 35F15 35C15 |
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