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Quelques applications de la positivité en théorie du transport. (Some applications of positivity to transport theory). (French) Zbl 0722.45009

The paper collects some results concerning the spectral theory of the linear transport operator (with continuous velocity dependence). Although many of these results were already known, the proofs here given are generally simpler than the previous ones. They are based on a comparison theorem on the spectral radii of positive operators which, when stated for the particular transport problem here sought, simply means that the fundamental eigenvalue of the transport operator for a bounded domain, is smaller than the fundamental eigenvalue of the bounded part of the same operator. It is shown, in particular, that the fundamental eigenvalue of the transport operator (i) is strictly dominant (ii) it increases strictly with the spatial domain and the collision operator (iii) the semigroup generated by the transport operator is irreducible. The theorem is also applied to show the existence and uniqueness of the solution of the one dimensional B.G.K. linear transport equation of the kinetic theory of gases; an adherent lower bound for the fundamental eigenvalue is also given.

MSC:

45K05 Integro-partial differential equations
45P05 Integral operators
45C05 Eigenvalue problems for integral equations
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C70 Transport processes in time-dependent statistical mechanics
82D75 Nuclear reactor theory; neutron transport
85A25 Radiative transfer in astronomy and astrophysics
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References:

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