Abstract
The time-evolution of charged particle swarms in a weakly ionized gas can be suitably modelled by the linear Boltzmann equation. In this work we discuss the time-dependent problem, the stationary problem as well as the long time behaviour of the particle distribution.
The paper is divided in three main parts. The first part is devoted to a simplified one-dimensional Boltzmann model of the Kač type, to study the velocity distribution of a spatially uniform diluted guest population of electrons moving within a host medium under the influence of a D.C. electric field. Necessary conditions and sufficient conditions are established for the existence, uniqueness and attractivity of a stationary nonnegative distribution corresponding to a specified concentration level. Conditions for the onset of the runaway process are established and the long time behaviour of the velocity distribution is studied within the framework of scattering theory.
The second part is devoted to the study of a non-homogeneous model where the collision frequency and the scattering kernel depend also on the space coordinates. A definition of “runaway” and a necessary condition for the suppression of runaways are given. The time-dependent problem is discussed and the long time behaviour of the solution is investigated. Also in this case, under physically reasonable assumptions on the collision frequency, we prove the existence of wave operators and the corresponding existence of travelling waves.
Finally, in the third part, we report some results about three-dimensional velocity systems.
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Frosali, G. (1991). Functional-analytic techniques in the study of time-dependent electron swarms in weakly ionized gases. In: Toscani, G., Boffi, V., Rionero, S. (eds) Mathematical Aspects of Fluid and Plasma Dynamics. Lecture Notes in Mathematics, vol 1460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091364
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DOI: https://doi.org/10.1007/BFb0091364
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