Properties of the Collision Operators of the Electron Boltzmann Equation for a Weakly Ionized Plasma. I. Representation and General Properties

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The compactness has been proved of the inscattering operator covering elastic, exciting and deexciting processes between electrons and heavy neutral particles. Starting point is the proof of the boundedness of the different inscattering operators for the individual collision processes using certain assumptions on the structure of the corresponding differential crow sections. Then, under somewhat more confined conditions the proof of compactness could be performed with the:aid of the square integrability of an eightfold iterated kernel. On the basis of this property it was possible to draw conclusions with regard to the speatrum of the total collision operator. They are related to the continuous as well as the discrete part of the spectrum and are discussed in detail. For elast,ic collisions additional statements can be deduced from our investigation.

“…as already discussed in [2]. In (6) u2 is an arbitrary constant; thus many cross section couraes are enclosed.…”

Section: P(v)mentioning

confidence: 99%

“…This proof includes the case of elastic collisions. According to the explanation given in [2] this means a restriction to weakinteractionswith 1113 < s 5 5.…”

Section: (24)mentioning

confidence: 99%

Properties of the Collision Operators of the Electron Boltzmann Equation for a Weakly Ionized Plasma. II. Compactness Proof and Statements on the Spectrum

Bartolomäus¹,

Wilhelm²

1982

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Remarks on the Calculation of Eigenvalues and Generalized Eigenfunctions Using a Finite Dimensional Approximation for the Inscattering Part of the Boltzmann Collision Operator

Bartolomäus¹,

Wilhelm²

1982

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From several points of view it is of advantage to know the properties of the collision operators in kinetic equations, e.g. in the well known Boltzmann equation, in particular for the purpose of solving eigenvalue problems. With regard to elastic, exciting and deexciting processes some attemps were recently made to investigate such operators in the Boltzmann equation describing the behaviour of electrons in weakly ionized plasmas.I n the following we will prove that in the case of a finite dimensional inscattering operator the eigenvalues and the corresponding eigenfunctions can be represented explicitly. Finite dimensional operators were used successfulty in special models to approximate the inscattering operators ; they possess the property of compactness and are well suitable for analytical or numerical calculations.The representation has been obtained by solving an adequate linear equation system. The generalized eigenfunctions correspond to the normal solutions used by CASE in the neutron transport theory. The regularization of the singular integrals which are necessary to obtain this solution will be given in detail. Further a velocity dependence of the collision frequency which need not be monotonous in the considered case and the dependence on the direction could be included.

Generalized Boundary Conditions in an Existence and Uniqueness Proof for the Solution of the Non‐Stationary Electron Boltzmann Equation by Means of Operator‐Semigroups

Bartolomäus

Wilhelm

1983

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Recently, based on the semigroup approach a new proof was presented of the existence of a unique solution of the non-stationary Boltzmann equation for the electron component of a collision dominated plasma. The proof underlies some restrictions which should be overcome to extend the validity range to other problems of physical interest. One of the restrictions is the boundary condition applied. The choice of the boundary condition is essential for the proof because i t determines the range of definition of the infinitesimal generator and thus the operator semigroup itself. The paper proves the existence of a unique solution for generalized boundary conditions, this solution takes non-negative values, which is necessary for D distribution function from the physical point of view.