Untersuchungen zu Eigenschaften des Boltzmannschen Stoßoperators der Elektronenkomponente eines Plasmas, welche für die Entwicklung nach Eigenfunktionen von Bedeutung sind

Wilhelm, J.; Bartolomäus, G.

doi:10.1002/ctpp.19760160304

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…Here v is the lumped velocity dependent collision frequency which characterhes the outscattering collisions, and Kf is the term which describes the contribution by inscattering collisions [17]. The Boltzmann-operator A will be considered in Banach spaces L,( V x R3), p 2 1 of measurable functions whose #h power is integrable.…”

Section: Statement Of the Problem And Some Premisesmentioning

confidence: 99%

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Existence and Uniqueness of the Solution of the Non‐stationary Boltzmann‐Equation for the Electrons in a Collision Dominated Plasma by Means of Operator Semigroups

Bartolomäus

Wilhelm

1981

Annalen der Physik

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Based on the semigroup approach a newproof.is presented of the existence of o unique solution of the non-stationary Boltzmann-equation for the electron component of a collision dominated plasma. All interactions can be included which yield bounded collision operators. The electric and magnetic fields were permitted to be inhomogeneous in space, and the investigations were performed for a bounded plasma. It has been shown that the Boltzmann-operator is the infiitesimal generator of a strongly continuous operator-semigroup which uniquely determines the nonnegative solution from a given initial function taking into account the given boundary conditions. Existenz und Eindeutigkeit der LSsung der nichtstationLen Boltzmann-Gleichung fur die Elektronen in einem sto Bbestimmten Plasma mit Hilfe von Operator-Halbgmppon Inhaltsubersicht. Im Folgenden wird ein neuer Beweis fur die Existenz einer eindeutigen Lijsung der nichtstationiiren Boltzmann-Gleichung fur die Elektronenkomponente eines stoI3bestimmten Plasmas auf der Grundlage der Halbgruppen-Theorie erbracht. Dabei lassen sich solche Wechselwirkungen mitberiicksichtigen, deren zugehorige StoDoperatoren beachriinkt sind. Elektrisches und magnetisches Feld werden als orteabhiingige Funktionen angeaehen und den Untersuchungen ein begrenztes Plasma zugrunde gelegt. Es wird gezeigt, d a B der Boltzmann-Operator infinitesimal eneugender Operator einer stark stetigen Operator-Halbgruppe ist, welche die nichtnegative Liisung unter vorgegebenen Randbedingungen aus der Anfangsverteilung eindeutig angibt.

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Section: Statement Of the Problem And Some Premisesmentioning

confidence: 99%

“…Finally, the functions y from (17) satisfy the boundary condition: For r E aV and v directed inwardly F t y equals zero for any t > 0 and thus, according to (17) y vanishes for such r and v . Hence we have proved that the elements of D(Tl), represented by (17), possess the properties demanded in the definition (6), i. e. that the assertion D(T,) 2 d is true.…”

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confidence: 99%

Existence and Uniqueness of the Solution of the Non‐stationary Boltzmann‐Equation for the Electrons in a Collision Dominated Plasma by Means of Operator Semigroups

Bartolomäus

Wilhelm

1981

Annalen der Physik

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“…n for deexciting processes [2]. Here f denotes the electron velocity distribution function, F the velocity distribution functions of the several kinds of heavy neutral particles, u the different differential cross sections, vf and a as well as g' and g in their original meaning denote the velocities and the modules of the relative velocities before and after a collision respectively.…”

Section: Introductionmentioning

confidence: 99%

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

Contrib Plasma Phys

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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.

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Motion of Electrons and Ions in a Weakly Ionized Gas in a Field. 1. Foundations of the Integral Theory

Braglia

1980

Contrib Plasma Phys

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A detailed analysis of an integral transport theory of electrons and ions in a gas in a field is presented. Different procedures allowing, in principle, the rigorous calculation of velocity distribution, drift velocity and diffusion coefficients are discussed. A derivation of a general expression for the transition probability, from a given velocity to another, which holds for any kind of collision, is given. Some questions linked to the convergence of iterative procedures of solution of some basic integral equations are also treated.

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Untersuchungen zu Eigenschaften des Boltzmannschen Stoßoperators der Elektronenkomponente eines Plasmas, welche für die Entwicklung nach Eigenfunktionen von Bedeutung sind

Cited by 5 publications

References 12 publications

Existence and Uniqueness of the Solution of the Non‐stationary Boltzmann‐Equation for the Electrons in a Collision Dominated Plasma by Means of Operator Semigroups

Existence and Uniqueness of the Solution of the Non‐stationary Boltzmann‐Equation for the Electrons in a Collision Dominated Plasma by Means of Operator Semigroups

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

Motion of Electrons and Ions in a Weakly Ionized Gas in a Field. 1. Foundations of the Integral Theory

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