Schur Complement and Continuous Spectrum in a Kinetic Plasma Model
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EMATICS
Schur Complement and Continuous Spectrum in a Kinetic Plasma Model S. A. Stepin Presented by RAS Academician V.P. Maslov December 4, 2019 Received December 6, 2019; revised February 18, 2020; accepted February 18, 2020
Abstract—The goal of this paper is spectral analysis of an evolutionary semigroup generator describing the dynamics of a rarefied two-component plasma subjected to a self-consistent electromagnetic field. For the problem in question, the spectrum is given in terms of the dispersion relationship and an effective approach to the calculation of the instability index is developed. Keywords: kinetic plasma model, generator of operator semigroup, Schur complement, dispersion relationship, instability index DOI: 10.1134/S1064562420030217
This paper deals with spectral analysis of a dynamical system representing a kinetic model of a rarefied two-component plasma consisting of particles of two types, namely, electrons and ions subjected to an electromagnetic field. The particle distribution in the phase space is characterized by distribution densities depending on time t and the coordinates and velocities of the particles. The system describing the dynamics of the plasma consists of Maxwell’s equations for the electromagnetic field components and Boltzmann kinetic equations for the distribution densities.
the magnetic field B(t, x) is degenerate. Then the considered system of equations has the form
In the collisionless approximation considered below, the charged particles interact via a self-consistent electromagnetic field, which, on the one hand, is induced by the motion of the charged particles and, on the other hand, affect the evolution of their distribution densities. By using this mathematical model, an unexpected phenomenon was found in [1], namely, wave damping in a collisionless plasma. This subject has been extensively studied by both physicists and mathematicians (see [2–6] and bibliography therein). Our goal is to perform spectral analysis of an integrodifferential operator generating an evolutionary semigroup describing the dynamics of an initial perturbation of the stationary solution to the system (cf. [7, 8]).
where e and Ze are charges; m and M are the masses of
For presentational simplicity, we consider a twodimensional single-particle phase space ( x, v ) when
( −)
( −)
( −)
∂f ∂f ∂f +v −eE = 0, ∂t ∂x m ∂v ∂f ( + ) ∂f (+) Ze ∂f (+) +v + E = 0, ∂t ∂x ∂v M ∞
∂E + 4πe v (Zf (+) − f (−) )d v = 0, ∂t −∞
electron and ion, respectively; f (∓) are their distribution densities; and E = E (t, x) is the electric field strength. Each of the kinetic equations can be written in the Liouville form (see [9]) and means that the total derivative of the distribution density along the trajectory of a particle of the corresponding type is zero. Stationary solutions of the kinetic equations have the form
2 f (−) = f0(−) mv − eφ( x) , 2 2 (+) (+ ) M v f = f0 + Zeφ( x) , 2 ∂φ . ∂x Below, by the unperturbed solution of the system, we mean its stationary solution indep
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