Existence, uniqueness and properties of the solutions of the Boltzmann kinetic equation for a weakly ionized gas. I.

Abstract

The Boltzmann kinetic equation for a weakly ionized gas in the presence of a time dependent exterior electric field and a static exterior magnetic field has been transformed into an integral equation. Existence and uniqueness theorems have been proved for inverse power-law potentials of the form A/rs with s≳3 and for a large class of initial distribution functions. For soft potentials (3<s⩽5), these theorems have been derived from the general properries of the integral operator. For hard potentials, 5<s⩽+∞, where no generla properties of the integral operator can be directly proved, an iteration procedure which constitutes the main part of the present work has been developed. In each case, some important properties of the solution have been established.

Integral Representation of the Linear Boltzmann Operator for Granular Gas Dynamics with Applications

“…It is well-known that, for non-dissipative interactions, i.e. when ǫ = 1, the gain part Q + can be written as an integral operator with explicit kernel [6,15] (see also [12,7] for similar results for the linearized Boltzmann equation). We prove that such a representation is still valid in the dissipative case:…”

“…We show in this section that, as it occurs for the classical Boltzmann equation, Q + turns out to be an integral operator with explicit kernel. The proof of such a result is based on well-known tools from the linear elastic scattering theory [6,12,15] while, in the dissipative case, similar calculations have been performed to derive a Carleman representation of the nonlinear Boltzmann operator in [17].…”

On singular mono‐energetic transport equations in slab geometry

“…This permits us to estimate, via Theorem 5.1, the essential type of the perturbed semigroup (e t(TH +K) ) t¿0 from which the asymptotic behaviour of the solution follows by standard methods. For physical models of singular transport equations we refer to References [2,7,15,17,19].…”

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

“…For a plasma an additional term caused by the Lorentz-force F = q(E + V X B ) appears in the kinetic equation. Also using an iterative approach related to the formal solution, MOLINET [8] and DRANGE [9] gave proofs of the existence and uniqueness. They used special and general interaction potentials respectively which ensure that the operator for inscattering collisions is bounded.…”

“…They used special and general interaction potentials respectively which ensure that the operator for inscattering collisions is bounded. In [8] the magnetic field B is assumed to be constant and the electric field E may depend on the time whilst in [9] E and B may vary in time and space where certain boundedness conditions must be fulfilled').…”