William Greenberg

Boundary value problems in abstract kinetic theory

This monograph is intended to be a reasonably self -contained and fairly complete exposition of rigorous results in abstract kinetic theory. Throughout, abstract kinetic equations refer to (an abstract formulation of) equations which describe transport of particles, momentum, energy, or, indeed, any transportable physical quantity. These include the equations of traditional (neutron) transport theory, radiative transfer, and rarefied gas dynamics, as well as a plethora of additional applications in various areas of physics, chemistry, biology and engineering. The mathematical problems addressed within the monograph deal with existence and uniqueness of solutions of initial-boundary value problems, as well as questions of positivity, continuity, growth, stability, explicit representation of solutions, and equivalence of various formulations of the transport equations under consideration. The reader is assumed to have a certain familiarity with elementary aspects of functional analysis, especially basic semigroup theory, and an effort is made to outline any more specialized topics as they are introduced. Over the past several years there has been substantial progress in developing an abstract mathematical framework for treating linear transport problems. The benefits of such an abstract theory are twofold: (i) a mathematically rigorous basis has been established for a variety of problems which were traditionally treated by somewhat heuristic distribution theory methods; and (ii) the results obtained are applicable to a great variety of disparate kinetic processes. Thus, numerous different systems of integrodifferential equations which model a variety of kinetic processes are themselves modelled by an abstract operator equation on a Hilbert (or Banach) space.

Generalized kinetic equations

AbstractWe study the abstract differential equation

Global Solutions of the Boltzmann Equation on a Lattice

T\frac{{\partial f}}{{\partial x}} + Af = 0

Functional analytic treatment of the transport equation

on a Hilbert space H, which represents a variety of different kinetic equations. T is assumed bounded and self-adjoint on H, and A (unbounded) positive self-adjoint and Fredholm. For partial range boundary conditions and 0≤x<∞, we prove existence and (non-) uniqueness theorems and give representations of the solution. Various examples from neutron transport, radiative transfer of polarized and unpolarized light, and electron transport are given.

A Global Existence Theorem for the Nonlinear BGK Equation

A functional analytic approach to the linearized collisionless Vlasov equation is presented utilizing a resolvent integration technique on the resolvent of the transport operator evaluated at a particular point. Formulae for the eigenfunction expansion are found for cases in which the plasma disperion function Λ has first and second order zeroes. Special care is taken in the study of real zeroes of Λ culminating in new results for this case. For a simple zero of Λ with nonvanishing imaginary part the van Kampen–Case discrete modes are reproduced. The results are used to obtain the solution to the initial value problem.

SPECTRAL PROPERTIES OF TRANSPORT EQUATIONS FOR SLAB GEOMETRY IN L1WITH REENTRY BOUNDARY CONDITIONS

The nonlinear Boltzmann equation with a discretized spatial variable is studied in a Banach space of absolutely integrable functions of the velocity variables. Conservation laws and positivity are utilized to extend weak local solutions to a global solution. This is shown to be a strong solution by analytic semigroup techniques.

A CLASS OF LINEAR KINETIC EQUATIONS IN A KREIN SPACE SETTING

By using the resolvent integration technique introduced by Larsen and Habetler, the one‐speed, isotropic scattering, neutron transport equation is treated in the infinite and semi‐infinite media. It is seen that the results previously obtained by Case’s ’’singular eigenfunction’’ approach are in agreement with those obtained by resolvent integration.

An Abstract Model for Radiative Transfer in an Atmosphere with Reflection by the Planetary Surface

Abstract We will discuss here only the neutron transport equation, although the methods we describe can be and, in fact, have been, equally well applied to other systems described by a linearized Boltzmann equation. Such systems might include electron oscillations in a plasma, radiative transport in stellar (or planetary) atmospheres, the dynamics of gases of sufficiently low density that the linearized Boltzmann equation is appropriate, and perhaps others.