Cornelis van der Mee

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.

Time Dependent Kinetic Equations: Method of Characteristics

Time dependent linear kinetic equations arise in a number of diverse applications in biology, chemistry and physics, as well as in various other modeling problems. Due to a tradition deeply rooted in classical mathematical physics and reinforced by the successes of quantum mechanics, such time dependent problems were initially attacked using the eigenfunction method. Yet, this method met with a relative lack of success, due to the nonnormal nature of the operators occurring in these kinetic problems, and it was supplanted by the semigroup approach, which for decades became the dominant method of time dependent kinetic theory. Despite its virtues, the semigroup approach is somewhat indirect, and is not naturally suited for treating linear evolution problems with time dependent operators, phase spaces and boundary conditions. This chapter will be devoted to an approach to these problems based on the method of characteristics.

Equivalence of Differential and Integral Formulations

Integral forms of transport equations first appeared in radiative transfer theory at the beginning of this century. If one considers radiative transfer with isotropic scattering in a layer of finite optical thickness τ, the boundary value problem may be written (cf. Section IX.1)

Elements of Linear Kinetic Theory

In this monograph we present a rigorous exposition of boundary value problems for an extensive class of kinetic equations. These equations describe the transport of particles or radiation through a host medium (possibly the vacuum) in the region Ω with velocities or frequencies belonging to a subset V ⊂ ℝn, and the solutions represent particle densities as a function of position and velocity, or radiative intensities as a function of position and frequency. At the boundary ∂Ω of the spatial domain Ω, particles or radiation may be incident from outside and may be exiting the medium, and reflection and absorption processes may be specified. One may also deal with particle or radiative sources within the region Ω.

Applications of the Initial Value Problem

In this chapter, we shall consider specific kinetic models related to the transport of neutrons and electrons, and to cellular growth. The first two sections will be devoted to neutron transport, with special attention to spectral properties of the full transport operator and implications to hydrodynamics. The third and fourth sections deal with electron transport. In the first of these, the Spencer-Lewis equation models the slowing down of electrons by a thermalizing medium. In the following, a linearized Boltzmann equation is presented, which describes the drift of electrons in a weakly ionized gas. Finally, in the last section, we will outline a biological model for the growth of cells, due to Rotenberg, Lebowitz and Rubinow.

Indefinite Sturm-Liouville Problems

In this chapter we shall discuss in some detail partial differential equations associated with self adjoint Sturm-Liouville boundary value problems with indefinite weights.

Semigroup Factorization and Reconstruction

In this chapter we shall continue our study of the theory of convolution equations and its applications to abstract kinetic equations. In the first section we will outline the classical method for solving Wiener-Hopf equations on a half line. This will reduce the half space problem to a factorization problem. In the second section we shall study the connection between the semigroups developed in Chapters II and III and the solution of convolution equations corresponding to abstract kinetic equations. In this way we will obtain an alternative way of defining these projections and semigroups. In the following section we will begin a study of explicit representations of the Wiener-Hopf factors of the symbol, which is important in the derivation of representations for the solutions of the half space and the finite slab problem, to be discussed in the next chapter. In the fourth section, we present some recent results on the treatment of nonregular collision operators. Finally, in the last section, we outline the extension of the previous theory to a Banach space setting.

Kinetic Equations on Finite Domains

In the previous three chapters we have analyzed in detail the existence and uniqueness theory for the abstract differential equation.

Albedo Operators, H-Equations and Representation of Solutions

In the previous chapters we have defined the albedo operator, which specifies the full boundary value of the solution of a half space problem in terms of partial range boundary data. In this section we shall construct, under the general assumptions of Section VII.2, the albedo operator in terms of certain special functions. These functions generalize the H-functions, which were first extensively studied by Chandrasekhar [89].

Conservative Kinetic Models

In the previous chapter we studied boundary value problems in half space geometry of the type with A strictly positive.