N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Stationary half-space solutions of the linearized Boltzmann equation are studied by energy estimates methods. We extend the results of Bardos, Caflisch and Nicolaenko for a gas of hard spheres to a general potential. Asymptotic behaviour is obtained for hard as well as soft potentials and compared to the case of hard spheres.

Stationary solutions of the linearized Boltzmann equation in a half-space

Boundaries occur naturally in physucal systems which satisfy the Vlasov-Maxwell system. Assume perfect conductor boundary conditions for Maxwell, and either specular reflection or partial absorption for Vlasov. Then weak solutions with finite energy exist for all time.

/pdf/global-weak-solutions-of-the-vlasov-maxwell-system-with-22xfwebhga.pdf

Global weak solutions of the Vlasov-Maxwell system with boundary conditions

These lecture notes present an extensive review of the factorization theory of matrix functions relative to a curve with emphasis on the developments of the last 20–25 years. The classes of functions considered range from rational and continuous matrix functions to matrix functions with almost periodic or even semi almost periodic entries. Also included are recent results about explicit factorization based on the state space method from systems theory, with examples from linear transport theory. Related applications to Riemann-Hilbert boundary value problems and the Fredholm theory of various classes of singular integral operators are described too. The applications also concern inversion of singular integral operators of different types, including Wiener-Hopf and Toeplitz operators.

An Overview of Matrix Factorization Theory and Operator Applications

The formation and propagation of singularities for the Boltzmann equation in bounded domains has been an important question in numerical studies as well as in theoretical studies. In this paper, we consider the nonlinear Boltzmann solution near Maxwellians under in-flow, diffuse, or bounce-back boundary conditions. We demonstrate that discontinuity is created at the non-convex part of the grazing boundary, and then it propagates only along the forward characteristics inside the domain before it hits on the boundary again.

https://www.ki-net.umd.edu/pubs/files/FRG-2010.singularity_formation_and_propagation(FRG).pdf

Formation and Propagation of Discontinuity for Boltzmann Equation in Non-Convex Domains

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.

Boundary Value Problems in Abstract Kinetic Theory

In the next several chapters we shall develop an existence and uniqueness theory for the Hilbert space boundary value problem.

Cornelis van der Mee

Papers

Boundary Value Problems in Abstract Kinetic Theory

Strictly Dissipative Kinetic Models