Book•
Boundary Value Problems in Abstract Kinetic Theory
10 Feb 1987-
TL;DR: In this paper, the authors present a survey of abstract kinetic theory and its application in various areas of physics, chemistry, biology, and engineering, including radiative transfer and rarefied gas dynamics.
Abstract: 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.
Citations
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TL;DR: Van Kampen as mentioned in this paper provides an extensive graduate-level introduction which is clear, cautious, interesting and readable, and could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes.
Abstract: 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.
3,647 citations
TL;DR: In this paper, the authors extend the results of Bardos, Caflisch and Nicolaenko for a gas of hard spheres to a general potential, and obtain asymptotic behavior for hard as well as soft potentials.
Abstract: 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.
99 citations
TL;DR: In this article, the authors assume perfect conductor boundary conditions for Vlasov and Maxwell, and either specular reflection or partial absorption for the latter for all-time weak solutions with finite energy.
Abstract: 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.
91 citations
01 Jan 2003
TL;DR: In this paper, the authors 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 with respect to matrix functions with almost periodic or even semi almost periodic entries.
Abstract: 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.
89 citations
TL;DR: In this paper, the formation and propagation of singularities for the Boltzmann equation in bounded domains has been studied in numerical studies as well as in theoretical studies, and it is shown 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.
Abstract: 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.
81 citations
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23,110 citations
TL;DR: Light scattering by small particles as mentioned in this paper, Light scattering by Small Particle Scattering (LPS), Light scattering with small particles (LSC), Light Scattering by Small Parts (LSP),
Abstract: Light scattering by small particles , Light scattering by small particles , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی
9,737 citations
Book•
01 Jan 1955
TL;DR: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable as discussed by the authors, which is a useful text in the application of differential equations as well as for the pure mathematician.
Abstract: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by the authors and probably contains more material than will ordinarily be covered in a one-year course. It is hoped that the book will be a useful text in the application of differential equations as well as for the pure mathematician.
7,071 citations
01 Jan 1954
TL;DR: In this article, a kinetic theory approach to collision processes in ionized and neutral gases is presented, which is adequate for the unified treatment of the dynamic properties of gases over a continuous range of pressures from the Knudsen limit to the high pressure limit where the aerodynamic equations are valid.
Abstract: A kinetic theory approach to collision processes in ionized and neutral gases is presented. This approach is adequate for the unified treatment of the dynamic properties of gases over a continuous range of pressures from the Knudsen limit to the high-pressure limit where the aerodynamic equations are valid. It is also possible to satisfy the correct microscopic boundary conditions. The method consists in altering the collision terms in the Boltzmann equation. The modified collision terms are constructed so that each collision conserves particle number, momentum, and energy; other characteristics such as persistence of velocities and angular dependence may be included. The present article illustrates the technique for a simple model involving the assumption of a collision time independent of velocity; this model is applied to the study of small amplitude oscillations of one-component ionized and neutral gases. The initial value problem for unbounded space is solved by performing a Fourier transformation on the space variables and a Laplace transformation on the time variable. For uncharged gases there results the correct adiabatic limiting law for sound-wave propagation at high pressures and, in addition, one obtains a theory of absorption and dispersion of sound for arbitrary pressures. For ionized gases the difference in the nature of the organization in the low-pressure plasma oscillations and in high-pressure sound-type oscillations is studied. Two important cases are distinguished. If the wavelengths of the oscillations are long compared to either the Debye length or the mean free path, a small change in frequency is obtained as the collision frequency varies from zero to infinity. The accompanying absorption is small; it reaches its maximum value when the collision frequency equals the plasma frequency. The second case refers to waves shorter than both the Debye length and the mean free path; these waves are characterized by a very heavy absorption.
6,004 citations