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Boundary Value Problems in Abstract Kinetic Theory
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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.read more
Citations
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Semigroup generation propertiesof streaming operators with noncontractive boundary conditions
TL;DR: This study presents c"0-semigroup generation results for the free streaming operator with abstractboundary conditions and establishes a general theorem that applies to the physical cases of Maxwell boundary conditions in the kinetic theory of gases, as well as to the nonlocal boundary conditions involved in transport-like equations from population dynamics.
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Regularity and time asymptotic behaviour of solutions to transport equations
Khalid Latrach,Bertrand Lods +1 more
TL;DR: In this paper, the authors studied the regularity properties of solutions to evolution transport problems, which are closely related to spectral properties of transport operators, using the geometrical properties of the functional spaces.
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Geometric Correction for Diffusive Expansion of Steady Neutron Transport Equation
TL;DR: In this paper, the diffusive limit of a steady neutron transport equation in a 2D unit disk with one-speed velocity was revisited and the classical result in [4] with Milne expansion was incorrect in $L^{\infty}$ and gave the right answer in studying the $\epsilon$-Milne expansion with geometric correction.
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Analytic solution of boundary-value problems for nonstationary model kinetic equations
A. V. Latyshev,A. A. Yushkanov +1 more
TL;DR: In this paper, a theory for constructing the solutions of boundary-value problems for nonstationary model kinetic equations is constructed, and an existence and uniqueness theorem for the expansion of the Laplace transform of the solution with respect to the eigenfunctions is proved.
Journal ArticleDOI
On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation
TL;DR: In this paper, the linear space-inhomogeneous Boltzmann equation for a distribution function in a bounded domain with general boundary conditions together with an external potential force is considered, and results on strong convergence to equilibrium, whent→∞, for general initial data are given.
References
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Light Scattering by Small Particles
H. C. Van de Hulst,V. Twersky +1 more
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),
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