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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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Transport equations and perturbations of boundary conditions
TL;DR: In this article, a new perturbation theorem for substochastic semigroups on abstract AL spaces was proposed, which can be applied to piecewise deterministic Markov processes and transport equations with abstract boundary conditions.
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Brownian motion in a uniform force field. Solution of a boundary value problem for the stationary equation
Anne Boutet de Monvel,Petre Dita +1 more
TL;DR: In this article, the distribution function of the one-dimensional Brownian motion in a uniform force field with Dirichlet boundary conditions is found by reducing it to an integral equation.
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The well-posedness of the abstract kinetic equation boundary value problems
TL;DR: In this paper, the abstract boundary value problem on Hilbert space H is studied, where the maximum positive/negative spectral projections of the self-adjoint operator T are used as the boundary value.
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Transport Equation with Boundary Conditions Related to the Interior Solution
TL;DR: In this paper, it was shown that the transport equation is governed by a strongly continuous semigroup, which is the same as the boundary operator of the boundary condition in this paper.
Journal ArticleDOI
Some spectral properties in Banach spaces and application to transport theory
TL;DR: In this article, the notion of Bochner measurability was used to establish some properties concerning some classes of $$C_0$$ -semigroups in Banach spaces, and applied them in the framework of transport theory in order to obtain compactness properties giving a good comprehension of the time asymptotic behavior of the solutions for the associated Cauchy problems.
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),
Book
Theory of Ordinary Differential Equations
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