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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
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Existence of minimal nonsquare J-symmetric factorizations for self-adjoint rational matrix functions
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Spectrum and resolvent of a partial integral operator
TL;DR: In this paper, the authors consider the partial integral operator and derive norms of the resolvents of compact operators and operators with the compact Hermitian components for the Hilbert space.
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Existence Result for the Kinetic Neutron Transport Problem with a General Albedo Boundary Condition
Richard Sanchez,Lahbib Bourhrara +1 more
TL;DR: In this paper, the authors present an existence result for the kinetic neutron transport equation with a general albedo boundary condition. But their proof is constructive in the sense that they build a sequence that converges to the solution of the problem by iterating on the Albedo term.
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A photon transport problem with a time-dependent point source
TL;DR: In this paper, the authors considered a time-dependent problem of photon transport in an interstellar cloud with a point photon source modeled by a Dirac δ functional and established the existence of a unique distributional solution to this problem by using the theory of continuous semigroups of operators on locally convex spaces coupled with a constructive approach for producing spaces of generalized functions.
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The solution of the nonlinear Boltzmann equation: A survey of analytic and computational methods
TL;DR: A survey and a critical overview of the mathematical results, analytic and computational, on the solution of the nonlinear Boltzmann equation can be found in this article, where the topics dealt with in this paper are the following: mathematical formulation of initial and/or boundary value problems, existence theorems, computational treatment of fluid dynamical 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),
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