W1,p REGULARITY IN TRANSPORT THEORY

1991 ◽  
Vol 01 (04) ◽  
pp. 477-499 ◽  
Author(s):  
MUSTAPHA MOKHTAR-KHARROUBI
Keyword(s):  
Fourier Analysis ◽  
Singular Integral ◽  
General Class ◽  
Transport Theory ◽  
Neutron Transport ◽  
Convex Domains ◽  
Integral Analysis ◽  
Neutron Transport Theory ◽  
Type C ◽  
Regularity Results

Let T be the streaming operator arising in neutron transport theory, T=−ν(∂/∂x)−σ(ν), with non-incoming boundary conditions in convex domains. The purpose of this paper is to give a general class [Formula: see text] of pair (C1, C2) of collision operators such that C1(λ−T)−1C2 maps continuously Lp into W1,p(1<p<+∞). The result is based on a singular integral analysis of Calderon-Zygmund type. In the case p=2 we obtain, by Fourier analysis, the same result under weaker assumptions. We also derive W1,p regularity results for operators of type C1(λ−T)−1.

Two-Group Neutron Transport Theory for Plane and Spherical Geometry

1970 ◽  
Vol 25 (10) ◽  
pp. 1370-1374 ◽  
Author(s):  
K. O. Thielheim ◽  
W. Blöcker
Keyword(s):  
Integral Equation ◽  
Singular Integral ◽  
Fredholm Integral Equation ◽  
Transport Theory ◽  
Neutron Transport ◽  
Spherical Geometry ◽  
Singular Integral Equations ◽  
Neutron Transport Theory ◽  
Critical Problems ◽  
Expansion Coefficients

Abstract Two-Group neutron transport theory is applied to critical problems in plane and spherical geometry. The neutron flux and the density transform for plane and spherical geometry respectively are expanded into singular eigenfunctions of the transport equation. With aid of the theory of singular integral equations the problem is reduced to one Fredholm integral equation for the expansion coefficients. The critical equations are presented as additional conditions.

scholarly journals Sobolev Regularity in Neutron Transport Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
Ahmed Zeghal
Keyword(s):  
Boundary Conditions ◽  
Lebesgue Measure ◽  
Periodic Boundary ◽  
Transport Theory ◽  
Neutron Transport ◽  
Periodic Boundary Conditions ◽  
Multidimensional Case ◽  
Neutron Transport Theory ◽  
Continuous Models ◽  
Regularity Results

The main purpose of this paper is to extend the W1,p regularity results in neutron transport theory, with respect to the Lebesgue measure due to Mokhtar-Kharroubi, (1991), and to abstract measures covering, in particular, the continuous models or multigroup models. The results are obtained for vacuum boundary conditions as well as periodic boundary conditions. H2 regularity results are derived when the velocity space is endowed with an appropriate class of measures (signed in the multidimensional case).

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