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Original Articles

Positivity and monotonicity properties of transport equations with spatially dependent cross sections

Pages 199-215 | Published online: 01 Dec 2006

References

  • Larsen , E. W. 1981 . Progress in Nucl. Energy , 8 : 203
  • the report on the Seventh International Conference on Transport Theory . See also
  • 1981 . TTSP , 10 : 115
  • We call f:(O,τ) → H uniformly Höllder continuous if there exist 0<α<1 and a constant M such that ‖ f(x)-f(y) ‖ < M|x-y|α for all 0<x, y<τ. The number α is called the Hölder exponent
  • Case , K. M. and Zweifel , P. F. 1967 . “Linear Transport Theory , Addison Wesley . See also Refs. 17 and 25
  • Busoni , G. , Mangiarotti , L. and Frosali , G. 1979 . Rend. Circ. Mat. Palenno , 28 ( 2 ) : 91
  • Angelescu , N. and Protopescu , V. 1977 . Rev. Roum. Phys. , 22 : 1055 There exists work related to the present article, namely critical multigroup problems dealt with using positive semigroup theory. See
  • Larsen , E. W. 1979 . J. Math. Phys. , 20 : 1775
  • zhu , Yang Ming and Guangtian , Zhu . 1978 . Sci. Sinica , 21 : 298
  • Melson , P. 1971 . J. Math. Anal. Appl. , 35 : 90 where a cone preservation method is used. We also mention
  • Busbridge , I. W. 1961 . Ap. J. , 133 : 198
  • Kelley , C. T. 1980 . J. Integral Equations , 2 : 155
  • Tezcan , C. 1981 . TTSP , 10 : 105
  • Pomraning , G. C. 1981 . J. Math. Phys. , 22 : 1623
  • Krein , M. G. and Rutman , M. A. 1950 . A.M. S. Transl. , 26 : 199
  • 1948 . Uspehi Matem. Nauk , 3 ( 1 ) : 3
  • Krasnoselskii , M. A. 1964 . “Positive solutions of operator equations , Groningen : Noordhoff .
  • We call Ei(x) = ∫∞ 1 z−1 e−z|x|dz the exponential integral function
  • It is easy to compute that ‖ H(x) ‖ = sup {μ−1e−|x|/μ:0<μ⩽1}, which equals l/(ex) for |x|<1 and e−|x| for |x| ⩾ 1
  • Kato , T. 1966 . “Perturbation Theory for Linear Operators , Springer Verlag . The solutions of the Cauchy problems (9a) and (9b) are based on Theorem IX 1.27 of this book.Analytic semigroups are discussed
  • We make abuse of notation. Here writing TF±, does not mean that T could be splitted off
  • Zaanen , A. C. 1967 . “Integration , Amsterdam : North-Holland . Strong measurability is defined with respect to Lebesque measure. See Section VI. 31
  • van der Mee , C. V.M. 1981 . “Semigroup and Factorization Methods in Transport Theory , Amsterdam : Mathematical Centre Tract 146 .
  • Busbridge , I. W. 1960 . “The Mathematics of Radiative Transfer , Cambridge : Cambridge University Press . Also the references given there
  • Soholev , V. V. 1956 . Soviet Phys. Doklady , 111 : 1000 The equation has been studied in Refs.6
  • Refs. 9 and 10, K is called normal if there exists δ > 0 such that x,y ε K and ‖x‖ = ‖y‖=1 imply ‖x+y‖ ⩾ δ. The equivalence of the two definitions is due to I. A. Bakhtin and the proof can be found in Ref. 10
  • Karlin , S. 1959 . J. Math. Mech. , 8 : 907
  • Gohberg , I. C. and Feldman , I. A. 1971 . Convolution Equations and Projection Methods for their Solution . A.M.S. Transl. Monographs , : 41 Providence, R. I
  • Gohberg , I. , Kaashoek , M. A. and Lay , D. C. 1978 . J. Func. Anal. , 28 : 102 fact, (I-λDNCN)−1 = I+λDN(I-λDCN2)−1 CN and (I-λDCN2)−1 = I+λDCN(I-λDNCN)−1 N imply the statement. Further, as NCN is self-adjoint, R has real eigenvalues only.See, for instance, Theorem 4.5
  • Note that c(y)H(x−y)Bψ(y) ε K, 0<y<σ. So ∫τ σ c(y)H(x−y)Bψ(y)dy = 0 would imply Bψ(y) = <ψ(y), e > = 0 (0<y<τ). and thus, by (13), ψ ≡ 0, which is a contradiction
  • ∫+∞ −∞ Ei(z)dz = 2∫∞ 1 z−2dz = 2. See also Ref. (11)
  • Hopf , E. 1934 . Mathematical Problems of Radiative Equilibrium , Cambridge : Cambridge University Press .
  • Here we use the subscript τ of R, which we have suppressed hitherto
  • van de Hulst , H. C. 1980 . Multiple Light Scattering , Pergamon Press . The function Ei2 (x) = ∫∞ 1 z−2 e−|x|zdz has been described in the first chapter of:, as well as in the appendix of: S. Chandrasekhar, “Radiative Transfer”, Second revised edition, Dover (1960).

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