Abstract
This paper is devoted to the study of the radiative transfer equations:
First, we prove a global existence theorem, which allows a blow-up of the opacity σv(ℰ) when ℰ→0. Thus, it extends Mercier's previous result [13]. This proof relies mainly on a nonlinear version of Hille-Yosida theorem: see Crandall-Ligett [9].
Then, we prove the uniqueness of the semigroup solving (TR), and some regularity results (in the class of functions with bounded variation).
Finally, we prove the convergence of some splitting algorithms associated to (TR).
Similar content being viewed by others
References
Barbu, V.: Non-linear semigroups theory and differential equations in Banach spaces. Noordhoff international publishing, (1976)
Badham, V.C., Larsen, E.W., Pomranning, G.C.: Asymptotic analysis of radiative transfer problems. J. Quant. Spectros. Radiat. Transfer29, 285–310 (1983)
Bardos, C.: Problèmes aux limites pour les équations aux dérivées partielles du ler ordre. Thèse Paris 1969. Ann. Sci. Ec. Norm. Super.4, 125–233 (1969)
Bénilan, Ph.: Equations d'évolution dans un espace de Banach quelconque et applications. Thèse Orsay 1972
Brézis, H., Pazy, A.: Convergence and approximation of semigroups of nonlinear operators in Banach spaces. J. Funct. Anal.9, 63–74 (1972)
Cessenat, M.: Théorèmes de traceL P pour des espaces de fonctions de la neutronique. Note C.R. Acad. Sci. Paris
Cessenat, M.: Théorèmes de trace pour des espaces de fonctions de la neutronique (2). Note C.R. Acad. Sci. Paris
Chandrasekhar, S.: Radiative transfer. Dover publications (1960)
Crandall, M.G., Ligett, T.S.: Generation of semigroups of nonlinear transformations on general Banach spaces. Am. J. Math.93, 265–297 (1971)
Evans, L.C.: Application of nonlinear semigroup theory to certain partial differential equations. In: Nonlinear evolution equations, pp. 265–297, M.G. Crandall (ed.). New York: Academic Press 1978
Golse, F., Perthame, B.: Existence globale d'une solution généralisée pour les équations du transfert radiatif. Note C.R. Acad. Sci. Paris299, Série I, 291–294 (1984)
Lions, P.L., Mercier, B.: Splitting algorithm for the sum of two maximal monotone operators. SIAM. J. Numer. Anal.16, 964–979 (1979)
Mercier, B.: Applications of accretive operator theory to the radiative transfer equations (to appear)
Mercier, B.: Influence de la troncature de l'opacité au voisinage dev=0, en relaxation rayonnement-matière. Note CEA 2387, January 1984
Pomranning, G.C.: Radiation hydrodynamics. Oxford: Pergamon Press
Treves, F.: Topological vector spaces, distributions and kernels. New York: Academic Press 1967
Author information
Authors and Affiliations
Additional information
Communicated by J. L. Lebowitz
Rights and permissions
About this article
Cite this article
Golse, F., Perthame, B. Generalized solutions of the radiative transfer equations in a singular case. Commun.Math. Phys. 106, 211–239 (1986). https://doi.org/10.1007/BF01454973
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01454973