zbMATH — the first resource for mathematics

The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation. (English) Zbl 0655.35075
Authors’ summary: We present an existence theory and an asymptotic analysis for the radiative transfer equations \[ (1)\quad \frac{\partial u_{\epsilon}}{\partial t}+\frac{\Omega \cdot \nabla_ xu_{\epsilon}}{\epsilon}+\frac{\sigma (\tilde u_{\epsilon})}{\epsilon^ 2}(u_{\epsilon}-\tilde u_{\epsilon})=0\quad in\quad X,\quad u_{\epsilon}|_{(\partial X\times S^ N)}=k,\quad u_{\epsilon}|_{t=0}=u_ 0, \] where \(u_{\epsilon}\equiv u_{\epsilon}(t,x,\Omega)\), \(t\in {\mathbb{R}}_+\), \(x\in X\subset {\mathbb{R}}^{N+1}\), \(\Omega \in S^ N\), and \(\tilde u_{\epsilon}(t,x)=1/| S^ N| \int u_{\epsilon}(t,x,\Omega)d\Omega.\) We prove that, even if \(\sigma\) has a singularity \((\sigma (0)=+\infty)\), (1) has a solution \(u_{\epsilon}\in L^{\infty}(R_+\times X\times S^ N)\). As \(\epsilon\) \(\to 0\), we show that \(u_{\epsilon}\) converges pointwise to a function \(u\in L^{\infty}(R_+\times X)\), solution of the degenerate parabolic equation \[ \partial u/\partial t-\Delta F(u)=0\quad in\quad X,\quad u|_{\partial X}=k,\quad u|_{t=0}=u_ 0. \] This is achieved without any monotonicity assumption on \(\sigma\) and therefore one cannot use the theory of nonlinear contraction semigroups.
Reviewer: U.F.Wodarzik

35Q99 Partial differential equations of mathematical physics and other areas of application
85A25 Radiative transfer in astronomy and astrophysics
35K65 Degenerate parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] Albertoni, G; Montagnini, B, On the spectrum of neutron transport equation in finite bodies, J. math. anal. appl., 13, 19-48, (1966) · Zbl 0144.48304
[2] Bardos, C; Golse, F; Perthame, B, The rosseland approximation for the radiative transfer equations, Comm. pure appl. math., 40, 691-721, (1987) · Zbl 0654.65095
[3] Bardos, C; Santos, R; Sentis, R, Diffusion approximation and computation of the critical size, Trans. amer. math. soc., 284, 2, 617-649, (1984) · Zbl 0508.60067
[4] Benilan, P, Equations d’évolution dans un espace de Banach quelconque et applications, Thesis, (1972), Orsay
[5] Benilan, P; Brézis, H; Crandall, M.G, A semilinear equation in L1(R), Ann. scuola norm. sup. Pisa, 2, 523-555, (1975) · Zbl 0314.35077
[6] Bertsch, M, Thesis, (1984), Leyden
[7] Bensoussan, A; Lions, J.L; Papanicolaou, G.C, Boundary layers and homogenization of transport processes, J. publ. res. inst. math. sci. Kyoto, 15, 53-157, (1979) · Zbl 0408.60100
[8] \scK. M. Case and P. F. Zweifel, “Linear Transport Theory,” Addison-Wesley, Reading, M.A. · Zbl 0162.58903
[9] Cessenat, M, Théorème de trace pour des espaces de fonctions de la neutronique, C. R. acad. sci. Paris, 300, 1, 89-92, (1985) · Zbl 0648.46028
[10] Chandrasekhar, S, Radiative transfer, (1960), Dover New York · Zbl 0037.43201
[11] Crandall, M.G; Ligett, T.S, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. math., 93, 265-297, (1971)
[12] Dautray, R; Lions, J.L, ()
[13] Evans, L.C, Applications of nonlinear semigroup theory to certain partial differential equations, ()
[14] Frisch, H; Faurobert, M, Boundary layer conditions for the transport of radiation in stars, Astron. astrophys., 140, (1984)
[15] Golse, F, The Milne problem for the radiative transfer equations (with frequency dependence), Trans. amer. math. soc., 303, 125-143, (1987) · Zbl 0656.35017
[16] Golse, F; Lions, P.L; Perthame, B; Sentis, R, Regularity of the moments of the solution of a transport equation, J. funct. anal., 16, (1988)
[17] Golse, F; Perthame, B, Generalized solutions of the radiative transfer equations in a singular case, Comm. math. phys., 106, 211-239, (1986) · Zbl 0614.35084
[18] Golse, F; Perthame, B; Sentis, R, Un résultat de compacité pour LES équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport, C. R. acad. sci. Paris, Sér. I math., 301, 341-344, (1985) · Zbl 0591.45007
[19] Kharroubi, M.Mokhtar, Propriétés spectrales de l’opérateur de transport dans le cas anisotrope, () · Zbl 0538.45009
[20] Larsen, E.W; Keller, J.B, Asymptotic solutions of neutrons transport problem, J. math. phys., 15, 75-81, (1974)
[21] Larsen, E.W; Pomraning, G.C; Badham, V.C, Asymptotic analysis of radiative transfer problems, J. quant. spectros. radiat. transfer, 29, 285-310, (1983)
[22] Mercier, B, Applications of accretive operators theory to the radiative transfer equations, SIAM J. math. anal., 18, 393-408, (1987) · Zbl 0654.47036
[23] Mihalas, B.W; Mihalas, D, Foundations of radiative hydrodynamics, (1984), Pergamon New York · Zbl 0651.76005
[24] Peletier, L.A, The porous media equation, (), 229-241 · Zbl 0497.76083
[25] Pomraning, G.C, Radiation hydrodynamics, (1983), Pergamon Elmsford, NY · Zbl 0779.76084
[26] Sentis, R, ()
[27] Sentis, R, Half-space problems for frequency dependent transport equation. application to the rosseland approximation, J. transport theory stat. phys., 16, 653-700, (1987) · Zbl 0644.76100
[28] Vidav, I, Spectrum of perturbed semigroups with application to transport theory, J. math. anal. appl., 30, 244-279, (1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.