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Luc Paquet; Raouf El Cheikh; Dominique Lochegnies; Norbert Siedow
Radiative Heating of a Glass Plate
MathS In Action, 5 no. 1 (2012), p. 1-30, doi: 10.5802/msia.6
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Class. Math.: 35K20, 35K55, 35K58, 35K90, 35Q20, 35Q60, 35Q80
Mots clés: elementary pencil of rays, Planck function, radiative transfer equation, glass plate, nonlinear heat-conduction equation, Stampacchia truncation method, Schauder theorem, Vitali theorem.

Résumé - Abstract

This paper aims to prove existence and uniqueness of a solution to the coupling of a nonlinear heat equation with nonlinear boundary conditions with the exact radiative transfer equation, assuming the absorption coefficient $\kappa (\lambda )$ to be piecewise constant and null for small values of the wavelength $\lambda $ as in the paper of N. Siedow, T. Grosan, D. Lochegnies, E. Romero, “Application of a New Method for Radiative Heat Tranfer to Flat Glass Tempering”, J. Am. Ceram. Soc., 88(8):2181-2187 (2005). An important observation is that for a fixed value of the wavelength $\lambda $, Planck function is a Lipschitz function with respect to the temperature. Using this fact, we deduce that the solution is at most unique. To prove existence of a solution, we define a fixed point problem related to our initial boundary value problem to which we apply Schauder theorem in a closed convex subset of the Banach separable space $L^{2}(0,t_{f};C([0,l]))$. We use also Stampacchia truncation method to derive lower and upper bounds on the solution.

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