Radiative Heating of a Glass Plate

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 κ ( λ ) to be piecewise constant and null for small values of the wavelength λ 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 λ , 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.