Radiative heating of a glass plate. (English) Zbl 1328.35240
Summary: 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 et al. [“Application of a new method for radiative heat transfer to flat glass tempering”, J. Am. Ceram. Soc., 88, No. 8, 2181–2187 (2005)]. An important observation is that for a fixed value of the wavelength \(\lambda\), the 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 the existence of a solution, we define a fixed point problem related to our initial boundary value problem to which we apply the Schauder theorem in a closed convex subset of the Banach separable space \(L^2\left(0,t_f;C([0,l])\right)\). We use also the Stampacchia truncation method to derive lower and upper bounds on the solution.
MSC:
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K55 | Nonlinear parabolic equations |
| 35K58 | Semilinear parabolic equations |
| 35K90 | Abstract parabolic equations |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
Keywords:
elementary pencil of rays; Planck function; radiative transfer equation; glass plate; nonlinear heat-conduction equation; Stampacchia truncation method; Schauder theorem; Vitali theoremReferences:
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