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Existence theory for radiative flows. (English) Zbl 1156.80371

Summary: We consider the coupling of radiative heat transfer equations and the energy equation for the temperature \(T\) of a compressible fluid occupying a bounded convex region \(D\) with smooth boundary. Using the technique of upper and lower sequences associated with integro-parabolic equations, we establish the existence and uniqueness of a solution \(T\), \(0\leq \Lambda_- \leq T(x,t) \leq \Lambda_+ < \infty\) with corresponding radiative intensity \(I(x,\Omega,\nu,t)\) where the total incident radiation satisfies \(\int_{S^2}I(x,\Omega,\nu,t)d\Omega = S_g B(\nu,t)+S_b B(\nu,T_b)\), and where \(S_b\) and \(S_g\) are positivity preserving linear operator, \(T_b\) is the external temperature of the boundary, and \(B\) is Planck’s function. We also establish certain energy estimates for \(T\).

MSC:

80A20 Heat and mass transfer, heat flow (MSC2010)
76R10 Free convection
78A40 Waves and radiation in optics and electromagnetic theory
45K05 Integro-partial differential equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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