The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation

We present an existence theory and an asymptotic analysis for the radiative transfer equations ∂u_ε ∂t+ Ω.{down triangle, open}_xu_ε ε+ σ(u_ε) ε²(u_ε-u_ε)=0 inX, (1) u_ε|_{(∂x × S^N)} = k, u_ε|_{t = 0} = u₀, where u_∈ u_∈(t, x, Ω), t ∈ R₊, x ∈ X ⊂ R^{n + 1}, Ω ∈; S^N, and u ̃_ε(t, x) = 1 |S^N| ∝ u_ε(t, x, Ω) dΩ. We prove that, even if σ has a singularity (σ(0) = +∞), (1) has a solution u_ε ε{lunate} L^∞(R₊ × X × S^N). As ε → 0, we show that u_ε converges pointwise to a function u ε{lunate} L^∞(R₊ × X), solution of the degenerate parabolic equation ∂u ∂t - ΔF(u) = 0 in X, u|_∂X = k, u|_{t = 0} = u₀. This is achieved without any monotonicity assumption on σ and therefore one cannot use the theory of nonlinear contraction semigroups.