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The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation. (English) Zbl 0655.35075
Authors’ summary: We present an existence theory and an asymptotic analysis for the radiative transfer equations \[ (1)\quad \frac{\partial u_{\epsilon}}{\partial t}+\frac{\Omega \cdot \nabla_ xu_{\epsilon}}{\epsilon}+\frac{\sigma (\tilde u_{\epsilon})}{\epsilon^ 2}(u_{\epsilon}-\tilde u_{\epsilon})=0\quad in\quad X,\quad u_{\epsilon}|_{(\partial X\times S^ N)}=k,\quad u_{\epsilon}|_{t=0}=u_ 0, \] where \(u_{\epsilon}\equiv u_{\epsilon}(t,x,\Omega)\), \(t\in {\mathbb{R}}_+\), \(x\in X\subset {\mathbb{R}}^{N+1}\), \(\Omega \in S^ N\), and \(\tilde u_{\epsilon}(t,x)=1/| S^ N| \int u_{\epsilon}(t,x,\Omega)d\Omega.\) We prove that, even if \(\sigma\) has a singularity \((\sigma (0)=+\infty)\), (1) has a solution \(u_{\epsilon}\in L^{\infty}(R_+\times X\times S^ N)\). As \(\epsilon\) \(\to 0\), we show that \(u_{\epsilon}\) converges pointwise to a function \(u\in L^{\infty}(R_+\times X)\), solution of the degenerate parabolic equation \[ \partial u/\partial t-\Delta F(u)=0\quad in\quad X,\quad u|_{\partial X}=k,\quad u|_{t=0}=u_ 0. \] This is achieved without any monotonicity assumption on \(\sigma\) and therefore one cannot use the theory of nonlinear contraction semigroups.
Reviewer: U.F.Wodarzik

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
85A25 Radiative transfer in astronomy and astrophysics
35K65 Degenerate parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
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