Bounded speed of propagation for solutions to radiative transfer equations
type: Item
instance of: scholarly article
title: Bounded speed of propagation for solutions to radiative transfer equations
zbMATH Open document ID: 0708.35090
DOI: 10.1007/BF02096931
author: Juan Luis Vázquez
author: Ben{ôı}t Perthame
published in: Communications in Mathematical Physics
publication date: 1990-01-01T00:00:00Z
review text: The Radiative Transfer Equation is the nonlinear transport equation \[ (RTE)\quad \partial_ tf+\frac{1}{\epsilon}v\cdot \nabla_ xf+\frac{1}{\epsilon^ 2}\sigma (\tilde f)(f-\tilde f)=0, \] where \(\tilde f(x,t)=\int f(x,v,t)dv\) denotes the average of f(x,...,t) on the unit sphere: \(| v| =1\). It describes the absorption and emission of photons in a hot medium. As the mean free path \(\epsilon\) goes to 0, \(f_{\epsilon}\) converges to a solution of the Porous Medium Equation \(\partial_ tu=\Delta F(u)\), with \(F'(u)=(N\sigma (u))^{-1}\). Since \(\sigma\) blows up at \(u=0\), solutions to the PME propagate with finite speed. Specifically if u(\(\cdot,0)\) has compact support in \({\mathbb{R}}^ N\) so does u(\(\cdot,t)\) for every \(t>0\) and the sets \(\Omega (t)=\{x:u(x,t)>0\}_{t>0}\) form an expanding family as t increases, and \(U\Omega (t)=R^ n\). We show in this paper that these propagation properties hold for the solutions \(f_{\epsilon}\) of the RTE for all small \(\epsilon\). Moreover, the growth of the support of \(f_{\epsilon}\) is uniform in \(\epsilon\). Our proofs rely on the construction of explicit solutions (of the travelling wave type) and subsolutions to the RTE. To our knowledge, this is the first example of a kinetic equation with high velocities where localized data propagate always with bounded speed. For Vlasov-Poisson equations, it has been proved recently (Pfaffelmoser; Lions and Perthame) that such a phenomenon can arise also.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 4163426
zbMATH Keywords: Radiative Transfer Equation
zbMATH Keywords: nonlinear transport equation
zbMATH Keywords: Porous Medium Equation
zbMATH Keywords: propagation properties
Wikidata QID: Q60502969
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4163426