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The Radiative Transfer Equation is the nonlinear transport equation(RTE)$$\partial _t f + \frac{1}{\varepsilon }v \cdot \nabla _x f + \frac{1}{{\varepsilon ^2 }}\sigma (\tilde f)(f - \tilde f) = 0,$$ where<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V6937206001P8U22_html\220_2005_Article_BF02096931_TeX2GIFIE1.gif" border="0" alt="$$\tilde f(x,{\mathbf{ }}t) = ff(x,{\mathbf{ }}v,t)dv$$" /> denotes the average off(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 ɛ goes to 0,fɛ converges to a solution of the Porous Medium Equation ∂tu=ΔF(u), withF′(u)=(Nσ(u))−1. Since σ blows up atu=0, solutions to the PME propagate with finite speed. Specifically ifu(·, 0) has compact support inRN so doesu(·, t) for everyt>0 and the sets Ω(t)={x∶u(x, t)>0}t>0 form an expanding family ast increases, andUΩ(t)=Rn. We show in this paper that these propagation properties hold for the solutionsfɛ of the RTE for all small ɛ. Moreover, the growth of the support offɛ is uniform in ɛ. 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, this arises only for particular initial data.

Bounded speed of propagation for solutions to radiative transfer equations