# Bounded speed of propagation for solutions to radiative transfer equations

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## Abstract

The Radiative Transfer Equation is the nonlinear transport equation where\(\tilde f(x,{\mathbf{ }}t) = ff(x,{\mathbf{ }}v,t)dv\) denotes the average of

$$\partial _t f + \frac{1}{\varepsilon }v \cdot \nabla _x f + \frac{1}{{\varepsilon ^2 }}\sigma (\tilde f)(f - \tilde f) = 0,$$

(RTE)

*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 ɛ goes to 0,*f*_{ɛ}converges to a solution of the Porous Medium Equation ∂_{ t }*u*=*ΔF(u)*, with*F′(u)=(N*σ(*u*))^{−1}. Since σ blows up at*u*=0, solutions to the PME propagate with finite speed. Specifically if*u*(·, 0) has compact support in**R**^{ N }so does*u(·, t)*for every*t*>0 and the sets Ω*(t)={x∶u(x, t)>0}*_{ t>0 }form an expanding family as*t*increases, and*U*Ω(*t*)=**R**^{ n }. We show in this paper that these propagation properties hold for the solutions*f*_{ɛ}of the RTE for all small ɛ. Moreover, the growth of the support of*f*_{ɛ}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.## Keywords

Porous Medium Kinetic Equation Free Path Transport Equation Compact Support
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