Communications in Mathematical Physics

, Volume 130, Issue 3, pp 457–469 | Cite as

Bounded speed of propagation for solutions to radiative transfer equations

  • Benoît Perthame
  • Juan Luis Vazquez


The Radiative Transfer Equation is the nonlinear transport equation
$$\partial _t f + \frac{1}{\varepsilon }v \cdot \nabla _x f + \frac{1}{{\varepsilon ^2 }}\sigma (\tilde f)(f - \tilde f) = 0,$$
where\(\tilde f(x,{\mathbf{ }}t) = ff(x,{\mathbf{ }}v,t)dv\) 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 ∂ t u=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 inR N 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)=R n . 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.


Porous Medium Kinetic Equation Free Path Transport Equation Compact Support 
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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Benoît Perthame
    • 1
  • Juan Luis Vazquez
    • 2
  1. 1.Département de MathématiquesUniversité d'OrléansOrleans Cedex 02France
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

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