ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract 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 $$\tilde f(x,{\mathbf{ }}t) = ff(x,{\mathbf{ }}v,t)dv$$ 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,f ɛ 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 solutionsf ɛ of the RTE for all small ɛ. Moreover, the growth of the support off 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02096931
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