Bounded speed of propagation for solutions to radiative transfer equations
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.
scientific article; zbMATH DE number 4163426
scientific article