Fundamental solution of the cauchy problem corresponding to the one-speed linear Boltzman equation for anisotropic media

Abstract

We consider the fundamental solution E (t,x,s;s ₀) of the Cauchy problem for the one-speed linear Boltzman equation (∂/∂t+c(s,grad _x)+γ)E(t,x,s;s ₀)=γν∫ f((s, s′))E(t,x,s′; s₀)ds′+Ωδ(t)δ(x)δ (s−s ₀) that is assumed to be valid for any (t,x)∈Rⁿ⁺¹; morevoer, for t<0 the condition E(t,x,s; s₀)=0 holds. By using the Fourier-laplace transform in space-time arguments, the problem reduces to the study of an integral equation in the variables. For 0<ν≤1, the uniqueness and existence of the solution of the original problem are proved for any fixeds in the space of tempered distributions with supports in the front space-time cone. If the scattering media are of isotropic type (f(.)=1), the solution of the integral equation is given in explicit form. In the approximation of “small mean-free paths,” various weak limits of the solution are obtained with the help of a Tauberian-type theorem, for distributions. Bibliography: 4 titles.