Fundamental solution of the Cauchy problem corresponding to one-speed linear Boltzman Equation for anisotropic media

Yu. B. Yanushanets

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 27, Tome 250 (1998), pp. 319-332

Résumé

We consider fundamental solution $E(t,\mathbf x,\mathbf s;\mathbf s_0)$ of the Cauchy problem for the one-speed linear Boltzman Equation $(\partial/\partial t +c(s,\operatorname{grad}_\mathbf x)+\gamma)E(t,\mathbf x,\mathbf s;\mathbf s_0)=\gamma\nu\int f\bigl((\mathbf s,\mathbf s')\bigr)E(t,\mathbf x,\mathbf s';\mathbf s_0)ds'+\Omega\delta(t)\delta(\mathbf x)\delta(\mathbf s-\mathbf s_0)$, assumed to be true for any $(t,\mathbf x)\in R^{n+1}$, while for $t0$ the condition $E(t,\mathbf x,\mathbf s;\mathbf s_0)=0$ holds. By using the Fourier–Laplace transform over space-time arguments the problem is reduced to investigation of an integral equation in the $\mathbf s$ argument. The uniqueness and existence of the initial problem for any fixed $\mathbf s$ within the space of tempered distributions with supports in the forward space-time cone are proved assuming $0\nu\le1$. If the scattering media are of isotropic type $f(.)=1$ the solution of the integral equation is given in the explicit form. In the limit of “small mean free paths” various weak limits of the solution are obtained with the help of tauberian type theorem for distributions.