Time dependent linear transport I. Existence, uniqueness, and continuous dependence

Abstract

This paper concerns the time dependent linear transport equation posed in a multidimensional rectangular parallelepiped with partially reflecting walls. We consider the continuous transport equation and the discrete ordinate equations simultaneously. Our boundary condition, partial specular reflection, includes both vacuum and reflecting boundaries as special cases. We define strong and weak solutions of the problem, strong solutions being solutions in the ordinary sense and weak solutions being distributions, and show that a weak solution is a strong solution if it has space and time derivatives almost everywhere. For weak solutions we establish existence, uniqueness, and continuous dependence upon the initial data and the other functions which define the problem.