The Boundary Convergence of the Steady Zero-Temperature-Driven Hard Spheres

We study the fundamental problem of two gas species in two dimensional velocity space whose molecules collide as hard circles in the presence of a flat boundary and with dependence on only one space dimension. The case of three-dimensional velocity space is a generalization. More speciffically the linear problem arising when the second gas dominates as a flow with constant velocity (and hence zero temperature) is considered. The boundary condition adopted consists of prescribing the outgoing velocity distribution at the wall. It is discovered that the presence of the boundary under general assumptions on the outgoing distribution ensures the convergence of a series of path integrals and thus a convenient representation for the solution is obtained.