  The free transport operator of probability density function $f(t,x,v)$ is one
the most fundamental operator which is widely used in many areas of PDE theory
including kinetic theory, in particular. When it comes to general boundary
problems in kinetic theory, however, it is well-known that high order
regularity is very hard to obtain in general. In this paper, we study the free
transport equation in a disk with the specular boundary condition. We obtain
initial-boundary compatibility conditions for $C^1_{t,x,v}$ and $C^2_{t,x,v}$
regularity of the solution. We also provide regularity estimates.
