On $C^2$ solution of the free-transport equation in a disk

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.