On c2 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 or-der regularity is very hard to obtain in general. In this paper, we study the free transport equation in a disk with the specular reflection boundary condi-tion. We obtain initial-boundary compatibility conditions for C1t,x,v and C2t,x,v regularity of the solution. We also provide regularity estimates.