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<p style='text-indent:20px;'>The free transport operator of probability density function <inline-formula><tex-math id="M2">\begin{document}$ f(t, x, v) $\end{document}</tex-math></inline-formula> is one the most fundamental operator which is widely used in many areas of PDE theory including kinetic theory, in particular. When it comes to general boundary problems in kinetic theory, however, it is well-known that high order regularity is very hard to obtain in general. In this paper, we study the free transport equation in a disk with the specular reflection boundary condition. We obtain initial-boundary compatibility conditions for <inline-formula><tex-math id="M3">\begin{document}$ C^{1}_{t, x, v} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ C^{2}_{t, x, v} $\end{document}</tex-math></inline-formula> regularity of the solution. We also provide regularity estimates.</p>