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Establishing global well-posedness and convergence toward equilibrium of theBoltzmann equation with specular reflection boundary condition has been one ofthe central questions in the subject of kinetic theory. Despite recentsignificant progress in this question when domains are strictly convex, asshown by Guo and Kim-Lee, the same question without the strict convexity ofdomains is still totally open in 3D. The major difficulty arises when abilliard map has an infinite number of bounces in a finite time interval orwhen the map fails to be Lipschitz continuous, both of which happen genericallywhen the domain is non-convex. In this paper, we develop a new method tocontrol a billiard map on a surface of revolution generated by revolving anyplanar analytic convex closed curve (e.g., typical shape of tokamak reactors'chamber). In particular, we classify and measure the size (to be small) of apullback set (along the billiard trajectory) of the infinite-bouncing andsingular-bouncing cases. As a consequence, we solve the open questionaffirmatively in such domains. To the best of our knowledge, this work is thefirst construction of global solutions to the hard-sphere Boltzmann equation ingeneric non-convex 3-dimensional domains. In Appendix, we introduce a novelmethod for constructive coercivity of a linearized collision operator $L$ whenthe specular boundary condition is imposed. In particular, this method worksfor a periodic cylindrical domain with an annulus cross-section.

Dynamical Billiard and a long-time behavior of the Boltzmann equation in general 3D toroidal domains