Abstract

Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation near the global Maxwellian with specular reflection boundary condition has been one of the central questions in kinetic theory. Despite recent significant progress in this question when domains are strictly convex, as shown by Guo, Kim and Lee, the same question without the strict convexity of domains is still totally open in the three-dimensional case. The major difficulty arises when a billiard map has an infinite number of bounces in a finite time interval or when the map fails to be Lipschitz continuous, both of which happen generically when the domain is nonconvex. We develop a new method to control a billiard map on a surface of revolution generated by revolving any planar analytic convex closed curve (e.g., typical shape of tokamak reactors’ chamber). In particular, we classify and measure the size (to be small) of a pullback set (along the billiard trajectory) of the infinite-bouncing and singular-bouncing cases. As a consequence, we solve the open question affirmatively in such domains. To the best of our knowledge, this work is the first construction of global solutions to the hard-sphere Boltzmann equation in generic nonconvex three-dimensional domains. In the appendix, we introduce a novel method for constructive coercivity of a linearized collision operator $L$ when the specular boundary condition is imposed. In particular, this method works for a periodic cylindrical domain with an annulus cross section.

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