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

TL;DR

This paper proves global well-posedness and convergence to equilibrium for the Boltzmann equation in certain non-convex 3D domains by controlling billiard dynamics, addressing a longstanding open problem in kinetic theory.

Contribution

It introduces a new method to control billiard maps in non-convex 3D domains, enabling the first construction of global solutions to the Boltzmann equation in such settings.

Findings

Controlled billiard maps in non-convex domains.

Established global solutions to Boltzmann in 3D non-convex domains.

Developed a new coercivity method for the linearized collision operator.

Abstract

Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant progress in this question when domains are strictly convex, as shown by Guo and Kim-Lee, the same question without the strict convexity of domains is still totally open in 3D. 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 non-convex. In this paper, 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 non-convex 3-dimensional domains. In 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.