动力弹子球与一般三维环面域中 Boltzmann 方程的长时间行为
Dynamical Billiard and a long-time behavior of the Boltzmann equation in general 3D toroidal domains
AI总结：
本文研究非凸三维环面域中镜面反射 Boltzmann 方程的整体适定性，通过分类并估计异常弹子球轨迹回拉集的小测度，首次构造硬球模型整体解并证明平衡收敛。
摘要：
在镜面反射边界条件下，建立 Boltzmann 方程的整体适定性及其向平衡态的收敛，一直是动理学理论的核心问题之一。尽管如 Guo 和 Kim-Lee 所示，近期在区域为严格凸的情形下对该问题取得了重大进展，但在三维中，不要求区域严格凸的同一问题仍完全开放。主要困难出现在弹子球映射在有限时间区间内发生无穷次反弹，或该映射不满足 Lipschitz 连续性时；当区域非凸时，这两种情况通常都会发生。本文发展了一种新方法，用于控制由任意平面解析凸闭曲线旋转生成的旋转曲面上的弹子球映射（例如托卡马克反应堆腔室的典型形状）。特别地，我们对无穷次反弹和奇异反弹情形的（沿弹子球轨迹的）回拉集进行分类，并度量其大小（证明其很小）。因此，我们在此类区域中肯定地解决了这一开放问题。据我们所知，这项工作首次在一般非凸三维区域中构造了硬球 Boltzmann 方程的整体解。附录中，我们引入了一种新方法，用于在施加镜面边界条件时构造线性化碰撞算子 $L$ 的强制性。特别地，该方法适用于截面为环形的周期柱形区域。
英文摘要：
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.
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