A basic question about the existence and stability of the Boltzmann equationin general non-convex domain with the specular reflection boundary conditionhas been widely open. In this paper, we consider cylindrical domains whosecross sections are general non-convex analytic planar domain. We establish theglobal-wellposedness and asymptotic stability of the Boltzmann equation withthe specular reflection boundary condition in such domains. Our method consistsof sharp classification of billiard trajectories which bounce infinitely manytimes or hit the boundary tangentially at some moment, and a delicateconstruction of an $\e$-tubular neighborhood of such trajectories. Analyticityof the boundary is crucially used. Away from such $\e$-tubular neighborhood, wecontrol the number of bounces of trajectories and its' distance from singularsets in a uniform fashion. The worst case, sticky grazing set, can be excludedby cutting off small portion of the temporal integration. Finally we apply amethod of \cite{KimLee} by the authors and achieve a pointwise estimate of theBoltzmann solutions.
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机译：关于具有镜面反射边界条件的一般非凸域中玻尔兹曼方程的存在性和稳定性的一个基本问题已经广为公开。在本文中，我们考虑其横截面为一般非凸解析平面域的圆柱域。我们在这样的域中建立了具有镜面反射边界条件的玻尔兹曼方程的整体适定性和渐近稳定性。我们的方法包括对台球轨迹进行清晰的分类，这些轨迹无数次反弹或在某个时刻切向触及边界，并且精细地构造了这种轨迹的\\ e $管状邻域。边界分析是至关重要的。远离这样的$ \ e $管状邻域，我们以统一的方式控制轨迹的反弹次数及其与奇异点的距离。最坏的情况是粘性放牧集，可以通过切除一小部分时间积分来排除。最后，我们采用作者的\ cite {KimLee}方法，并获得了玻尔兹曼解的逐点估计。
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