We study the rate of convergence to equilibrium for a collisionless (Knudsen) gas enclosed in a vessel in dimension $ n \in \{2,3\} $. By semigroup arguments, we prove that in the $ L^1 $ norm, the polynomial rate of convergence $ \frac{1}{(t+1)^{n-}} $ given in [25], [17] and [18] can be extended to any $ C^2 $ domain, with standard assumptions on the initial data. This is to our knowledge, the first quantitative result in collisionless kinetic theory in dimension equal to or larger than 2 relying on deterministic arguments that does not require any symmetry of the domain, nor a monokinetic regime. The dependency of the rate with respect to the initial distribution is detailed. Our study includes the case where the temperature at the boundary varies. The demonstrations are adapted from a deterministic version of a subgeometric Harris' theorem recently established by Cañizo and Mischler [7]. We also compare our model with a free-transport equation with absorbing boundary.

中文翻译：

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半群方法求解无碰撞气体的收敛速度

我们研究了封装在尺寸为$ n \ in \ {2,3 \} $的容器中的无碰撞（Knudsen）气体的平衡收敛速率。通过半群论证，我们证明在$ L ^ 1 $范数中，多项式收敛速度$ \ frac {1} {（t + 1）^ {n-}} $在[25]，[17]和[18岁可以扩展到任何$ C ^ 2 $域，并且对初始数据有标准的假设。据我们所知，无碰撞动力学理论中第一个定量结果等于或大于2，它依赖于确定论点，该论点不需要域的任何对称性，也不需要单动力学机制。详细说明了速率相对于初始分布的依赖性。我们的研究包括边界温度变化的情况。这些示范取材于Cañizo和Mischler最近建立的亚几何哈里斯定理的确定性版本[7]。我们还将模型与具有吸收边界的自由运输方程进行比较。