ON MULTI-DIMENSIONAL HYPOCOERCIVE BGK MODELS
ON MULTI-DIMENSIONAL HYPOCOERCIVE BGK MODELS
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DOI：
10.3934/krm.2018038
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发表时间：
2018-08-01
影响因子：
1
通讯作者：
Carlen, Eric A.
中科院分区：
文献类型：
--
作者：
Achleitner, Franz;Arnold, Anton;Carlen, Eric A.
本文研究了连续相空间中一类线性化BGK模型的hypovervical性。我们开发的熵泛函，使我们能够证明指数松弛与明确的和物理上有意义的速率平衡的方法。事实上，我们不仅估计了指数速率，而且还估计了第二个时间尺度，它决定了在L-1距离上开始看到指数弛豫之前必须等待的时间。当从相空间中高度集中的初始数据开始时，这种等待时间现象在指数衰减“开始”之前有一个很长的平台期，这在奥尔德斯和迪亚科尼斯关于马尔可夫链的工作中很常见，但在我们的连续相空间中是新的设置。我们的策略是基于熵和谱的方法，我们引入了一个新的“hypocolvity指数”，这是相关的模型，我们的类型涉及跳跃过程，而不仅仅是扩散。在我们的方法的核心是一个分解技术，使我们能够适应李雅普诺夫的直接方法，我们的连续相空间设置，以构建我们的熵泛函。这些用于获得线性化BGK模型的精确信息。最后，我们还证明了非线性BGK模型的局部渐近稳定性。
We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically meaningful rates. In fact, we not only estimate the exponential rate, but also the second time scale governing the time one must wait before one begins to see the exponential relaxation in the L-1 distance. This waiting time phenomenon, with a long plateau before the exponential decay "kicks in" when starting from initial data that is well-concentrated in phase space, is familiar from work of Aldous and Diaconis on Markov chains, but is new in our continuous phase space setting. Our strategies are based on the entropy and spectral methods, and we introduce a new "index of hypocoercivity" that is relevant to models of our type involving jump processes and not only diffusion. At the heart of our method is a decomposition technique that allows us to adapt Lyapunov's direct method to our continuous phase space setting in order to construct our entropy functionals. These are used to obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.