任意空间维度下光滑速度场的最优混合与增强耗散
Optimal mixing and enhanced dissipation for a smooth velocity field in any spatial dimension
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中文总结 AI 辅助
本文在任意维度环面上构造光滑随机速度场，证明其作为通用指数混合器，并在小分子扩散下以最优速率增强耗散，方法基于随机动力系统和两点马尔可夫过程的均匀遍历性。
中文摘要 AI 辅助
本文考虑d维环面上的对流扩散方程，其中d为任意正整数。首先，对于相关的输运方程，我们构造了一个显式的时空光滑、无散度、随机速度场，该速度场是通用指数混合器。其次，我们将分子扩散系数设为（小的）正值，并证明同一速度场以最优速率增强耗散。我们的构造基于随机动力系统观点，并依赖于与二维Pierrehumbert模型相关的两点马尔可夫过程的均匀遍历性。
英文摘要
In this note, we consider the advection-diffusion equation on the $d$-dimensional torus, where $d$ is arbitrary. First, for the associated transport equation, we provide an explicit example of a space-time smooth, divergence-free, random velocity field that is a universal exponential mixer. Second, we set the molecular diffusivity to a (small) positive value, and we show that the same velocity field is dissipation enhancing with optimal rate. Our construction is based on the Random Dynamical Systems viewpoint, and relies on the uniform ergodicity of the two-point Markov process associated with the two-dimensional Pierrehumbert model.
发表机构
- Imperial College London（帝国理工学院）
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