We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a time-scale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation time-scales. The proofs are based on a contradiction argument that takes advantage of the cascading mechanism due to mixing, an estimate of the distance between the inviscid and viscous dynamics, and of an optimization step in the frequency cut-off. Thanks to the generality and robustness of our approach, we are able to apply our abstract results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation time-scales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation time-scale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion.

中文翻译：

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增强耗散时间尺度与混合率的关系

我们研究了不同线性流体动力学模型中的扩散和混合，主要与不可压缩流动有关。在这种情况下，混合是一种纯粹的平流效应，它导致能量转移到高频。当存在扩散时，混合会增强耗散力。这种现象被称为增强耗散，即比纯扩散更快的时间尺度识别。我们在负 Sobolev 范数的衰减和增强的耗散时间尺度方面建立了定量混合率之间的精确联系。证明基于一个矛盾论证，该论证利用了由于混合引起的级联机制、对无粘性和粘性动力学之间距离的估计以及频率截止中的优化步骤。由于我们方法的通用性和稳健性，我们能够将我们的抽象结果应用于许多问题。例如，我们证明接触 Anosov 流服从对数快速耗散时间尺度。据我们所知，这是第一个比多项式更快地引起增强耗散时间尺度的流的例子。其他应用包括平面和径向设置中的被动标量演化以及分数扩散。