The main objective of this paper is to study the time decay of transport-diffusion equation with inhomogeneous localized damping in the multi-dimensional torus. The drift is governed by an autonomous Lipschitz vector field and the diffusion by the standard heat equation with small viscosity parameter $\nu$. In the first part we deal with the inviscid case and show some results on the time decay of the energy using in a crucial way the ergodicity and the unique ergodicity of the flow generated by the drift. In the second part we analyze the same problem with small viscosity and provide quite similar results on the exponential decay uniformly with respect to the viscosity in some logarithmic time scaling of the type $t \in [0, C_0 \: \mathrm{ln}(1 / \nu)]$.

中文翻译：

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遍历性对具有局部阻尼的输运扩散方程的影响

本文的主要目的是研究多维环面中具有非均匀局部阻尼的运输扩散方程的时间衰减。漂移由自主的Lipschitz矢量场控制，扩散由具有小的粘度参数$ \ nu $的标准热方程控制。在第一部分中，我们处理了无粘性的情况，并以关键的方式使用了漂移产生的流动的遍历性和唯一遍历性，显示了能量随时间衰减的一些结果。在第二部分中，我们分析了具有较小粘度的相同问题，并在某些类型为$ t \ in [0，C_0 \：\ mathrm {ln}的对数时间尺度中，就粘度的指数衰减均匀地提供了非常相似的结果。 （1 / \ nu）] $。