We study the long-time dynamics of 2D linear Fokker–Planck equations driven by a drift that can be decomposed in the sum of a large shear component and the gradient of a regular potential depending on one spatial variable. The problem can be interpreted as that of a passive scalar advected by a slightly compressible shear flow, and undergoing small diffusion. For the corresponding stochastic differential equation, we give explicit homogenization rates in terms of a family of time-scales depending on the parameter measuring the strength of the incompressible perturbation. This is achieved by exploiting an auxiliary Poisson problem, and by computing the related effective diffusion coefficients. Regarding the long-time behavior of the solution of the Fokker–Planck equation, we provide explicit decay rates to the unique invariant measure by employing a quantitative version of the classical hypocoercivity scheme. From a fluid mechanics perspective, this turns out to be equivalent to quantifying the phenomenon of enhanced diffusion for slightly compressible shear flows.

中文翻译：

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弱可压缩剪切流驱动的Fokker-Planck方程的均质化和矫顽力

我们研究了由漂移驱动的二维线性Fokker-Planck方程的长期动力学，该漂移可以分解成较大的剪切分量和规则势的梯度之和，具体取决于一个空间变量。该问题可以解释为被动标量的问题，该被动标量通过略微可压缩的剪切流平移，并经历很小的扩散。对于相应的随机微分方程，我们根据测量不可压缩扰动强度的参数，根据一系列时标给出明确的均化率。这是通过利用辅助泊松问题并通过计算相关的有效扩散系数来实现的。关于福克-普朗克方程解的长期行为，通过采用经典的低矫顽力方案的定量版本，我们为唯一不变性度量提供了明确的衰减率。从流体力学的角度来看，这等效于量化对于略微可压缩的剪切流的扩散增强现象。