论文标题
由弱压缩剪切流驱动的Fokker-Planck方程的均质化和低调性
Homogenization and hypocoercivity for Fokker-Planck equations driven by weakly compressible shear flows
论文作者
论文摘要
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，该漂移可以通过一个大型剪切分量的总和分解，并根据一个空间变量而具有常规电位的梯度。该问题可以解释为被动标量略有可压缩的剪切流以及经历小扩散的标量。对于相应的随机微分方程，我们根据测量不可压缩扰动的强度的参数，根据时间尺度的家族给出明确的同质化速率。这是通过利用辅助泊松问题和计算相关有效扩散系数来实现的。关于Fokker-Planck方程解决方案解决方案的长期行为，我们通过采用经典低调方案的定量版本来为唯一不变的度量提供明确的衰减率。从流体力学的角度来看，这相当于量化增强扩散现象的略有压缩剪切流的现象。
We study the long-time dynamics of two-dimensional 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 behaviour 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.
