Homogenization and hypocoercivity for Fokker–Planck equations driven by weakly compressible shear flows

Autor: Grigorios A. Pavliotis, Michele Coti Zelati
Rok vydání: 2020
Předmět:
Spatial variable
DISSIPATION ENHANCEMENT
0199 Other Mathematical Sciences
Mathematics
Applied
homogenization
hypocoercivity
math.PR
01 natural sciences
Homogenization (chemistry)
010305 fluids & plasmas
PERIODIC HOMOGENIZATION
Stochastic differential equation
Mathematics - Analysis of PDEs
0102 Applied Mathematics
shear flows
0103 physical sciences
FOS: Mathematics
0101 mathematics
math.AP
Physics
enhanced diffusion
Science & Technology
TIME-SCALES
0103 Numerical and Computational Mathematics
Applied Mathematics
Probability (math.PR)
010102 general mathematics
Mathematical analysis
Fokker-Planck equation
Fluid mechanics
35B27
35K15
60J60
76F25
DIFFUSION
TRANSPORT
Physical Sciences
Compressibility
Fokker–Planck equation
Invariant measure
Shear flow
Mathematics - Probability
Mathematics
Analysis of PDEs (math.AP)
Zdroj: IMA Journal of Applied Mathematics. 85:951-979
ISSN: 1464-3634
0272-4960
DOI: 10.1093/imamat/hxaa035
Popis: 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.
Comment: 22 pages, 5 figures
Databáze: OpenAIRE
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