Effective diffusion in thin structures via generalized gradient systems and EDP-convergence

Thomas Frenzel; Matthias Liero

doi:10.3934/dcdss.2020345

Article Contents

2021, Volume 14, Issue 1: 395-425. Doi: 10.3934/dcdss.2020345

This issue Previous Article Perturbed minimizing movements of families of functionals Next Article Stochastic homogenization of

$ \Lambda $

-convex gradient flows

Effective diffusion in thin structures via generalized gradient systems and EDP-convergence

Thomas Frenzel^, and
Matthias Liero

Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Mohrenstraße 39, 10117 Berlin, Germany

^* Corresponding author: Thomas Frenzel
^* Corresponding author: Thomas Frenzel

Received: June 2019

Revised: December 2019

Early access: May 2020

Published: January 2021

T.F. was partially supported by Deutsche Forschungsgemeinschaft (DFG) via the SFB 1114 Scaling Cascades in Complex Systems (subproject C05 "Effective models for materials and interfaces with multiple scales"). M.L. was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689) via project AA2-1.

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Abstract

The notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker–Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelin-de Donder kinetics.

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Mathematics Subject Classification: Primary: 35K20, 35K10; Secondary: 35K57, 49S99.

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Figure 1. Sketch of the domain $ \Omega_ \varepsilon\subset \mathbb{R}^d $ with cross section $ \Sigma\subset \mathbb{R}^{d-1} $. The diameter of $ \Sigma $ is considered to be large compared to the thickness of the domain $ \Omega_ \varepsilon $. The domain is decomposed into a top layer $ \Omega_ \varepsilon^+ $, center layer $ \Omega_ \varepsilon^0 $, and bottom layer $ \Omega_ \varepsilon^- $, whose thicknesses are given by $ \varepsilon $, $ \varepsilon^{1+\delta} $ for a fixed $ \delta>0 $, and $ \varepsilon $, respectively

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