January  2021, 14(1): 395-425. doi: 10.3934/dcdss.2020345

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

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

* Corresponding author: Thomas Frenzel

Received  June 2019 Revised  December 2019 Published  May 2020

Fund Project: 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

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.

Citation: Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345
References:
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G. Dal MasoG. Franzina and D. Zucco, Transmission conditions obtained by homogenisation, Nonlinear Anal., 177 (2018), 361-386.  doi: 10.1016/j.na.2018.04.015.  Google Scholar

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P. Dondl, T. Frenzel and A. Mielke, A gradient system with a wiggly energy and relaxed EDP-convergence, ESAIM Control Optim. Calc. Var., 25 (2019), 45pp. doi: 10.1051/cocv/2018058.  Google Scholar

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A. Mielke, A. Montefusco and M. A. Peletier, Exploring families of energy-dissipation landscapes via tilting – three types of EDP convergence, preprint, arXiv: 2001.01455. Google Scholar

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A. MielkeM. A. Peletier and D. R. M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Anal., 41 (2014), 1293-1327.  doi: 10.1007/s11118-014-9418-5.  Google Scholar

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M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720.  doi: 10.1137/060665452.  Google Scholar

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F. Otto, Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141 (1998), 63-103.  doi: 10.1007/s002050050073.  Google Scholar

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F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.  Google Scholar

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[32]

M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker Inc., New York, 1991.  Google Scholar

[33]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.  Google Scholar

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S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31 (2011), 1427-1451.  doi: 10.3934/dcds.2011.31.1427.  Google Scholar

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U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642.  doi: 10.1137/070684574.  Google Scholar

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A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations, 47 (2013), 273-317.  doi: 10.1007/s00526-012-0519-y.  Google Scholar

show all references

References:
[1] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.   Google Scholar
[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. doi: 10.1007/b137080.  Google Scholar

[3]

S. ArnrichA. MielkeM. A. PeletierG. Savaré and M. Veneroni, Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction, Calc. Var. Partial Differential Equations, 44 (2012), 419-454.  doi: 10.1007/s00526-011-0440-9.  Google Scholar

[4]

H. Attouch and C. Picard, Comportement limite de problèmes de transmission unilateraux à travers des grilles de forme quelconque, Rend. Sem. Mat. Univ. Politec. Torino, 45 (1987), 71-85.   Google Scholar

[5]

A. Braides, A handbook of $\Gamma$-convergence, in Handbook of Differential Equations: Stationary Partial Differential Equations, Elsevier, 2006,101–213. Google Scholar

[6]

A. Braides, Local Minimization, Variational Evolution and $\Gamma$-Convergence, Lecture Notes in Mathematics, 2094, Springer, Cham, 2014. doi: 10.1007/978-3-319-01982-6.  Google Scholar

[7]

G. Dal MasoG. Franzina and D. Zucco, Transmission conditions obtained by homogenisation, Nonlinear Anal., 177 (2018), 361-386.  doi: 10.1016/j.na.2018.04.015.  Google Scholar

[8]

P. Dondl, T. Frenzel and A. Mielke, A gradient system with a wiggly energy and relaxed EDP-convergence, ESAIM Control Optim. Calc. Var., 25 (2019), 45pp. doi: 10.1051/cocv/2018058.  Google Scholar

[9]

M. Duchoň and P. Maličký, A Helly theorem for functions with values in metric spaces, Tatra Mt. Math. Publ., 44 (2009), 159-168.  doi: 10.2478/v10127-009-0056-z.  Google Scholar

[10]

M. Feinberg, On chemical kinetics of a certain class, Arch. Rational Mech. Anal., 46 (1972), 1-41.  doi: 10.1007/BF00251866.  Google Scholar

[11]

T. Frenzel, On the Derivation of Effective Gradient Systems via EDP-Convergence, Ph.D thesis, Humboldt Universität in Berlin, 2019. Google Scholar

[12]

A. N. GorbanI. V. KarlinV. B. Zmievskii and S. V. Dymova, Reduced description in the reaction kinetics, Phys. A: Statistical Mech. Appl., 275 (2000), 361-379.  doi: 10.1016/S0378-4371(99)00402-1.  Google Scholar

[13] M. Grmela, Chapter 2 - Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering.in Advances in Chemical Engineering, Advances in Chemical Engineering, 39,, Academic Press, 2010.  doi: 10.1016/S0065-2377(10)39002-8.  Google Scholar
[14]

R. JordanD. Kinderlehrer and F. Otto, Free energy and the Fokker-Planck equation. Landscape paradigms in physics and biology, Phys. D, 107 (1997), 265-271.  doi: 10.1016/S0167-2789(97)00093-6.  Google Scholar

[15]

M. LieroA. MielkeM. A. Peletier and D. R. M. Renger, On microscopic origins of generalized gradient structures, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1-35.  doi: 10.3934/dcdss.2017001.  Google Scholar

[16]

S. Lisini, Absolutely Continuous Curves in Wasserstein Spaces with Applications to Continuity Equation and to Nonlinear Diffusion Equations, Ph.D thesis, Universita degli Studi di Pavia, 2008. Google Scholar

[17]

S. Lisini, Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces, ESAIM Control Optim. Calc. Var., 15 (2009), 712-740.  doi: 10.1051/cocv:2008044.  Google Scholar

[18]

J. Lott, Some geometric calculations on Wasserstein space, Comm. Math. Phys., 277 (2008), 423-437.  doi: 10.1007/s00220-007-0367-3.  Google Scholar

[19]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.  doi: 10.1088/0951-7715/24/4/016.  Google Scholar

[20]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 479-499.  doi: 10.3934/dcdss.2013.6.479.  Google Scholar

[21]

A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems, in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Lect. Notes Appl. Math. Mech., 3, Springer, 2016,187–249. doi: 10.1007/978-3-319-26883-5_3.  Google Scholar

[22]

A. Mielke, A. Montefusco and M. A. Peletier, Exploring families of energy-dissipation landscapes via tilting – three types of EDP convergence, preprint, arXiv: 2001.01455. Google Scholar

[23]

A. MielkeM. A. Peletier and D. R. M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Anal., 41 (2014), 1293-1327.  doi: 10.1007/s11118-014-9418-5.  Google Scholar

[24]

A. MielkeR. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calc. Var. Partial Differential Equations, 46 (2013), 253-310.  doi: 10.1007/s00526-011-0482-z.  Google Scholar

[25]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720.  doi: 10.1137/060665452.  Google Scholar

[26]

L. Onsager, Reciprocal relations in irreversible processes. I, Phys. Rev., 37 (1931), 405-426.  doi: 10.1103/PhysRev.37.405.  Google Scholar

[27]

L. Onsager, Reciprocal relations in irreversible processes. II, Phys. Rev., 38 (1931), 2265-2279.  doi: 10.1103/PhysRev.38.2265.  Google Scholar

[28]

L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. (2), 91 (1953), 1505-1512.  doi: 10.1103/PhysRev.91.1505.  Google Scholar

[29]

F. Otto, Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141 (1998), 63-103.  doi: 10.1007/s002050050073.  Google Scholar

[30]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.  Google Scholar

[31]

M. A. Peletier, Variational modelling: Energies, gradient flows, and large deviations, preprint, arXiv: 1402.1990v1. Google Scholar

[32]

M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker Inc., New York, 1991.  Google Scholar

[33]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.  Google Scholar

[34]

S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31 (2011), 1427-1451.  doi: 10.3934/dcds.2011.31.1427.  Google Scholar

[35]

U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642.  doi: 10.1137/070684574.  Google Scholar

[36]

A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations, 47 (2013), 273-317.  doi: 10.1007/s00526-012-0519-y.  Google Scholar

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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