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  January 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345
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. 
[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.

[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.

[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. 

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.
[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[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.

[31]

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

[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.

[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.

[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.

[35]

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

[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.

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. 
[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.

[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.

[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. 

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.
[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[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.

[31]

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

[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.

[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.

[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.

[35]

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

[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.

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
[1]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216

[2]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056

[3]

Jonathan Zinsl. The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 919-933. doi: 10.3934/dcdss.2017047

[4]

Mingying Zhong. Diffusion limit and the optimal convergence rate of the Vlasov-Poisson-Fokker-Planck system. Kinetic and Related Models, 2022, 15 (1) : 1-26. doi: 10.3934/krm.2021041

[5]

Anton Arnold, Beatrice Signorello. Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022009

[6]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems and Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033

[7]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks and Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027

[8]

Karoline Disser, Matthias Liero. On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks and Heterogeneous Media, 2015, 10 (2) : 233-253. doi: 10.3934/nhm.2015.10.233

[9]

Matthias Erbar, Max Fathi, Vaios Laschos, André Schlichting. Gradient flow structure for McKean-Vlasov equations on discrete spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6799-6833. doi: 10.3934/dcds.2016096

[10]

C.Y. Wang, M.X. Li. Convergence property of the Fletcher-Reeves conjugate gradient method with errors. Journal of Industrial and Management Optimization, 2005, 1 (2) : 193-200. doi: 10.3934/jimo.2005.1.193

[11]

Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507

[12]

Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427

[13]

Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389

[14]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic and Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165

[15]

Siniša Slijepčević. Extended gradient systems: Dimension one. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 503-518. doi: 10.3934/dcds.2000.6.503

[16]

Peter Benner, Tobias Breiten, Carsten Hartmann, Burkhard Schmidt. Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations. Journal of Computational Dynamics, 2020, 7 (1) : 1-33. doi: 10.3934/jcd.2020001

[17]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393

[18]

Nora Merabet. Global convergence of a memory gradient method with closed-form step size formula. Conference Publications, 2007, 2007 (Special) : 721-730. doi: 10.3934/proc.2007.2007.721

[19]

Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014

[20]

Gaohang Yu, Lutai Guan, Guoyin Li. Global convergence of modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property. Journal of Industrial and Management Optimization, 2008, 4 (3) : 565-579. doi: 10.3934/jimo.2008.4.565

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (187)
  • HTML views (299)
  • Cited by (1)

Other articles
by authors

[Back to Top]