×

Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. (English) Zbl 1458.35039

Summary: 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.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
35K10 Second-order parabolic equations
35K57 Reaction-diffusion equations
35Q84 Fokker-Planck equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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. · Zbl 1090.35002
[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. · Zbl 1270.35055
[3] S. Arnrich; A. Mielke; M. A. Peletier; G. Savaré; M. Veneroni, Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction, Calc. Var. Partial Differential Equations, 44, 419-454 (2012) · Zbl 0694.49006
[4] H. Attouch; C. Picard, Comportement limite de problèmes de transmission unilateraux à travers des grilles de forme quelconque, Rend. Sem. Mat. Univ. Politec. Torino, 45, 71-85 (1987) · Zbl 1195.35002
[5] A. Braides, A handbook of \(\Gamma \)-convergence, in Handbook of Differential Equations: Stationary Partial Differential Equations, Elsevier, 2006,101-213. · Zbl 1316.49002
[6] A. Braides, Local Minimization, Variational Evolution and \(\Gamma \)-Convergence, Lecture Notes in Mathematics, 2094, Springer, Cham, 2014. · Zbl 1410.35026
[7] G. Dal Maso; G. Franzina; D. Zucco, Transmission conditions obtained by homogenisation, Nonlinear Anal., 177, 361-386 (2018) · Zbl 1444.35101
[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. · Zbl 1287.28003
[9] M. Duchoň; P. Maličký, A Helly theorem for functions with values in metric spaces, Tatra Mt. Math. Publ., 44, 159-168 (2009)
[10] M. Feinberg, On chemical kinetics of a certain class, Arch. Rational Mech. Anal., 46, 1-41 (1972)
[11] T. Frenzel, On the Derivation of Effective Gradient Systems via EDP-Convergence, Ph.D thesis, Humboldt Universität in Berlin, 2019.
[12] A. N. Gorban; I. V. Karlin; V. B. Zmievskii; S. V. Dymova, Reduced description in the reaction kinetics, Phys. A: Statistical Mech. Appl., 275, 361-379 (2000) · Zbl 1029.82507
[13] M. Liero; A. Mielke; M. A. Peletier; D. R. M. Renger, On microscopic origins of generalized gradient structures, Discrete Contin. Dyn. Syst. Ser. S, 10, 1-35 (2017) · Zbl 1515.35127
[14] R. Jordan; D. Kinderlehrer; F. Otto, Free energy and the Fokker-Planck equation. Landscape paradigms in physics and biology, Phys. D, 107, 265-271 (1997)
[15] M. Liero; A. Mielke; M. A. Peletier; D. R. M. Renger, On microscopic origins of generalized gradient structures, Discrete Contin. Dyn. Syst. Ser. S, 10, 1-35 (2017) · Zbl 1178.35201
[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. · Zbl 1144.58007
[17] S. Lisini, Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces, ESAIM Control Optim. Calc. Var., 15, 712-740 (2009) · Zbl 1227.35161
[18] J. Lott, Some geometric calculations on Wasserstein space, Comm. Math. Phys., 277, 423-437 (2008) · Zbl 1262.35127
[19] A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24, 1329-1346 (2011)
[20] A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete Contin. Dyn. Syst. Ser. S, 6, 479-499 (2013)
[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. · Zbl 1304.35692
[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. · Zbl 1270.35289
[23] A. Mielke; M. A. Peletier; 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, 1293-1327 (2014) · Zbl 1145.35017
[24] A. Mielke; R. Rossi; G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calc. Var. Partial Differential Equations, 46, 253-310 (2013) · JFM 57.1168.10
[25] M. Neuss-Radu; W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39, 687-720 (2007) · Zbl 0004.18303
[26] L. Onsager, Reciprocal relations in irreversible processes. I, Phys. Rev., 37, 405-426 (1931) · Zbl 0053.15106
[27] L. Onsager, Reciprocal relations in irreversible processes. II, Phys. Rev., 38, 2265-2279 (1931) · Zbl 0905.35068
[28] L. Onsager; S. Machlup, Fluctuations and irreversible processes, Phys. Rev. (2), 91, 1505-1512 (1953) · Zbl 0984.35089
[29] F. Otto, Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141, 63-103 (1998)
[30] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26, 101-174 (2001)
[31] M. A. Peletier, Variational modelling: Energies, gradient flows, and large deviations, preprint, arXiv: 1402.1990v1. · Zbl 1065.49011
[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. · Zbl 1239.35015
[33] E. Sandier; S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57, 1627-1672 (2004) · Zbl 1194.35214
[34] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31, 1427-1451 (2011) · Zbl 1304.47073
[35] U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47, 1615-1642 (2008) · Zbl 1194.35214
[36] A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations, 47, 273-317 (2013) · Zbl 1304.47073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.