American Institute of Mathematical Sciences

January  2021, 14(1): 427-453. doi: 10.3934/dcdss.2020328

Stochastic homogenization of $\Lambda$-convex gradient flows

 1 Fakultät für Mathematik, Technische Universität München, Garching, Germany 2 Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany 3 Faculty of Mathematics, Technische Universität Dresden, Dresden, Germany

* Corresponding author: Mario Varga

This paper is dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  May 2019 Revised  October 2019 Published  January 2021 Early access  April 2020

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $\Lambda$-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $p$-Laplace operator with $p\in (1, \infty)$. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ($\Lambda$-)convex functionals.

Citation: Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $\Lambda$-convex gradient flows. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328
References:
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Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. [8] V. I. Bogachev, Measure Theory. Vol. I, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. [9] A. Bourgeat, A. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51. [10] H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Vol. 5, Elsevier, 1973. [11] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [12] D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of nonlinear integrals via the periodic unfolding method, C. R. Math. Acad. Sci. Paris, 339 (2004), 77-82.  doi: 10.1016/j.crma.2004.03.028. [13] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9. [14] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148. [15] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, Optimal transportation, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 413 (2014), 100–144. arXiv: 1009.3737. [16] F. Delarue and R. Rhodes, Stochastic homogenization of quasilinear PDEs with a spatial degeneracy, Asymptotic Analysis, 61 (2009), 61-90.  doi: 10.3233/ASY-2008-0925. [17] Y. Efendiev and A. Pankov, Homogenization of nonlinear random parabolic operators, Advances in Differential Equations, 10 (2005), 1235-1260. [18] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI., 2010. doi: 10.1090/gsm/019. [19] A. Faggionato, Random walks and exclusion processes among random conductances on random infinite clusters: Homogenization and hydrodynamic limit, Electron. J. Probab., 13 (2008), 2217-2247.  doi: 10.1214/EJP.v13-591. [20] T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal., 91 (2012), 1129-1154.  doi: 10.1080/00036811.2011.625016. [21] G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286. [22] H. Hanke and D. Knees, A phase-field damage model based on evolving microstructure, Asymptot. Anal., 101 (2017), 149-180.  doi: 10.3233/ASY-161396. [23] M. Heida, An extension of the stochastic two-scale convergence method and application, Asymptot. Anal., 72 (2011), 1-30.  doi: 10.3233/ASY-2010-1022. [24] M. 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Sci., 463 (2007), 2287-2307.  doi: 10.1098/rspa.2007.1876. [30] A. Mielke, Deriving amplitude equations via evolutionary $\Gamma$-convergence, Discrete Contin. Dyn. Syst., 35 (2015), 2679-2700.  doi: 10.3934/dcds.2015.35.2679. [31] A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, 3 (2016), 187-249.  doi: 10.1007/978-3-319-26883-5_3. [32] A. Mielke, S. Reichelt and M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw. Heterog. Media, 9 (2014), 353-382.  doi: 10.3934/nhm.2014.9.353. [33] A. Mielke, R. 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. [34] A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), 642-668.  doi: 10.1137/060672790. [35] S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, PhD thesis, Technische Universität München, 2010. [36] S. Neukamm and M. Varga, Stochastic unfolding and homogenization of spring network models, Multiscale Model. Simul., 16 (2018), 857-899.  doi: 10.1137/17M1141230. [37] S. Neukamm, M. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, arXiv: 1905.02945. [38] 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. [39] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043. [40] G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. Ⅰ, Ⅱ (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam-New York, 27 (1981), 835-873. [41] R. T. Rockafellar, Convex integral functionals and duality, Contributions to Nonlinear Functional Analysis, Academic Press, New York, (1971), 215–236. [42] R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614.  doi: 10.1051/cocv:2006013. [43] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Vol. 153, Springer Science & Business Media, 2013. [44] 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. [45] M. Sango and J. L. Woukeng, Stochastic $\Sigma$-convergence and applications, Dyn. Partial Differ. Equ., 8 (2011), 261–310. arXiv: 1106.0409. doi: 10.4310/DPDE.2011.v8.n4.a1. [46] 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. [47] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997. [48] U. Stefanelli, The Brézis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642.  doi: 10.1137/070684574. [49] M. Varga, Stochastic Unfolding and Homogenization of Evolutionary Gradient Systems, Dissertation, TU Dresden, 2019. Available from: https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-349342. [50] A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397.  doi: 10.1051/cocv:2006012. [51] J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales, Ann. Mat. Pura Appl., 189 (2010), 357-379.  doi: 10.1007/s10231-009-0112-y. [52] E. Zeidler, Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5020-3. [53] V. V. Zhikov, S. M. Kozlov and O. A. Oleǐnik, Averaging of parabolic operators, Trudy Moskovskogo Matematicheskogo Obshchestva, 45 (1982), 182–236. [54] V. V. Zhikov and A. L. Pyatnitskiǐ, Homogenization of random singular structures and random measures, Izv. Math., 70 (2006), 19-67.  doi: 10.1070/IM2006v070n01ABEH002302.

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References:
 [1] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Studies in Economic Theory, 4. Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-03004-2. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [3] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. [4] K. T. Andrews and S. Wright, Stochastic homogenization of elliptic boundary-value problems with $L^{p}$-data, Asymptot. Anal., 17 (1998), 165-184. [5] H. Attouch, Convergence de fonctionnelles convexes, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 1–40. [6] H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. [7] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. [8] V. I. Bogachev, Measure Theory. Vol. I, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. [9] A. Bourgeat, A. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51. [10] H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Vol. 5, Elsevier, 1973. [11] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [12] D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of nonlinear integrals via the periodic unfolding method, C. R. Math. Acad. Sci. Paris, 339 (2004), 77-82.  doi: 10.1016/j.crma.2004.03.028. [13] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9. [14] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148. [15] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, Optimal transportation, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 413 (2014), 100–144. arXiv: 1009.3737. [16] F. Delarue and R. Rhodes, Stochastic homogenization of quasilinear PDEs with a spatial degeneracy, Asymptotic Analysis, 61 (2009), 61-90.  doi: 10.3233/ASY-2008-0925. [17] Y. Efendiev and A. Pankov, Homogenization of nonlinear random parabolic operators, Advances in Differential Equations, 10 (2005), 1235-1260. [18] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI., 2010. doi: 10.1090/gsm/019. [19] A. Faggionato, Random walks and exclusion processes among random conductances on random infinite clusters: Homogenization and hydrodynamic limit, Electron. J. Probab., 13 (2008), 2217-2247.  doi: 10.1214/EJP.v13-591. [20] T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal., 91 (2012), 1129-1154.  doi: 10.1080/00036811.2011.625016. [21] G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286. [22] H. Hanke and D. Knees, A phase-field damage model based on evolving microstructure, Asymptot. Anal., 101 (2017), 149-180.  doi: 10.3233/ASY-161396. [23] M. Heida, An extension of the stochastic two-scale convergence method and application, Asymptot. Anal., 72 (2011), 1-30.  doi: 10.3233/ASY-2010-1022. [24] M. Heida, Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities, Applicable Analysis, 91 (2012), 1243-1264.  doi: 10.1080/00036811.2011.567191. [25] V. V. Jikov, S. M. Kozlov and O. A. Oleǐnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5. [26] A. Y. Kruger, On Fréchet subdifferentials, J. Math. Sci. (N.Y.), 116 (2003), 3325-3358.  doi: 10.1023/A:1023673105317. [27] M. Liero and S. Reichelt, Homogenization of Cahn-Hilliard-type equations via evolutionary $\Gamma$-convergence, Nonlinear Differ. Equat. Appl., 25 (2018), Art. 6, 31 pp. doi: 10.1007/s00030-018-0495-9. [28] D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86. [29] P. Mathieu and A. Piatnitski, Quenched invariance principles for random walks on percolation clusters, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2287-2307.  doi: 10.1098/rspa.2007.1876. [30] A. Mielke, Deriving amplitude equations via evolutionary $\Gamma$-convergence, Discrete Contin. Dyn. Syst., 35 (2015), 2679-2700.  doi: 10.3934/dcds.2015.35.2679. [31] A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, 3 (2016), 187-249.  doi: 10.1007/978-3-319-26883-5_3. [32] A. Mielke, S. Reichelt and M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw. Heterog. Media, 9 (2014), 353-382.  doi: 10.3934/nhm.2014.9.353. [33] A. Mielke, R. 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. [34] A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), 642-668.  doi: 10.1137/060672790. [35] S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, PhD thesis, Technische Universität München, 2010. [36] S. Neukamm and M. Varga, Stochastic unfolding and homogenization of spring network models, Multiscale Model. Simul., 16 (2018), 857-899.  doi: 10.1137/17M1141230. [37] S. Neukamm, M. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, arXiv: 1905.02945. [38] 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. [39] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043. [40] G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. Ⅰ, Ⅱ (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam-New York, 27 (1981), 835-873. [41] R. T. Rockafellar, Convex integral functionals and duality, Contributions to Nonlinear Functional Analysis, Academic Press, New York, (1971), 215–236. [42] R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614.  doi: 10.1051/cocv:2006013. [43] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Vol. 153, Springer Science & Business Media, 2013. [44] 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. [45] M. Sango and J. L. Woukeng, Stochastic $\Sigma$-convergence and applications, Dyn. Partial Differ. Equ., 8 (2011), 261–310. arXiv: 1106.0409. doi: 10.4310/DPDE.2011.v8.n4.a1. [46] 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. [47] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997. [48] U. Stefanelli, The Brézis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642.  doi: 10.1137/070684574. [49] M. Varga, Stochastic Unfolding and Homogenization of Evolutionary Gradient Systems, Dissertation, TU Dresden, 2019. Available from: https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-349342. [50] A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397.  doi: 10.1051/cocv:2006012. [51] J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales, Ann. Mat. Pura Appl., 189 (2010), 357-379.  doi: 10.1007/s10231-009-0112-y. [52] E. Zeidler, Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5020-3. [53] V. V. Zhikov, S. M. Kozlov and O. A. Oleǐnik, Averaging of parabolic operators, Trudy Moskovskogo Matematicheskogo Obshchestva, 45 (1982), 182–236. [54] V. V. Zhikov and A. L. Pyatnitskiǐ, Homogenization of random singular structures and random measures, Izv. Math., 70 (2006), 19-67.  doi: 10.1070/IM2006v070n01ABEH002302.
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