# American Institute of Mathematical Sciences

March  2020, 19(3): 1669-1695. doi: 10.3934/cpaa.2020061

## Homogenization of a locally periodic time-dependent domain

 Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

* Corresponding author

Received  February 2019 Revised  August 2019 Published  November 2019

We consider the homogenization of a Robin boundary value problem in a locally periodic perforated domain which is also time-dependent. We aim at justifying the homogenization limit, that we derive through asymptotic expansion technique. More exactly, we obtain the so-called corrector homogenization estimate that specifies the convergence rate. The major challenge is that the media is not cylindrical and changes over time. We also show the existence and uniqueness of solutions of the microscopic problem.

Citation: Morteza Fotouhi, Mohsen Yousefnezhad. Homogenization of a locally periodic time-dependent domain. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1669-1695. doi: 10.3934/cpaa.2020061
##### References:
 [1] G. A. Afrouzi and K. Brown, On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions, Proceedings of the American Mathematical Society, 127 (1999), 125-130.  doi: 10.1090/S0002-9939-99-04561-X. [2] G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.  doi: 10.1137/0523084. [3] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Model. Simul., 4 (2005), 790-812.  doi: 10.1137/040611239. [4] R. Barreira, C. M. Elliot and A. Madzvamuse, The surface finite element method for pattern formation on evolving biological surfaces, J. Math. Biol., 63 (2011), 1095-1119.  doi: 10.1007/s00285-011-0401-0. [5] J. Calvo, N. Matteo and O. Giandomenico, Parabolic equations in time-dependent domains, Journal of Evolution Equations, 17 (2017), 781-804.  doi: 10.1007/s00028-016-0336-4. [6] M. A. J. Chaplain, M. Ganesh and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth, J. Math. Biol., 42 (2001), 387-423.  doi: 10.1007/s002850000067. [7] G. A. Chechkin and A. L. Piatnitski, Homogenization of boundary-value problem in a locally periodic perforated domain, Appl. Anal., 71 (1999), 215-235.  doi: 10.1080/00036819908840714. [8] G. A. Chechkin, A. L. Piatnitski and A. S. Shamev, Homogenization: Methods and Applications, American Mathematical Soc., Vol. 234, 2007. [9] E. J. Crampin, W. W. Hackborn and P. K. Maini, Pattern formation in reaction-diffusion models with nonuniform domain growth, Bulletin of Mathematical Biology, 64 (2002), 747-769.  doi: 10.1006/bulm.2002.0295. [10] S. Dobberschutz, Homogenization Techniques for Lower Dimensional Structures, Doctoral dissertation, Bremen, Universität Bremen, Diss., 2012. [11] T. Giorgi and R. Smits, Eigenvalue estimates and critical temperature in zero fields for enhanced surface superconductivity, Zeitschrift für angewandte Mathematik und Physik, 58 (2007), 224-245.  doi: 10.1007/s00033-005-0049-y. [12] L. G. Harrison, S. Wehner and D. M. Holloway, Complex morphogenesis of surfaces: theory and experiment on coupling of reaction-diffusion patterning to growth, Faraday Discuss, 120 (2001), 277-294.  doi: 10.1039/b103246c. [13] U. Hornung, Homogenization and Porous Media, Springer Science & Business Media, Vol. 6, 2012. doi: 10.1007/978-1-4612-1920-0. [14] T. Hou, X. H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Mathematics of Computation, 68 (1999), 913-943.  doi: 10.1090/S0025-5718-99-01077-7. [15] J. E. Marsden and T. J. Hughes, Mathematical Foundations of Elasticity, Dover, 1994. [16] S. A. Meier, Two-Scale Models for Reactive Transport and Evolving Microstructure, PhD thesis, Universität Bremen, 2008. [17] S. A. Meier and M. Böhm, A note on the construction of function spaces for distributed-microstructure models with spatially varying cell geometry, Int. J. Numer. Anal. Model, 5 (2008), 109-125. [18] A. Muntean and T. L. Van Noorden, Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity., European Journal of Applied Mathematics, 24 (2013), 657-677.  doi: 10.1017/S0956792513000090. [19] C. Nicholson and S. Hrabětová, Brain extracellular space: The final frontier of neuroscience, Biophysical Journal, 113 (2017), 2133-2142.  doi: 10.1016/j.bpj.2017.06.052. [20] C. Nicholson, P. Kamali-Zare and L. Tao, Brain extracellular space as a diffusion barrier,, Computing and Visualization in Science, 14 (2011), 309-325.  doi: 10.1007/s00791-012-0185-9. [21] R. Nittka, Inhomogeneous parabolic Neumann problems, Czechoslovak Mathematical Journal, 64 (2014), 703-742.  doi: 10.1007/s10587-014-0127-4. [22] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623.  doi: 10.1137/0520043. [23] F. Paronetto, An existence result for evolution equations in non-cylindrical domains, Nonlinear Differential Equations and Applications, 20 (2013), 1723-1740.  doi: 10.1007/s00030-013-0227-0. [24] R. G. Plaza, F. Sánchez-Garduño, P. Padilla, R. A. Barrio and P. K. Maini, The effect of growth and curvature on pattern formation, Journal of Dynamics and Differential Equations, 16 (2004), 1093-1121.  doi: 10.1007/s10884-004-7834-8. [25] J. Pruss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhauser, Vol. 105, 2016. doi: 10.1007/978-3-319-27698-4. [26] M. Ptashnyk, Two-scale convergence for locally periodic microstructures and homogenization of plywood structures, Multiscale Modeling & Simulation, 11 (2013), 92–117. doi: 10.1137/120862338. [27] M. Ptashnyk, Locally periodic unfolding method and two-scale convergence on surfaces of locally periodic microstructures, Multiscale Modeling & Simulation, 13 (2015), 1061–1105. doi: 10.1137/140978405. [28] N. Ray, T. van Noorden, F. Frank and P. Knabner, Multiscale modeling of colloid and fluid dynamics in porous media including an evolving microstructure, Transp. Porous Media, 95 (2012), 669-696.  doi: 10.1007/s11242-012-0068-z. [29] N. Ray, T. L. van Noorden, F. A. Radu, W. Friess and P. Knabner, Drug release from collagen matrices including an evolving microstructure, ZAMM Z. Angew. Math. Mech., 93 (2013), 811-822.  doi: 10.1002/zamm.201200196. [30] S. Reichelt., Two-Scale Homogenization of Systems of Nonlinear Parabolic Equations, PhD thesis, University of Berlin, 2015. [31] R. Schulz and P. Knabner, Derivation and analysis of an effective model for biofilm growth in evolving porous media, Math. Methods Appl. Sci., 40 (2016), 2930-2948.  doi: 10.1002/mma.4211. [32] R. Schulz and P. Knabner, An effective model for biofilm growth made by chemotactical bacteria in evolving porous media, SIAM Journal on Applied Mathematics, 77 (2017), 1653-1677.  doi: 10.1137/16M108817X. [33] V. A. Solonnikov., On the boundary value problems for linear parabolic systems of differential equations of general form. Proceedings of the Steklov Institute of Mathematics, 83 (1965). (English translation by American Mathematical Society, 1967) [34] T. L. Van Noorden, Crystal precipitation and dissolution in a porous medium: effective equations and numerical experiments, Multiscale Modeling & Simulation, 7 (2009), 1220–1236. doi: 10.1137/080722096. [35] T. L. Van Noorden and A. Muntean, Homogenization of a locally periodic medium with areas of low and high diffusivity, European Journal of Applied Mathematics, 22 (2011), 493-516.  doi: 10.1017/S0956792511000209. [36] E. Weinan, Principles of Multiscale Modeling, Cambridge University Press, 2011. [37] M. Yousefnezhad, M. Fotouhi, K. Vejdani and P. Kamali-Zare, Unified model of brain tissue microstructure dynamically binds diffusion and osmosis with extracellular space geometry, Physical Review E, 94 (2016), 032411. doi: 10.1103/PhysRevE.94.032411.

show all references

##### References:
 [1] G. A. Afrouzi and K. Brown, On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions, Proceedings of the American Mathematical Society, 127 (1999), 125-130.  doi: 10.1090/S0002-9939-99-04561-X. [2] G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.  doi: 10.1137/0523084. [3] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Model. Simul., 4 (2005), 790-812.  doi: 10.1137/040611239. [4] R. Barreira, C. M. Elliot and A. Madzvamuse, The surface finite element method for pattern formation on evolving biological surfaces, J. Math. Biol., 63 (2011), 1095-1119.  doi: 10.1007/s00285-011-0401-0. [5] J. Calvo, N. Matteo and O. Giandomenico, Parabolic equations in time-dependent domains, Journal of Evolution Equations, 17 (2017), 781-804.  doi: 10.1007/s00028-016-0336-4. [6] M. A. J. Chaplain, M. Ganesh and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth, J. Math. Biol., 42 (2001), 387-423.  doi: 10.1007/s002850000067. [7] G. A. Chechkin and A. L. Piatnitski, Homogenization of boundary-value problem in a locally periodic perforated domain, Appl. Anal., 71 (1999), 215-235.  doi: 10.1080/00036819908840714. [8] G. A. Chechkin, A. L. Piatnitski and A. S. Shamev, Homogenization: Methods and Applications, American Mathematical Soc., Vol. 234, 2007. [9] E. J. Crampin, W. W. Hackborn and P. K. Maini, Pattern formation in reaction-diffusion models with nonuniform domain growth, Bulletin of Mathematical Biology, 64 (2002), 747-769.  doi: 10.1006/bulm.2002.0295. [10] S. Dobberschutz, Homogenization Techniques for Lower Dimensional Structures, Doctoral dissertation, Bremen, Universität Bremen, Diss., 2012. [11] T. Giorgi and R. Smits, Eigenvalue estimates and critical temperature in zero fields for enhanced surface superconductivity, Zeitschrift für angewandte Mathematik und Physik, 58 (2007), 224-245.  doi: 10.1007/s00033-005-0049-y. [12] L. G. Harrison, S. Wehner and D. M. Holloway, Complex morphogenesis of surfaces: theory and experiment on coupling of reaction-diffusion patterning to growth, Faraday Discuss, 120 (2001), 277-294.  doi: 10.1039/b103246c. [13] U. Hornung, Homogenization and Porous Media, Springer Science & Business Media, Vol. 6, 2012. doi: 10.1007/978-1-4612-1920-0. [14] T. Hou, X. H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Mathematics of Computation, 68 (1999), 913-943.  doi: 10.1090/S0025-5718-99-01077-7. [15] J. E. Marsden and T. J. Hughes, Mathematical Foundations of Elasticity, Dover, 1994. [16] S. A. Meier, Two-Scale Models for Reactive Transport and Evolving Microstructure, PhD thesis, Universität Bremen, 2008. [17] S. A. Meier and M. Böhm, A note on the construction of function spaces for distributed-microstructure models with spatially varying cell geometry, Int. J. Numer. Anal. Model, 5 (2008), 109-125. [18] A. Muntean and T. L. Van Noorden, Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity., European Journal of Applied Mathematics, 24 (2013), 657-677.  doi: 10.1017/S0956792513000090. [19] C. Nicholson and S. Hrabětová, Brain extracellular space: The final frontier of neuroscience, Biophysical Journal, 113 (2017), 2133-2142.  doi: 10.1016/j.bpj.2017.06.052. [20] C. Nicholson, P. Kamali-Zare and L. Tao, Brain extracellular space as a diffusion barrier,, Computing and Visualization in Science, 14 (2011), 309-325.  doi: 10.1007/s00791-012-0185-9. [21] R. Nittka, Inhomogeneous parabolic Neumann problems, Czechoslovak Mathematical Journal, 64 (2014), 703-742.  doi: 10.1007/s10587-014-0127-4. [22] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623.  doi: 10.1137/0520043. [23] F. Paronetto, An existence result for evolution equations in non-cylindrical domains, Nonlinear Differential Equations and Applications, 20 (2013), 1723-1740.  doi: 10.1007/s00030-013-0227-0. [24] R. G. Plaza, F. Sánchez-Garduño, P. Padilla, R. A. Barrio and P. K. Maini, The effect of growth and curvature on pattern formation, Journal of Dynamics and Differential Equations, 16 (2004), 1093-1121.  doi: 10.1007/s10884-004-7834-8. [25] J. Pruss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhauser, Vol. 105, 2016. doi: 10.1007/978-3-319-27698-4. [26] M. Ptashnyk, Two-scale convergence for locally periodic microstructures and homogenization of plywood structures, Multiscale Modeling & Simulation, 11 (2013), 92–117. doi: 10.1137/120862338. [27] M. Ptashnyk, Locally periodic unfolding method and two-scale convergence on surfaces of locally periodic microstructures, Multiscale Modeling & Simulation, 13 (2015), 1061–1105. doi: 10.1137/140978405. [28] N. Ray, T. van Noorden, F. Frank and P. Knabner, Multiscale modeling of colloid and fluid dynamics in porous media including an evolving microstructure, Transp. Porous Media, 95 (2012), 669-696.  doi: 10.1007/s11242-012-0068-z. [29] N. Ray, T. L. van Noorden, F. A. Radu, W. Friess and P. Knabner, Drug release from collagen matrices including an evolving microstructure, ZAMM Z. Angew. Math. Mech., 93 (2013), 811-822.  doi: 10.1002/zamm.201200196. [30] S. Reichelt., Two-Scale Homogenization of Systems of Nonlinear Parabolic Equations, PhD thesis, University of Berlin, 2015. [31] R. Schulz and P. Knabner, Derivation and analysis of an effective model for biofilm growth in evolving porous media, Math. Methods Appl. Sci., 40 (2016), 2930-2948.  doi: 10.1002/mma.4211. [32] R. Schulz and P. Knabner, An effective model for biofilm growth made by chemotactical bacteria in evolving porous media, SIAM Journal on Applied Mathematics, 77 (2017), 1653-1677.  doi: 10.1137/16M108817X. [33] V. A. Solonnikov., On the boundary value problems for linear parabolic systems of differential equations of general form. Proceedings of the Steklov Institute of Mathematics, 83 (1965). (English translation by American Mathematical Society, 1967) [34] T. L. Van Noorden, Crystal precipitation and dissolution in a porous medium: effective equations and numerical experiments, Multiscale Modeling & Simulation, 7 (2009), 1220–1236. doi: 10.1137/080722096. [35] T. L. Van Noorden and A. Muntean, Homogenization of a locally periodic medium with areas of low and high diffusivity, European Journal of Applied Mathematics, 22 (2011), 493-516.  doi: 10.1017/S0956792511000209. [36] E. Weinan, Principles of Multiscale Modeling, Cambridge University Press, 2011. [37] M. Yousefnezhad, M. Fotouhi, K. Vejdani and P. Kamali-Zare, Unified model of brain tissue microstructure dynamically binds diffusion and osmosis with extracellular space geometry, Physical Review E, 94 (2016), 032411. doi: 10.1103/PhysRevE.94.032411.
Schematic representation of a locally periodic heterogeneous medium in a time slice
Schematic a non-cylindrical domain approximated by a family of cylindrical domains
 [1] Cristian Barbarosie, Anca-Maria Toader. Optimization of bodies with locally periodic microstructure by varying the periodicity pattern. Networks and Heterogeneous Media, 2014, 9 (3) : 433-451. doi: 10.3934/nhm.2014.9.433 [2] Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279 [3] Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations and Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015 [4] Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485 [5] Mohamed Jleli, Bessem Samet. Blow-up for semilinear wave equations with time-dependent damping in an exterior domain. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3885-3900. doi: 10.3934/cpaa.2020143 [6] Erik Kropat. Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 51-76. doi: 10.3934/naco.2017003 [7] Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2659-2677. doi: 10.3934/dcds.2015.35.2659 [8] Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems and Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039 [9] Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic and Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011 [10] Masahiro Kubo, Noriaki Yamazaki. Periodic stability of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles. Conference Publications, 2007, 2007 (Special) : 614-623. doi: 10.3934/proc.2007.2007.614 [11] Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240 [12] Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053 [13] Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141 [14] G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks and Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1 [15] Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure and Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969 [16] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [17] Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control and Related Fields, 2013, 3 (2) : 143-160. doi: 10.3934/mcrf.2013.3.143 [18] Leonardo J. Colombo, María Emma Eyrea Irazú, Eduardo García-Toraño Andrés. A note on Hybrid Routh reduction for time-dependent Lagrangian systems. Journal of Geometric Mechanics, 2020, 12 (2) : 309-321. doi: 10.3934/jgm.2020014 [19] Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems and Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041 [20] Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068

2021 Impact Factor: 1.273