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. BarreiraC. 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. CalvoN. 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. ChaplainM. 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. CrampinW. 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. HarrisonS. 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. HouX. 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. NicholsonP. 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. PlazaF. Sánchez-GarduñoP. PadillaR. 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. RayT. van NoordenF. 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. RayT. L. van NoordenF. A. RaduW. 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. BarreiraC. 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. CalvoN. 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. ChaplainM. 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. CrampinW. 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. HarrisonS. 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. HouX. 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. NicholsonP. 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. PlazaF. Sánchez-GarduñoP. PadillaR. 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. RayT. van NoordenF. 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. RayT. L. van NoordenF. A. RaduW. 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.

Figure 1.  Schematic representation of a locally periodic heterogeneous medium in a time slice
Figure 2.  Schematic a non-cylindrical domain approximated by a family of cylindrical domains
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