Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data
Authors:
Vishnu Raveendran, Ida de Bonis, Emilio N. M. Cirillo and Adrian Muntean
Journal:
Quart. Appl. Math. 83 (2025), 19-57
MSC (2020):
Primary 35B27, 35B45, 35Q92
DOI:
https://doi.org/10.1090/qam/1687
Published electronically:
February 8, 2024
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.
References
- A. Abdulle and M. E. Huber, Numerical homogenization method for parabolic advection-diffusion multiscale problems with large compressible flows, Numer. Math. 136 (2017), no. 3, 603–649. MR 3660297, DOI 10.1007/s00211-016-0854-6
- E. Acerbi, V. Chiadò Piat, G. Dal Maso, and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal. 18 (1992), no. 5, 481–496. MR 1152723, DOI 10.1016/0362-546X(92)90015-7
- Grégoire Allaire, Andro Mikelić, and Andrey Piatnitski, Homogenization approach to the dispersion theory for reactive transport through porous media, SIAM J. Math. Anal. 42 (2010), no. 1, 125–144. MR 2596548, DOI 10.1137/090754935
- G. Allaire, S. Desroziers, G. Enchéry, and F. Ouaki, A multiscale finite element method for transport modelling, CD-ROM Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering, Vienna University of Technology, Austria, 2012.
- G. Allaire and H. Hutridurga, Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA J. Appl. Math. 77 (2012), no. 6, 788–815. MR 2999138, DOI 10.1093/imamat/hxs049
- Grégoire Allaire and Harsha Hutridurga, Upscaling nonlinear adsorption in periodic porous media—homogenization approach, Appl. Anal. 95 (2016), no. 10, 2126–2161. MR 3523144, DOI 10.1080/00036811.2015.1038254
- Grégoire Allaire, Andro Mikelić, and Andrey Piatnitski, Homogenization approach to the dispersion theory for reactive transport through porous media, SIAM J. Math. Anal. 42 (2010), no. 1, 125–144. MR 2596548, DOI 10.1137/090754935
- G. Allaire, I. Pankratova, and A. Piatnitski, Homogenization and concentration for a diffusion equation with large convection in a bounded domain, J. Funct. Anal. 262 (2012), no. 1, 300–330. MR 2852263, DOI 10.1016/j.jfa.2011.09.014
- Jacob Bear and Leonid G. Fel, A phenomenological approach to modeling transport in porous media, Transp. Porous Media 92 (2012), no. 3, 649–665. MR 2899436, DOI 10.1007/s11242-011-9926-3
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829, DOI 10.1007/978-0-387-70914-7
- Emilio N. M. Cirillo, Ida de Bonis, Adrian Muntean, and Omar Richardson, Upscaling the interplay between diffusion and polynomial drifts through a composite thin strip with periodic microstructure, Meccanica 55 (2020), no. 11, 2159–2178. MR 4173437, DOI 10.1007/s11012-020-01253-8
- Emilio N. M. Cirillo, Oleh Krehel, Adrian Muntean, Rutger van Santen, and Aditya Sengar, Residence time estimates for asymmetric simple exclusion dynamics on strips, Phys. A 442 (2016), 436–457. MR 3412980, DOI 10.1016/j.physa.2015.09.037
- Martin Hairer and Charles Manson, Periodic homogenization with an interface: the multi-dimensional case, Ann. Probab. 39 (2011), no. 2, 648–682. MR 2789509, DOI 10.1214/10-AOP564
- Patrick Henning and Mario Ohlberger, The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift, Netw. Heterog. Media 5 (2010), no. 4, 711–744. MR 2740530, DOI 10.3934/nhm.2010.5.711
- Patrick Henning and Mario Ohlberger, A note on homogenization of advection-diffusion problems with large expected drift, Z. Anal. Anwend. 30 (2011), no. 3, 319–339. MR 2819498, DOI 10.4171/ZAA/1437
- Thomas Holding, Harsha Hutridurga, and Jeffrey Rauch, Convergence along mean flows, SIAM J. Math. Anal. 49 (2017), no. 1, 222–271. MR 3598785, DOI 10.1137/16M1068657
- H. Hutridurga, Homogenization of complex flows in porous media and applications, Ph.D. thesis, École Polytechnique, Palaiseau, France, 2013.
- Ekeoma R. Ijioma and Adrian Muntean, Fast drift effects in the averaging of a filtration combustion system: a periodic homogenization approach, Quart. Appl. Math. 77 (2019), no. 1, 71–104. MR 3897920, DOI 10.1090/qam/1509
- O. Iliev, A. Mikelić, T. Prill, and A. Sherly, Homogenization approach to the upscaling of a reactive flow through particulate filters with wall integrated catalyst, Adv. Water Resour. 146 (2020), 103779.
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 244627
- S. Liu and J. H. Masliyah, Dispersion in Porous Media, Handbook of Porous Media (2nd ed.) (K. Vafai, ed.), CRC Press, 2005, pp. 81–141.
- Dag Lukkassen, Gabriel Nguetseng, and Peter Wall, Two-scale convergence, Int. J. Pure Appl. Math. 2 (2002), no. 1, 35–86. MR 1912819
- Eduard Maru ić-Paloka and Andrey L. Piatnitski, Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection, J. London Math. Soc. (2) 72 (2005), no. 2, 391–409. MR 2156660, DOI 10.1112/S0024610705006824
- D. W. McLaughlin, G. C. Papanicolaou, and O. R. Pironneau, Convection of microstructure and related problems, SIAM J. Appl. Math. 45 (1985), no. 5, 780–797. MR 804006, DOI 10.1137/0145046
- F. Ouaki, Étude de schémas multi-échelles pour la simulation de réservoir, Ph.D. thesis, École Polytechnique, Palaiseau, France, 2013.
- Iryna Pankratova and Andrey Piatnitski, Homogenization of convection-diffusion equation in infinite cylinder, Netw. Heterog. Media 6 (2011), no. 1, 111–126. MR 2777012, DOI 10.3934/nhm.2011.6.111
- Andrey Piatnitski and Mariya Ptashnyk, Homogenization of biomechanical models of plant tissues with randomly distributed cells, Nonlinearity 33 (2020), no. 10, 5510–5542. MR 4151416, DOI 10.1088/1361-6544/ab95ab
- Vishnu Raveendran, Emilio N. M. Cirillo, Ida de Bonis, and Adrian Muntean, Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer, Quart. Appl. Math. 80 (2022), no. 1, 157–200. MR 4360553, DOI 10.1090/qam/1607
- Vishnu Raveendran, Emilio N. M. Cirillo, and Adrian Muntean, Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift, Quart. Appl. Math. 80 (2022), no. 4, 641–667. MR 4489000, DOI 10.1090/qam/1622
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 365062
- Y. Zhang, Quantitative estimates in elliptic homogenization of non-divergence form with unbounded drift and an interface, preprint arXiv:2301.01411 (2023), 1–34.
References
- A. Abdulle and M. E. Huber, Numerical homogenization method for parabolic advection-diffusion multiscale problems with large compressible flows, Numer. Math. 136 (2017), no. 3, 603–649. MR 3660297, DOI 10.1007/s00211-016-0854-6
- E. Acerbi, V. Chiadò Piat, G. Dal Maso, and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal. 18 (1992), no. 5, 481–496. MR 1152723, DOI 10.1016/0362-546X(92)90015-7
- Grégoire Allaire, Andro Mikelić, and Andrey Piatnitski, Homogenization approach to the dispersion theory for reactive transport through porous media, SIAM J. Math. Anal. 42 (2010), no. 1, 125–144. MR 2596548, DOI 10.1137/090754935
- G. Allaire, S. Desroziers, G. Enchéry, and F. Ouaki, A multiscale finite element method for transport modelling, CD-ROM Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering, Vienna University of Technology, Austria, 2012.
- G. Allaire and H. Hutridurga, Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA J. Appl. Math. 77 (2012), no. 6, 788–815. MR 2999138, DOI 10.1093/imamat/hxs049
- Grégoire Allaire and Harsha Hutridurga, Upscaling nonlinear adsorption in periodic porous media—homogenization approach, Appl. Anal. 95 (2016), no. 10, 2126–2161. MR 3523144, DOI 10.1080/00036811.2015.1038254
- Grégoire Allaire, Andro Mikelić, and Andrey Piatnitski, Homogenization approach to the dispersion theory for reactive transport through porous media, SIAM J. Math. Anal. 42 (2010), no. 1, 125–144. MR 2596548, DOI 10.1137/090754935
- G. Allaire, I. Pankratova, and A. Piatnitski, Homogenization and concentration for a diffusion equation with large convection in a bounded domain, J. Funct. Anal. 262 (2012), no. 1, 300–330. MR 2852263, DOI 10.1016/j.jfa.2011.09.014
- Jacob Bear and Leonid G. Fel, A phenomenological approach to modeling transport in porous media, Transp. Porous Media 92 (2012), no. 3, 649–665. MR 2899436, DOI 10.1007/s11242-011-9926-3
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
- Emilio N. M. Cirillo, Ida de Bonis, Adrian Muntean, and Omar Richardson, Upscaling the interplay between diffusion and polynomial drifts through a composite thin strip with periodic microstructure, Meccanica 55 (2020), no. 11, 2159–2178. MR 4173437, DOI 10.1007/s11012-020-01253-8
- Emilio N. M. Cirillo, Oleh Krehel, Adrian Muntean, Rutger van Santen, and Aditya Sengar, Residence time estimates for asymmetric simple exclusion dynamics on strips, Phys. A 442 (2016), 436–457. MR 3412980, DOI 10.1016/j.physa.2015.09.037
- Martin Hairer and Charles Manson, Periodic homogenization with an interface: the multi-dimensional case, Ann. Probab. 39 (2011), no. 2, 648–682. MR 2789509, DOI 10.1214/10-AOP564
- Patrick Henning and Mario Ohlberger, The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift, Netw. Heterog. Media 5 (2010), no. 4, 711–744. MR 2740530, DOI 10.3934/nhm.2010.5.711
- Patrick Henning and Mario Ohlberger, A note on homogenization of advection-diffusion problems with large expected drift, Z. Anal. Anwend. 30 (2011), no. 3, 319–339. MR 2819498, DOI 10.4171/ZAA/1437
- Thomas Holding, Harsha Hutridurga, and Jeffrey Rauch, Convergence along mean flows, SIAM J. Math. Anal. 49 (2017), no. 1, 222–271. MR 3598785, DOI 10.1137/16M1068657
- H. Hutridurga, Homogenization of complex flows in porous media and applications, Ph.D. thesis, École Polytechnique, Palaiseau, France, 2013.
- Ekeoma R. Ijioma and Adrian Muntean, Fast drift effects in the averaging of a filtration combustion system: a periodic homogenization approach, Quart. Appl. Math. 77 (2019), no. 1, 71–104. MR 3897920, DOI 10.1090/qam/1509
- O. Iliev, A. Mikelić, T. Prill, and A. Sherly, Homogenization approach to the upscaling of a reactive flow through particulate filters with wall integrated catalyst, Adv. Water Resour. 146 (2020), 103779.
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 244627
- S. Liu and J. H. Masliyah, Dispersion in Porous Media, Handbook of Porous Media (2nd ed.) (K. Vafai, ed.), CRC Press, 2005, pp. 81–141.
- Dag Lukkassen, Gabriel Nguetseng, and Peter Wall, Two-scale convergence, Int. J. Pure Appl. Math. 2 (2002), no. 1, 35–86. MR 1912819
- Eduard Maru ić-Paloka and Andrey L. Piatnitski, Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection, J. London Math. Soc. (2) 72 (2005), no. 2, 391–409. MR 2156660, DOI 10.1112/S0024610705006824
- D. W. McLaughlin, G. C. Papanicolaou, and O. R. Pironneau, Convection of microstructure and related problems, SIAM J. Appl. Math. 45 (1985), no. 5, 780–797. MR 804006, DOI 10.1137/0145046
- F. Ouaki, Étude de schémas multi-échelles pour la simulation de réservoir, Ph.D. thesis, École Polytechnique, Palaiseau, France, 2013.
- Iryna Pankratova and Andrey Piatnitski, Homogenization of convection-diffusion equation in infinite cylinder, Netw. Heterog. Media 6 (2011), no. 1, 111–126. MR 2777012, DOI 10.3934/nhm.2011.6.111
- Andrey Piatnitski and Mariya Ptashnyk, Homogenization of biomechanical models of plant tissues with randomly distributed cells, Nonlinearity 33 (2020), no. 10, 5510–5542. MR 4151416, DOI 10.1088/1361-6544/ab95ab
- Vishnu Raveendran, Emilio N. M. Cirillo, Ida de Bonis, and Adrian Muntean, Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer, Quart. Appl. Math. 80 (2022), no. 1, 157–200. MR 4360553, DOI 10.1090/qam/1607
- Vishnu Raveendran, Emilio N. M. Cirillo, and Adrian Muntean, Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift, Quart. Appl. Math. 80 (2022), no. 4, 641–667. MR 4489000
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 365062
- Y. Zhang, Quantitative estimates in elliptic homogenization of non-divergence form with unbounded drift and an interface, preprint arXiv:2301.01411 (2023), 1–34.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2020):
35B27,
35B45,
35Q92
Retrieve articles in all journals
with MSC (2020):
35B27,
35B45,
35Q92
Additional Information
Vishnu Raveendran
Affiliation:
Department of Mathematics and Computer Science, Karlstad University, Universitetsgatan 2, 651 88 Karlstad, Sweden
MR Author ID:
1483393
ORCID:
0000-0001-5168-0841
Email:
vishnu.raveendran@kau.se
Ida de Bonis
Affiliation:
Dipartimento di Pianificazione, Design, Tecnologia dell’Architettura, Sapienza Università di Roma, via Flaminia 72, 00196 Roma RM, Italy
MR Author ID:
1072338
Email:
ida.debonis@uniroma1.it
Emilio N. M. Cirillo
Affiliation:
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa, 16, 00161 Roma RM, Italy
MR Author ID:
606246
ORCID:
0000-0003-3673-2054
Email:
emilio.cirillo@uniroma1.it
Adrian Muntean
Affiliation:
Department of Mathematics and Computer Science, Karlstad University, Universitetsgatan 2, 651 88 Karlstad, Sweden
MR Author ID:
684703
ORCID:
0000-0002-1160-0007
Email:
adrian.muntean@kau.se
Keywords:
Homogenization,
reaction-diffusion equations with large nonlinear drift,
two-scale convergence with drift,
strong convergence in moving coordinates,
effective dispersion tensors for reactive flow in porous media
Received by editor(s):
September 9, 2023
Received by editor(s) in revised form:
January 7, 2024
Published electronically:
February 8, 2024
Additional Notes:
The work of the first and fourth authors is partially supported by the Swedish Research Council’s project “Homogenization and dimension reduction of thin heterogeneous layers” (grant nr. VR 2018-03648).
Article copyright:
© Copyright 2024
Brown University
