• Previous Article
    Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$
  • DCDS-B Home
  • This Issue
  • Next Article
    Separable Lyapunov functions for monotone systems: Constructions and limitations
October  2015, 20(8): 2527-2551. doi: 10.3934/dcdsb.2015.20.2527

On the homogenization of multicomponent transport

1. 

CMAP, CNRS UMR 7641, École Polytechnique, Route de Saclay, Palaiseau F91128

2. 

DPMMS, CMS, University of Cambridge, Wilberforce road, Cambridge CB3 0WB, United Kingdom

Received  November 2014 Revised  March 2015 Published  August 2015

This paper is devoted to the homogenization of weakly coupled cooperative parabolic systems in strong convection regime with purely periodic coefficients. Our approach is to factor out oscillations from the solution via principal eigenfunctions of an associated spectral problem and to cancel any exponential decay in time of the solution using the principal eigenvalue of the same spectral problem. We employ the notion of two-scale convergence with drift in the asymptotic analysis of the factorized model as the lengthscale of the oscillations tends to zero. This combination of the factorization method and the method of two-scale convergence is applied to upscale an adsorption model for multicomponent flow in an heterogeneous porous medium.
Citation: Grégoire Allaire, Harsha Hutridurga. On the homogenization of multicomponent transport. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2527-2551. doi: 10.3934/dcdsb.2015.20.2527
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.

[2]

G. Allaire, Periodic homogenization and effective mass theorems for the Schrödinger equation, in Quantum Transport-Modelling, Analysis and Asymptotics (eds. N. Ben Abdallah and G. Frosali), Lecture Notes in Mathematics, 1946, Springer, 2008, 1-44. doi: 10.1007/978-3-540-79574-2.

[3]

G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91-117. doi: 10.1016/S0045-7825(99)00112-7.

[4]

G. Allaire, Y. Capdeboscq, A. Piatnitski, V. Siess and M. Vanninathan, Homogenization of periodic systems with large potentials, Arch. Rat. Mech. Anal., 174 (2004), 179-220. doi: 10.1007/s00205-004-0332-7.

[5]

G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media (May 1995) (eds. A. Bourgeat, et al.), World Scientific Pub., Singapore, 1996, 15-25.

[6]

G. Allaire, A. Mikelic and A. Piatnitski, Homogenization approach to the dispersion theory for reactive transport through porous media, SIAM J. Math. Anal., 42 (2010), 125-144. doi: 10.1137/090754935.

[7]

G. Allaire and A. L. Raphael, Homogenization of a convection diffusion model with reaction in a porous medium, C. R. Math. Acad. Sci. Paris, 344 (2007), 523-528. doi: 10.1016/j.crma.2007.03.008.

[8]

L. Boudin, B. Grec and F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations, Discrete Contin. Dyn. Syst. B, 17 (2012), 1427-1440. doi: 10.3934/dcdsb.2012.17.1427.

[9]

Y. Capdeboscq, Homogenization of a neutronic multigroup evolution model, Asymptot. Anal., 24 (2000), 143-165.

[10]

Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 567-594. doi: 10.1017/S0308210500001785.

[11]

C. Conca, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures et Appl., 64 (1985), 31-75.

[12]

L. Desvillettes, Th. Lepoutre and A. Moussa, Entropy, duality and cross diffusion, SIAM Journal on Mathematical Analysis, 46 (2014), 820-853. doi: 10.1137/130908701.

[13]

L. Desvillettes and A. Trescases, New results for triangular reaction cross diffusion system, J. Math. Anal. Appl., 430 (2015), 32-59. doi: 10.1016/j.jmaa.2015.03.078.

[14]

P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms, in Multi Scale Problems and Asymptotic Analysis, GAKUTO International Series, Math. Sci. Appl., 24, Gakkōtosho, Tokyo, 2005, 153-165.

[15]

U. Hornung, Homogenization and Porous Media, Interdiscip. Appl. Math., 6, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1920-0.

[16]

H. Hutridurga, Homogenization of Complex Flows in Porous Media and Applications, Doctoral Thesis, École Polytechnique, Palaiseau, 2013.

[17]

V. V. Jikov, Asymptotic behavior and stabilization of solutions of a second order parabolic equation with lowest terms, Transc. Moscow Math. Soc., 2 (1984), 69-99.

[18]

S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Transc. Moscow Math. Soc., 2 (1984), 101-126.

[19]

M. Marion and R. Temam, Global existence for fully nonlinear reaction-diffusion systems describing multicomponent reactive flows, preprint, arXiv:1310.2624, (2013). doi: 10.1016/j.matpur.2015.02.003.

[20]

E. Marusic-Paloka and A. Piatnitski, Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating co-efficients and strong convection, Journal of London Math. Soc., 72 (2005), 391-409. doi: 10.1112/S0024610705006824.

[21]

S. Mirrahimi and P. E. Souganidis, A homogenization approach for the motion of motor proteins, NoDEA, 20 (2013), 129-147. doi: 10.1007/s00030-012-0156-3.

[22]

E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity, Math. Nachr., 173 (1995), 259-286. doi: 10.1002/mana.19951730115.

[23]

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.

[24]

G. Sweers, Strong positivity in $C(\bar\Omega)$ for elliptic systems, Math. Z., 209 (1992), 251-272. doi: 10.1007/BF02570833.

[25]

M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.

[2]

G. Allaire, Periodic homogenization and effective mass theorems for the Schrödinger equation, in Quantum Transport-Modelling, Analysis and Asymptotics (eds. N. Ben Abdallah and G. Frosali), Lecture Notes in Mathematics, 1946, Springer, 2008, 1-44. doi: 10.1007/978-3-540-79574-2.

[3]

G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91-117. doi: 10.1016/S0045-7825(99)00112-7.

[4]

G. Allaire, Y. Capdeboscq, A. Piatnitski, V. Siess and M. Vanninathan, Homogenization of periodic systems with large potentials, Arch. Rat. Mech. Anal., 174 (2004), 179-220. doi: 10.1007/s00205-004-0332-7.

[5]

G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media (May 1995) (eds. A. Bourgeat, et al.), World Scientific Pub., Singapore, 1996, 15-25.

[6]

G. Allaire, A. Mikelic and A. Piatnitski, Homogenization approach to the dispersion theory for reactive transport through porous media, SIAM J. Math. Anal., 42 (2010), 125-144. doi: 10.1137/090754935.

[7]

G. Allaire and A. L. Raphael, Homogenization of a convection diffusion model with reaction in a porous medium, C. R. Math. Acad. Sci. Paris, 344 (2007), 523-528. doi: 10.1016/j.crma.2007.03.008.

[8]

L. Boudin, B. Grec and F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations, Discrete Contin. Dyn. Syst. B, 17 (2012), 1427-1440. doi: 10.3934/dcdsb.2012.17.1427.

[9]

Y. Capdeboscq, Homogenization of a neutronic multigroup evolution model, Asymptot. Anal., 24 (2000), 143-165.

[10]

Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 567-594. doi: 10.1017/S0308210500001785.

[11]

C. Conca, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures et Appl., 64 (1985), 31-75.

[12]

L. Desvillettes, Th. Lepoutre and A. Moussa, Entropy, duality and cross diffusion, SIAM Journal on Mathematical Analysis, 46 (2014), 820-853. doi: 10.1137/130908701.

[13]

L. Desvillettes and A. Trescases, New results for triangular reaction cross diffusion system, J. Math. Anal. Appl., 430 (2015), 32-59. doi: 10.1016/j.jmaa.2015.03.078.

[14]

P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms, in Multi Scale Problems and Asymptotic Analysis, GAKUTO International Series, Math. Sci. Appl., 24, Gakkōtosho, Tokyo, 2005, 153-165.

[15]

U. Hornung, Homogenization and Porous Media, Interdiscip. Appl. Math., 6, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1920-0.

[16]

H. Hutridurga, Homogenization of Complex Flows in Porous Media and Applications, Doctoral Thesis, École Polytechnique, Palaiseau, 2013.

[17]

V. V. Jikov, Asymptotic behavior and stabilization of solutions of a second order parabolic equation with lowest terms, Transc. Moscow Math. Soc., 2 (1984), 69-99.

[18]

S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Transc. Moscow Math. Soc., 2 (1984), 101-126.

[19]

M. Marion and R. Temam, Global existence for fully nonlinear reaction-diffusion systems describing multicomponent reactive flows, preprint, arXiv:1310.2624, (2013). doi: 10.1016/j.matpur.2015.02.003.

[20]

E. Marusic-Paloka and A. Piatnitski, Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating co-efficients and strong convection, Journal of London Math. Soc., 72 (2005), 391-409. doi: 10.1112/S0024610705006824.

[21]

S. Mirrahimi and P. E. Souganidis, A homogenization approach for the motion of motor proteins, NoDEA, 20 (2013), 129-147. doi: 10.1007/s00030-012-0156-3.

[22]

E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity, Math. Nachr., 173 (1995), 259-286. doi: 10.1002/mana.19951730115.

[23]

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.

[24]

G. Sweers, Strong positivity in $C(\bar\Omega)$ for elliptic systems, Math. Z., 209 (1992), 251-272. doi: 10.1007/BF02570833.

[25]

M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.

[1]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279

[2]

María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks and Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004

[3]

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

[4]

Markus Gahn. Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6511-6531. doi: 10.3934/dcdsb.2019151

[5]

Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91

[6]

Danielle Hilhorst, Hideki Murakawa. Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium. Networks and Heterogeneous Media, 2014, 9 (4) : 669-682. doi: 10.3934/nhm.2014.9.669

[7]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400

[8]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks and Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006

[9]

Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281

[10]

Hakima Bessaih, Yalchin Efendiev, Razvan Florian Maris. Stochastic homogenization for a diffusion-reaction model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5403-5429. doi: 10.3934/dcds.2019221

[11]

Vsevolod Laptev. Deterministic homogenization for media with barriers. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 29-44. doi: 10.3934/dcdss.2015.8.29

[12]

Oleh Krehel, Toyohiko Aiki, Adrian Muntean. Homogenization of a thermo-diffusion system with Smoluchowski interactions. Networks and Heterogeneous Media, 2014, 9 (4) : 739-762. doi: 10.3934/nhm.2014.9.739

[13]

Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks and Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001

[14]

Yangyang Shi, Hongjun Gao. Homogenization for stochastic reaction-diffusion equations with singular perturbation term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2401-2426. doi: 10.3934/dcdsb.2021137

[15]

Michael Eden, Michael Böhm. Homogenization of a poro-elasticity model coupled with diffusive transport and a first order reaction for concrete. Networks and Heterogeneous Media, 2014, 9 (4) : 599-615. doi: 10.3934/nhm.2014.9.599

[16]

Mario Ohlberger, Ben Schweizer. Modelling of interfaces in unsaturated porous media. Conference Publications, 2007, 2007 (Special) : 794-803. doi: 10.3934/proc.2007.2007.794

[17]

Shifeng Geng, Zhen Wang. Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant. Communications on Pure and Applied Analysis, 2012, 11 (2) : 475-500. doi: 10.3934/cpaa.2012.11.475

[18]

Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211

[19]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242

[20]

Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]