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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.

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##### 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.
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