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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 & 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.  doi: 10.1137/0523084.  Google Scholar [2] G. Allaire, Periodic homogenization and effective mass theorems for the Schrödinger equation,, in Quantum Transport-Modelling, (1946), 1.  doi: 10.1007/978-3-540-79574-2.  Google Scholar [3] G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion,, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91.  doi: 10.1016/S0045-7825(99)00112-7.  Google Scholar [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.  doi: 10.1007/s00205-004-0332-7.  Google Scholar [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, (1995), 15.   Google Scholar [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.  doi: 10.1137/090754935.  Google Scholar [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.  doi: 10.1016/j.crma.2007.03.008.  Google Scholar [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.  doi: 10.3934/dcdsb.2012.17.1427.  Google Scholar [9] Y. Capdeboscq, Homogenization of a neutronic multigroup evolution model,, Asymptot. Anal., 24 (2000), 143.   Google Scholar [10] Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 567.  doi: 10.1017/S0308210500001785.  Google Scholar [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.   Google Scholar [12] L. Desvillettes, Th. Lepoutre and A. Moussa, Entropy, duality and cross diffusion,, SIAM Journal on Mathematical Analysis, 46 (2014), 820.  doi: 10.1137/130908701.  Google Scholar [13] L. Desvillettes and A. Trescases, New results for triangular reaction cross diffusion system,, J. Math. Anal. Appl., 430 (2015), 32.  doi: 10.1016/j.jmaa.2015.03.078.  Google Scholar [14] P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms,, in Multi Scale Problems and Asymptotic Analysis, (2005), 153.   Google Scholar [15] U. Hornung, Homogenization and Porous Media,, Interdiscip. Appl. Math., (1997).  doi: 10.1007/978-1-4612-1920-0.  Google Scholar [16] H. Hutridurga, Homogenization of Complex Flows in Porous Media and Applications,, Doctoral Thesis, (2013).   Google Scholar [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.   Google Scholar [18] S. Kozlov, Reducibility of quasiperiodic differential operators and averaging,, Transc. Moscow Math. Soc., 2 (1984), 101.   Google Scholar [19] M. Marion and R. Temam, Global existence for fully nonlinear reaction-diffusion systems describing multicomponent reactive flows,, preprint, (2013).  doi: 10.1016/j.matpur.2015.02.003.  Google Scholar [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.  doi: 10.1112/S0024610705006824.  Google Scholar [21] S. Mirrahimi and P. E. Souganidis, A homogenization approach for the motion of motor proteins,, NoDEA, 20 (2013), 129.  doi: 10.1007/s00030-012-0156-3.  Google Scholar [22] E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity,, Math. Nachr., 173 (1995), 259.  doi: 10.1002/mana.19951730115.  Google Scholar [23] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608.  doi: 10.1137/0520043.  Google Scholar [24] G. Sweers, Strong positivity in $C(\bar\Omega)$ for elliptic systems,, Math. Z., 209 (1992), 251.  doi: 10.1007/BF02570833.  Google Scholar [25] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains,, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239.  doi: 10.1007/BF02838079.  Google Scholar

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References:
 [1] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.  doi: 10.1137/0523084.  Google Scholar [2] G. Allaire, Periodic homogenization and effective mass theorems for the Schrödinger equation,, in Quantum Transport-Modelling, (1946), 1.  doi: 10.1007/978-3-540-79574-2.  Google Scholar [3] G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion,, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91.  doi: 10.1016/S0045-7825(99)00112-7.  Google Scholar [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.  doi: 10.1007/s00205-004-0332-7.  Google Scholar [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, (1995), 15.   Google Scholar [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.  doi: 10.1137/090754935.  Google Scholar [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.  doi: 10.1016/j.crma.2007.03.008.  Google Scholar [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.  doi: 10.3934/dcdsb.2012.17.1427.  Google Scholar [9] Y. Capdeboscq, Homogenization of a neutronic multigroup evolution model,, Asymptot. Anal., 24 (2000), 143.   Google Scholar [10] Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 567.  doi: 10.1017/S0308210500001785.  Google Scholar [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.   Google Scholar [12] L. Desvillettes, Th. Lepoutre and A. Moussa, Entropy, duality and cross diffusion,, SIAM Journal on Mathematical Analysis, 46 (2014), 820.  doi: 10.1137/130908701.  Google Scholar [13] L. Desvillettes and A. Trescases, New results for triangular reaction cross diffusion system,, J. Math. Anal. Appl., 430 (2015), 32.  doi: 10.1016/j.jmaa.2015.03.078.  Google Scholar [14] P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms,, in Multi Scale Problems and Asymptotic Analysis, (2005), 153.   Google Scholar [15] U. Hornung, Homogenization and Porous Media,, Interdiscip. Appl. Math., (1997).  doi: 10.1007/978-1-4612-1920-0.  Google Scholar [16] H. Hutridurga, Homogenization of Complex Flows in Porous Media and Applications,, Doctoral Thesis, (2013).   Google Scholar [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.   Google Scholar [18] S. Kozlov, Reducibility of quasiperiodic differential operators and averaging,, Transc. Moscow Math. Soc., 2 (1984), 101.   Google Scholar [19] M. Marion and R. Temam, Global existence for fully nonlinear reaction-diffusion systems describing multicomponent reactive flows,, preprint, (2013).  doi: 10.1016/j.matpur.2015.02.003.  Google Scholar [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.  doi: 10.1112/S0024610705006824.  Google Scholar [21] S. Mirrahimi and P. E. Souganidis, A homogenization approach for the motion of motor proteins,, NoDEA, 20 (2013), 129.  doi: 10.1007/s00030-012-0156-3.  Google Scholar [22] E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity,, Math. Nachr., 173 (1995), 259.  doi: 10.1002/mana.19951730115.  Google Scholar [23] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608.  doi: 10.1137/0520043.  Google Scholar [24] G. Sweers, Strong positivity in $C(\bar\Omega)$ for elliptic systems,, Math. Z., 209 (1992), 251.  doi: 10.1007/BF02570833.  Google Scholar [25] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains,, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239.  doi: 10.1007/BF02838079.  Google Scholar
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