American Institute of Mathematical Sciences

March  2011, 6(1): 111-126. doi: 10.3934/nhm.2011.6.111

Homogenization of convection-diffusion equation in infinite cylinder

 1 Narvik University College, Postbox 385, 8505 Narvik, Norway 2 Narvik University College, HiN, Postbox 385, 8505 Narvik, Norway, and, P.N. Lebedev Physical Institute RAS, Leninski prospect, 53, Moscow, 117924

Received  February 2010 Revised  May 2010 Published  March 2011

The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion equation stated in a thin infinite cylindrical domain with homogeneous Neumann boundary condition on the lateral boundary. It is shown that homogenization result holds in moving coordinates, and that the solution admits an asymptotic expansion which consists of the interior expansion being regular in time, and an initial layer.
Citation: Iryna Pankratova, Andrey Piatnitski. Homogenization of convection-diffusion equation in infinite cylinder. Networks and Heterogeneous Media, 2011, 6 (1) : 111-126. doi: 10.3934/nhm.2011.6.111
References:
 [1] G. Allaire and A. Raphael, Homogenization of a convection-diffusion model with reaction in a porous medium, (English, French summary) C. R. Math. Acad. Sci. Paris, 344 (2007), 523-528. [2] D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291. [3] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 607-694. [4] N. S. Bakhvalov and G. P. Panasenko, "Homogenization: Averaging Processes in Periodic Media," Kluwer, Dordrecht/Boston/London, 1989. [5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structure," Studies in Mathematics and its Applications, 5. North-Holland Publishing Co., Amsterdam-New York, 1978. [6] P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms, Multi Scale Problems and Asymptotic Analysis, GAKUTO Internat. Ser. Math. Sci. Appl., 24 (2005), 153-165. [7] M. V. Kozlova and G. P. Panasenko, Averaging a three-dimensional problem of elasticity theory in a nonhomogeneous rod, Comput. Math. Math. Phys., 31 (1992), 128-131. [8] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1967. [9] I. Pankratova and A. Piatnitski, On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder, DCDS-B, 11 (2009), 935-970. doi: 10.3934/dcdsb.2009.11.935. [10] L. Trabucho and J. M. Viaño, Derivation of generalized models for linear elastic beams by asymptotic expansion methods, Applications of multiple scaling in mechanis (Paris, 1986), Rech. Math. Appl., 4, Masson, Paris (1987), 302-315. [11] Z. Tutek, A homogenized model of rod in linear elasticity, Applications of multiple scaling in mechanis (Paris, 1986), Rech. Math. Appl., 4, Masson, Paris (1987), 302-315. [12] V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals," Springer-Verlag, Berlin, 1994. [13] V. V. Zhikov, On an extension and an application of the two-scale convergence method, Sb. Math., 191 (2000), 973-1014. doi: 10.1070/SM2000v191n07ABEH000491.

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References:
 [1] G. Allaire and A. Raphael, Homogenization of a convection-diffusion model with reaction in a porous medium, (English, French summary) C. R. Math. Acad. Sci. Paris, 344 (2007), 523-528. [2] D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291. [3] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 607-694. [4] N. S. Bakhvalov and G. P. Panasenko, "Homogenization: Averaging Processes in Periodic Media," Kluwer, Dordrecht/Boston/London, 1989. [5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structure," Studies in Mathematics and its Applications, 5. North-Holland Publishing Co., Amsterdam-New York, 1978. [6] P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms, Multi Scale Problems and Asymptotic Analysis, GAKUTO Internat. Ser. Math. Sci. Appl., 24 (2005), 153-165. [7] M. V. Kozlova and G. P. Panasenko, Averaging a three-dimensional problem of elasticity theory in a nonhomogeneous rod, Comput. Math. Math. Phys., 31 (1992), 128-131. [8] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1967. [9] I. Pankratova and A. Piatnitski, On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder, DCDS-B, 11 (2009), 935-970. doi: 10.3934/dcdsb.2009.11.935. [10] L. Trabucho and J. M. Viaño, Derivation of generalized models for linear elastic beams by asymptotic expansion methods, Applications of multiple scaling in mechanis (Paris, 1986), Rech. Math. Appl., 4, Masson, Paris (1987), 302-315. [11] Z. Tutek, A homogenized model of rod in linear elasticity, Applications of multiple scaling in mechanis (Paris, 1986), Rech. Math. Appl., 4, Masson, Paris (1987), 302-315. [12] V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals," Springer-Verlag, Berlin, 1994. [13] V. V. Zhikov, On an extension and an application of the two-scale convergence method, Sb. Math., 191 (2000), 973-1014. doi: 10.1070/SM2000v191n07ABEH000491.
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