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 & 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.   Google Scholar

[2]

D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations,, Arch. Rational Mech. Anal., 25 (1967), 81.  doi: 10.1007/BF00281291.  Google Scholar

[3]

D. G. Aronson, Non-negative solutions of linear parabolic equations,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 607.   Google Scholar

[4]

N. S. Bakhvalov and G. P. Panasenko, "Homogenization: Averaging Processes in Periodic Media,", Kluwer, (1989).   Google Scholar

[5]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structure,", Studies in Mathematics and its Applications, (1978).   Google Scholar

[6]

P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms,, Multi Scale Problems and Asymptotic Analysis, 24 (2005), 153.   Google Scholar

[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.   Google Scholar

[8]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, 23 (1967).   Google Scholar

[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.  doi: 10.3934/dcdsb.2009.11.935.  Google Scholar

[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, 4 (1987), 302.   Google Scholar

[11]

Z. Tutek, A homogenized model of rod in linear elasticity,, Applications of multiple scaling in mechanis (Paris, 4 (1987), 302.   Google Scholar

[12]

V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals,", Springer-Verlag, (1994).   Google Scholar

[13]

V. V. Zhikov, On an extension and an application of the two-scale convergence method,, Sb. Math., 191 (2000), 973.  doi: 10.1070/SM2000v191n07ABEH000491.  Google Scholar

show all references

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.   Google Scholar

[2]

D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations,, Arch. Rational Mech. Anal., 25 (1967), 81.  doi: 10.1007/BF00281291.  Google Scholar

[3]

D. G. Aronson, Non-negative solutions of linear parabolic equations,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 607.   Google Scholar

[4]

N. S. Bakhvalov and G. P. Panasenko, "Homogenization: Averaging Processes in Periodic Media,", Kluwer, (1989).   Google Scholar

[5]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structure,", Studies in Mathematics and its Applications, (1978).   Google Scholar

[6]

P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms,, Multi Scale Problems and Asymptotic Analysis, 24 (2005), 153.   Google Scholar

[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.   Google Scholar

[8]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, 23 (1967).   Google Scholar

[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.  doi: 10.3934/dcdsb.2009.11.935.  Google Scholar

[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, 4 (1987), 302.   Google Scholar

[11]

Z. Tutek, A homogenized model of rod in linear elasticity,, Applications of multiple scaling in mechanis (Paris, 4 (1987), 302.   Google Scholar

[12]

V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals,", Springer-Verlag, (1994).   Google Scholar

[13]

V. V. Zhikov, On an extension and an application of the two-scale convergence method,, Sb. Math., 191 (2000), 973.  doi: 10.1070/SM2000v191n07ABEH000491.  Google Scholar

[1]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159

[2]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044

[3]

Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031

[4]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017

[5]

Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281

[6]

Alain Damlamian, Klas Pettersson. Homogenization of oscillating boundaries. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 197-219. doi: 10.3934/dcds.2009.23.197

[7]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[8]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[9]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385

[10]

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

[11]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[12]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[13]

Peter Frolkovič, Karol Mikula, Jooyoung Hahn, Dirk Martin, Branislav Basara. Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 865-879. doi: 10.3934/dcdss.2020350

[14]

Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106

[15]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[16]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[17]

Ke Su, Yumeng Lin, Chun Xu. A new adaptive method to nonlinear semi-infinite programming. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021012

[18]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328

[19]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[20]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

2019 Impact Factor: 1.053

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]