American Institute of Mathematical Sciences

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

Homogenization of convection-diffusion equation in infinite cylinder

Iryna Pankratova 1, and  Andrey Piatnitski 2,

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.
Keywords: infinite cylinder, asymptotic behaviour., Homogenization, convection-diffusion equation.
Mathematics Subject Classification: Primary: 35B27, 35B40, 35K20; Secondary: 35B2.
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]

Iryna Pankratova, Andrey Piatnitski. On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 935-970. doi: 10.3934/dcdsb.2009.11.935
[2]

Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133
[3]

Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems & Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063
[4]

Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575
[5]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
[6]

Qiang Du, Zhan Huang, Richard B. Lehoucq. Nonlocal convection-diffusion volume-constrained problems and jump processes. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 373-389. doi: 10.3934/dcdsb.2014.19.373
[7]

Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11
[8]

M. González, J. Jansson, S. Korotov. A posteriori error analysis of a stabilized mixed FEM for convection-diffusion problems. Conference Publications, 2015, 2015 (special) : 525-532. doi: 10.3934/proc.2015.0525
[9]

Holger Heumann, Ralf Hiptmair. Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1471-1495. doi: 10.3934/dcds.2011.29.1471
[10]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496
[11]

Youngmok Jeon, Eun-Jae Park. Cell boundary element methods for convection-diffusion equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 309-319. doi: 10.3934/cpaa.2006.5.309
[12]

Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663
[13]

Huan-Zhen Chen, Zhao-Jie Zhou, Hong Wang, Hong-Ying Man. An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 325-341. doi: 10.3934/dcdsb.2011.15.325
[14]

Antti Lipponen, Aku Seppänen, Jari Hämäläinen, Jari P. Kaipio. Nonstationary inversion of convection-diffusion problems - recovery from unknown nonstationary velocity fields. Inverse Problems & Imaging, 2010, 4 (3) : 463-483. doi: 10.3934/ipi.2010.4.463
[15]

Catherine Choquet, Marie-Christine Néel. From particles scale to anomalous or classical convection-diffusion models with path integrals. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 207-238. doi: 10.3934/dcdss.2014.7.207
[16]

Huiqing Zhu, Runchang Lin. $L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1493-1505. doi: 10.3934/dcdsb.2013.18.1493
[17]

Lili Ju, Wensong Wu, Weidong Zhao. Adaptive finite volume methods for steady convection-diffusion equations with mesh optimization. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 669-690. doi: 10.3934/dcdsb.2009.11.669
[18]

Tomás Caraballo, Antonio M. Márquez-Durán, Rivero Felipe. Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1817-1833. doi: 10.3934/dcdsb.2017108
[19]

Kin Ming Hui. Collasping behaviour of a singular diffusion equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2165-2185. doi: 10.3934/dcds.2012.32.2165
[20]

Toru Sasaki, Takashi Suzuki. Asymptotic behaviour of the solutions to a virus dynamics model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 525-541. doi: 10.3934/dcdsb.2017206

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences