# American Institute of Mathematical Sciences

February  2009, 23(1&2): 281-298. doi: 10.3934/dcds.2009.23.281

## Multiscale analysis for convection dominated transport equations

 1 Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States 2 Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

Received  February 2008 Revised  July 2008 Published  September 2008

In this paper, we perform a systematic multiscale analysis forconvection dominated transport equations with a weak diffusion and ahighly oscillatory velocity field. The paper primarily focuses onupscaling linear transport equations. But we also discuss brieflyhow to upscale two-phase miscible flows, in which casethe concentration equation is coupled to the pressure equationin a nonlinear fashion. For the problem we consider here,the local Peclet number is of order $O(\epsilon^{-m+1})$ with $m \in[2,\infty]$ being any integer, where $\epsilon$ characterizes thesmall scale in the heterogeneous media. Due to the presence of thenonlocal memory effect, upscaling a convection dominated transportequation is known to be very difficult. One of the key ideas inderiving a well-posed homogenized equation for the convectiondominated transport equation is to introduce a projection operatorwhich projects the fluctuation onto a suitable subspace. Thisprojection operator corresponds to averaging along the streamlinesof the flow. In the case of linear convection dominated transportequations, we prove the well-posedness of the homogenized equationsand establish rigorous error estimates for our multiscale expansion.
Citation: Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281
 [1] Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks and Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020 [2] Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks and Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006 [3] Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217 [4] Bilal Saad, Mazen Saad. Numerical analysis of a non equilibrium two-component two-compressible flow in porous media. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 317-346. doi: 10.3934/dcdss.2014.7.317 [5] Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157 [6] Clément Cancès. On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types. Networks and Heterogeneous Media, 2010, 5 (3) : 635-647. doi: 10.3934/nhm.2010.5.635 [7] Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93 [8] Brahim Amaziane, Mladen Jurak, Leonid Pankratov, Anja Vrbaški. Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 629-665. doi: 10.3934/dcdsb.2018037 [9] Ting Zhang. The modeling error of well treatment for unsteady flow in porous media. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2171-2185. doi: 10.3934/dcdsb.2015.20.2171 [10] Helmut Abels, Yutaka Terasawa. Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022117 [11] Theodore Tachim Medjo. A two-phase flow model with delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137 [12] Dieter Bothe, Jan Prüss. Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition the isothermal incompressible case. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 673-696. doi: 10.3934/dcdss.2017034 [13] T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665 [14] Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198 [15] Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control and Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006 [16] Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. doi: 10.3934/dcds.2021090 [17] K. F. C. Yiu, L. L. Xie, K. L. Mak. Analysis of bullwhip effect in supply chains with heterogeneous decision models. Journal of Industrial and Management Optimization, 2009, 5 (1) : 81-94. doi: 10.3934/jimo.2009.5.81 [18] Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541 [19] Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265 [20] Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks and Heterogeneous Media, 2017, 12 (4) : 619-642. doi: 10.3934/nhm.2017025

2020 Impact Factor: 1.392