January & 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 & Continuous Dynamical Systems, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281
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