# American Institute of Mathematical Sciences

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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