Homogenization: In mathematics or physics?

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In mathematics, homogenization theory considers the limitations of the sequences of the problems and their solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size between the micro scale and macro scale. So what is considered is a sequence of problems in axed domain while the characteristic size in micro scale tends to zero. But in the real physics or engineering situations, the micro scale of a medium isxed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to innity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what sense we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in H1, while in standard homogenization theory, the source term is assumed to be at least compacted in H1. A real example is also given to show the validation of our observation and results.





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Xu, S, X Yue and C Zhang (2016). Homogenization: In mathematics or physics?. Discrete and Continuous Dynamical Systems - Series S, 9(5). pp. 1575–1590. 10.3934/dcdss.2016064 Retrieved from https://hdl.handle.net/10161/27456.

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Shixin Xu

Assistant Professor of Mathematics at Duke Kunshan University

Shixin Xu is an Assistant Professor of Mathematics.   His research interests are machine learning and data-driven models for diseases,  multiscale modeling of complex fluids, Neurovascular coupling, homogenization theory, and numerical analysis.  The current projects he is working on are

  • image data-based for the prediction of hemorrhagic transformation in acute ischemic stroke,
  • electrodynamics modeling of saltatory conduction along a myelinated axon
  • electrochemical modeling
  • fluid-structure interaction with mass transportation and reaction

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