Homogenization: In mathematics or physics?

Shixin  Xu; Xingye Yue; Changrong  Zhang

doi:10.3934/dcdss.2016064

Article Contents

2016, Volume 9, Issue 5: 1575-1590. Doi: 10.3934/dcdss.2016064

This issue Previous Article About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows Next Article

Homogenization: In mathematics or physics?

1.
Department of Mathematics, Soochow University, Suzhou 215006
2.
High Speed Aerodynamics Institute, China Aerodynamisc Development and Research Center, Mianyang 622661

Received: October 31, 2014

Revised: August 31, 2015

Published: September 2016

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Abstract

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 a fixed 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 is fixed 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 infinity. 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 $H^{-1}$, while in standard homogenization theory, the source term is assumed to be at least compacted in $H^{-1}$. A real example is also given to show the validation of our observation and results.

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Mathematics Subject Classification: Primary: 35B27, 35B40; Secondary: 35M30.

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