## Homogenization: In mathematics or physics?

Publication ,  Journal Article
Xu, S; Yue, X; Zhang, C
Published in: Discrete and Continuous Dynamical Systems - Series S
October 1, 2016

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.

## Published In

Discrete and Continuous Dynamical Systems - Series S

1937-1179

1937-1632

October 1, 2016

9

5

## Start / End Page

1575 / 1590

• 4904 Pure mathematics
• 4901 Applied mathematics
• 0102 Applied Mathematics
• 0101 Pure Mathematics

### Citation

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Xu, S., Yue, X., & Zhang, C. (2016). Homogenization: In mathematics or physics? Discrete and Continuous Dynamical Systems - Series S, 9(5), 1575–1590. https://doi.org/10.3934/dcdss.2016064
Xu, S., X. Yue, and C. Zhang. “Homogenization: In mathematics or physics?Discrete and Continuous Dynamical Systems - Series S 9, no. 5 (October 1, 2016): 1575–90. https://doi.org/10.3934/dcdss.2016064.
Xu S, Yue X, Zhang C. Homogenization: In mathematics or physics? Discrete and Continuous Dynamical Systems - Series S. 2016 Oct 1;9(5):1575–90.
Xu, S., et al. “Homogenization: In mathematics or physics?Discrete and Continuous Dynamical Systems - Series S, vol. 9, no. 5, Oct. 2016, pp. 1575–90. Scopus, doi:10.3934/dcdss.2016064.
Xu S, Yue X, Zhang C. Homogenization: In mathematics or physics? Discrete and Continuous Dynamical Systems - Series S. 2016 Oct 1;9(5):1575–1590.

## Published In

Discrete and Continuous Dynamical Systems - Series S

1937-1179

1937-1632

October 1, 2016

9

5

1575 / 1590