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Incompressible limit of the Hookean elastodynamics in a bounded domain

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Abstract

Previously, the incompressible limit of the equations of Hookean elastodynamics has been established in the whole space. The present paper is devoted to the study of the incompressible limit in a bounded domain. Here, the main difficulty lies in the interactions between the large operator and the boundary. By designing some delicate semi-norms and energy estimates, we obtain the uniform estimates of the solutions. As a result, the incompressible limit is established by classical compactness argument.

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Acknowledgements

The research was supported by the Open Project of Key Laboratory No. CSSXKFKTQ202007, School of Mathematical Sciences, Chongqing Normal University. The first author was supported by the Science and Technology Research Program of Chongqing Municipal Educaton Commission (Grant No. KJQN201900543), the Natural Science Foundation of Chongqing (Grant No. cstc2020jcyj-msxmX0709 and No. cstc2020jcyj-jqX0022) and the National Natural Science Foundation of China (Grant No. 12001073). The second author was supported by the National Natural Science Foundation of China (Grant No. 12001506) and the Natural Science Foundation of Shandong Province (Grant No. ZR2020QA014).

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  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, China

    Guowei Liu

  2. School of Mathematical Sciences and College of Oceanic and Atmospheric Sciences, Ocean University of China, Qingdao, China

    Xin Xu

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  1. Guowei Liu
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  2. Xin Xu
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Correspondence to Xin Xu.

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Liu, G., Xu, X. Incompressible limit of the Hookean elastodynamics in a bounded domain. Z. Angew. Math. Phys. 72, 81 (2021). https://doi.org/10.1007/s00033-021-01523-9

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  • DOI: https://doi.org/10.1007/s00033-021-01523-9

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