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Low Mach number limit of multidimensional steady flows on the airfoil problem

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Abstract

In this paper, we justify the low Mach number limit of the steady irrotational Euler flows for the airfoil problem, which is the first result for the low Mach number limit of the steady Euler flows in an exterior domain. The uniform estimates on the compressibility parameter \(\varepsilon \), which is singular for the flows, are established via a variational approach based on the compressible–incompressible difference functions. The limit is on the Hölder space and is unique. Moreover, the convergence rate is of order \(\varepsilon ^2\). It is noticeable that, due to the feature of the airfoil problem, the extra force dominates the asymptotic decay rate of the compressible flow to the infinity. And the effect of extra force vanishes in the limiting process from compressible flows to the incompressible ones, as the Mach number goes to zero.

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Acknowledgements

The authors would like to thank Professor Song Jiang for valuable suggestions. The research of Mingjie Li is supported by the NSFC Grant No. 11671412. The research of Tian-Yi Wang was supported in part by the NSFC Grant Nos. 11601401 and 11971024 and the Fundamental Research Funds for the Central Universities(WUT: 2017 IVA 072 and 2017 IVB 066). The research of Wei Xiang was supported in part by the Grants Council of the HKSAR, China (Project Nos. CityU 21305215, CityU 11332916, CityU 11304817 and CityU 11303518).

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Authors and Affiliations

  1. College of Science, Minzu University of China, Beijing, 100081, People’s Republic of China

    Mingjie Li

  2. Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan, 430070, Hubei, People’s Republic of China

    Tian-Yi Wang

  3. Gran Sasso Science Institute, viale Francesco Crispi, 7, 67100, L’Aquila, Italy

    Tian-Yi Wang

  4. City University of Hong Kong, Kowloon Tong, Hong Kong, People’s Republic of China

    Wei Xiang

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  1. Mingjie Li
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  2. Tian-Yi Wang
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  3. Wei Xiang
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Correspondence to Tian-Yi Wang.

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Communicated by L. Ambrosio.

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Li, M., Wang, TY. & Xiang, W. Low Mach number limit of multidimensional steady flows on the airfoil problem. Calc. Var. 59, 68 (2020). https://doi.org/10.1007/s00526-020-1720-z

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