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Two Dimensional Subsonic Euler Flows Past a Wall or a Symmetric Body

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Abstract

The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal velocity in the upstream, there exists a critical value \({\rho_{\rm cr}}\) such that if the incoming density in the upstream is larger than \({\rho_{\rm cr}}\), then there exists a subsonic flow past a wall. Furthermore, \({\rho_{\rm cr}}\) is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than \({\rho_{\rm cr}}\). The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the difference between the velocity field and its far field states.

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Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350108, Fujian, People’s Republic of China

    Chao Chen

  2. Department of Mathematics, Sichuan University, Chengdu, 610064, People’s Republic of China

    Lili Du

  3. Department of Mathematics, Institute of Natural Sciences, Ministry of Education Key Laboratory of Scientific and Engineering Computing, SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Chunjing Xie

  4. The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Zhouping Xin

Authors
  1. Chao Chen
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  2. Lili Du
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  3. Chunjing Xie
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  4. Zhouping Xin
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Corresponding author

Correspondence to Chunjing Xie.

Additional information

Communicated by A. Bressan

Chen is supported by NSFC Grant 11301079. Du is supported in part by NSFC Grant 11571243, PCSIRT (IRT1273) and Sichuan Youth Science and Technology Foundation 2014JQ0003. Xie is supported in part by NSFC Grants 11201297, 11422105, and 11511140276, Shanghai Chenguang Program, and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning. Xin is supported in part by Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants CUHK4041/11P, and CUHK4048/13P, a Focus Area Grant from The Chinese University of Hong Kong, and a CAS-Croucher Joint Grant.

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Chen, C., Du, L., Xie, C. et al. Two Dimensional Subsonic Euler Flows Past a Wall or a Symmetric Body. Arch Rational Mech Anal 221, 559–602 (2016). https://doi.org/10.1007/s00205-016-0968-0

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  • DOI: https://doi.org/10.1007/s00205-016-0968-0

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