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Steady Subsonic Ideal Flows Through an Infinitely Long Nozzle with Large Vorticity

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Abstract

In this paper, the existence, uniqueness, and far field behavior of a class of subsonic flows with large vorticity for the steady Euler equations in infinitely long nozzles are established. More precisely, for any given convex horizontal velocity of incoming flow in the upstream, there exists a critical value m cr , if the mass flux is larger than m cr , then there exists a unique smooth subsonic Euler flow through the infinitely long nozzle. This well-posedness result is proved by a new observation for the method developed in Xie and Xin (SIAM J Math Anal 42:751–784, 2010) to deal with the Euler system. Furthermore, the optimal convergence rates of the subsonic flows at far fields are obtained via the maximum principle and an elaborate choice of the comparison functions.

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Authors and Affiliations

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

    Lili Du

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

    Chunjing Xie

  3. Institute of Mathematical Science and Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

    Zhouping Xin

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

Correspondence to Chunjing Xie.

Additional information

Communicated by W. Schlag

Du is supported in part by NSFC grant 11171236, PCSIRT (IRT1273) and Fundamental Research Funds for the Central Universities. Xie is supported in part by NSFC grants 11241001, 11201297, Shanghai Chenguang Program, Shanghai Pujiang program 12PJ1405200, and a startup grant from Shanghai Jiao Tong University. Xin is supported in parts by Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants CUHK 4042/08P and CUHK 4041/11P, and a grant from Croucher Foundation.

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Du, L., Xie, C. & Xin, Z. Steady Subsonic Ideal Flows Through an Infinitely Long Nozzle with Large Vorticity. Commun. Math. Phys. 328, 327–354 (2014). https://doi.org/10.1007/s00220-014-1951-y

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  • DOI: https://doi.org/10.1007/s00220-014-1951-y

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