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Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations

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Abstract

A compactness framework is established for approximate solutions to subsonic-sonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework we develop applies for both non-homentropic and rotational flows. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional subsonic-sonic full Euler flows through infinitely long nozzles.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Gui-Qiang Chen

  2. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Gui-Qiang Chen & Tian-Yi Wang

  3. Academy of Mathematics and Systems Science, Academia Sinica, Beijing, 100190, People’s Republic of China

    Gui-Qiang Chen

  4. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Academia Sinica, Beijing, 100190, People’s Republic of China

    Fei-Min Huang & Tian-Yi Wang

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

    Tian-Yi Wang

Authors
  1. Gui-Qiang Chen
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  2. Fei-Min Huang
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  3. Tian-Yi Wang
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Correspondence to Gui-Qiang Chen.

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Communicated by C. Dafermos

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Chen, GQ., Huang, FM. & Wang, TY. Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations. Arch Rational Mech Anal 219, 719–740 (2016). https://doi.org/10.1007/s00205-015-0905-7

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  • DOI: https://doi.org/10.1007/s00205-015-0905-7

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