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Transonic Shocks for the Full Compressible Euler System in a General Two-Dimensional De Laval Nozzle

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Abstract

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Hölder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.

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Authors and Affiliations

  1. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Jun Li

  2. Department of Mathematics & IMS, Nanjing University, Nanjing, 210093, People’s Republic of China

    Jun Li

  3. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Zhouping Xin

  4. Department of Mathematics & IMS, Nanjing University, Nanjing, 210093, People’s Republic of China

    Huicheng Yin

Authors
  1. Jun Li
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  2. Zhouping Xin
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  3. Huicheng Yin
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Correspondence to Zhouping Xin.

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Communicated by C. Dafermos

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Li, J., Xin, Z. & Yin, H. Transonic Shocks for the Full Compressible Euler System in a General Two-Dimensional De Laval Nozzle. Arch Rational Mech Anal 207, 533–581 (2013). https://doi.org/10.1007/s00205-012-0580-x

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  • DOI: https://doi.org/10.1007/s00205-012-0580-x

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