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

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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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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Keywords

  • Euler Equation
  • Strong Convergence
  • Entropy Function
  • Distributional Sense
  • Young Measure