Incompressible limit of solutions of multidimensional steady compressible Euler equations

  • Gui-Qiang G. Chen
  • Feimin Huang
  • Tian-Yi Wang
  • Wei Xiang
Open Access

DOI: 10.1007/s00033-016-0629-z

Cite this article as:
Chen, GQ.G., Huang, F., Wang, TY. et al. Z. Angew. Math. Phys. (2016) 67: 75. doi:10.1007/s00033-016-0629-z


A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.


Multidimensional Incompressible limit Steady flow Euler equations Compressible flow Full Euler flow Homentropic flow Compactness framework Strong convergence 

Mathematics Subject Classification

35Q31 35M30 35L65 76N10 76G25 35B40 35D30 
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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Gui-Qiang G. Chen
    • 1
    • 2
    • 3
  • Feimin Huang
    • 1
  • Tian-Yi Wang
    • 1
    • 3
    • 4
    • 5
    • 6
  • Wei Xiang
    • 7
  1. 1.Academy of Mathematics and Systems Science, Academia SinicaBeijingPeople’s Republic of China
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China
  3. 3.Mathematical InstituteUniversity of OxfordOxfordUK
  4. 4.Department of Mathematics, School of ScienceWuhan University of TechnologyWuhanPeople’s Republic of China
  5. 5.Gran Sasso Science InstituteL’AquilaItaly
  6. 6.The Institute of Mathematical SciencesThe Chinese University of Hong KongShatinHong Kong
  7. 7.Department of MathematicsCity University of Hong KongKowloonHong Kong, People’s Republic of China

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