SpringerLink
Log in

Low Mach number limit of a compressible Euler-Korteweg model

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

This article deals with the low Mach number limit of the compressible Euler-Korteweg equations. It is justified rigorously that solutions of the compressible Euler-Korteweg equations converge to those of the incompressible Euler equations as the Mach number tends to zero. Furthermore, the desired convergence rates are also obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Antonelli, P. Marcati: The quantum hydrodynamics system in two space dimensions. Arch. Ration. Mech. Anal. 203 (2012), 499–527.

    Article  MATH  Google Scholar 

  2. K. Asano: On the incompressible limit of the compressible Euler equation. Japan J. Appl. Math. 4 (1987), 455–488.

    Article  MATH  Google Scholar 

  3. C. Audiard: Dispersive smoothing for the Euler-Korteweg model. SIAM J. Math. Anal. 44 (2012), 3018–3040.

    Article  MATH  Google Scholar 

  4. C. Audiard, B. Haspot: Global well-posedness of the Euler-Korteweg system for small irrotational data. Commun. Math. Phys. 351 (2017), 201–247.

    Article  MATH  Google Scholar 

  5. S. Benzoni-Gavage, R. Danchin, S. Descombes: Well-posedness of one-dimensional Korteweg models. Electron. J. Differ. Equ. 2006 (2006), Article ID 59, 35 pages.

  6. S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness for the Euler-Korteweg model in several space dimensions. Indiana Univ. Math. J. 56 (2007), 1499–1579.

    Article  MATH  Google Scholar 

  7. Y. Brenier: Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equations 25 (2000), 737–754.

    Article  MATH  Google Scholar 

  8. D. Bresch, B. Desjardins, B. Ducomet: Quasi-neutral limit for a viscous capillary model of plasma. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22 (2005), 1–9.

    Article  MATH  Google Scholar 

  9. R. Carles, R. Danchin, J.-C. Saut: Madelung, Gross-Pitaevskii and Korteveg. Nonlinearity 25 (2012), 2843–2873.

    Article  MATH  Google Scholar 

  10. R. M. Colombo, G. Guerra, V. Schleper: The compressible to incompressible limit of one dimensional Euler equations: The non smooth case. Arch. Ration. Mech. Anal. 219 (2016), 701–718.

    Article  MATH  Google Scholar 

  11. D. Donatelli, E. Feireisl, P. Marcati: Well/ill posedness for the Euler-Korteweg-Poisson system and related problems. Commun. Partial Differ. Equations 40 (2015), 1314–1335.

    Article  MATH  Google Scholar 

  12. E. Feireisl, A. Novotný: Inviscid incompressible limits of the full Navier-Stokes-Fourier system. Commun. Math. Phys. 321 (2013), 605–628.

    Article  MATH  Google Scholar 

  13. T. Hmidi: The low Mach number limit for the isentropic Euler system with axisymmetric initial data. J. Inst. Math. Jussieu 12 (2013), 335–389.

    Article  MATH  Google Scholar 

  14. M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, V. Schweikhard: Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics. Phys. Rev. A 74 (2006), Article ID 023623.

  15. T. Iguchi: The incompressible limit and the initial layer of the compressible Euler equation in ℝ n+ . Math. Methods Appl. Sci. 20 (1997), 945–958.

    Article  MATH  Google Scholar 

  16. D. Jamet, D. Torres, J. U. Brackbill: On the theory and computation of surface tension: The elimination of parasitic currents through energy conservation in the second-gradient method. J. Comput. Phys. 182 (2002), 262–276.

    Article  MATH  Google Scholar 

  17. S. Jiang, Q. Ju, F. Li: Incompressible limit of the nonisentropic ideal magnetohydrodynamic equations. SIAM J. Math. Anal. 48 (2016), 302–319.

    Article  MATH  Google Scholar 

  18. A. Jüngel, C.-K. Lin, K.-C. Wu: An asymptotic limit of a Navier-Stokes system with capillary effects. Commun. Math. Phys. 329 (2014), 725–744.

    Article  MATH  Google Scholar 

  19. S. Klainerman, A. Majda: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34 (1981), 481–524.

    Article  MATH  Google Scholar 

  20. P.-L. Lions, N. Masmoudi: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl., IX. Sér. 77 (1998), 585–627.

    Article  MATH  Google Scholar 

  21. J. Liu, J. B. Schneider, J. P. Gollub: Three-dimensional instabilities of film flows. Phys. Fluids 7 (1995), 55–67.

    Article  Google Scholar 

  22. P. Noble, J.-P. Vila: Stability theory for difference approximations of Euler-Korteweg equations and application to thin film flows. SIAM J. Numer. Anal. 52 (2014), 2770–2791.

    Article  MATH  Google Scholar 

  23. S. Ukai: The incompressible limit and the initial layer of the compressible Euler equation. J. Math. Kyoto Univ. 26 (1986), 323–331.

    MATH  Google Scholar 

  24. D. Wang, C. Yu: Incompressible limit for the compressible flow of liquid crystals. J. Math. Fluid Mech. 16 (2014), 771–786.

    Article  MATH  Google Scholar 

  25. W.-A. Yong: A note on the zero Mach number limit of compressible Euler equations. Proc. Am. Math. Soc. 133 (2005), 3079–3085.

    Article  MATH  Google Scholar 

Download references

Funding

The research has been supported by the Natural Science Foundation of Henan Province (No. 202300410277).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, Henan Province, 450045, P. R. China

    Yajie Wang & Jianwei Yang

Authors
  1. Yajie Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianwei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianwei Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Yang, J. Low Mach number limit of a compressible Euler-Korteweg model. Appl Math 68, 99–108 (2023). https://doi.org/10.21136/AM.2022.0067-21

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/AM.2022.0067-21

Keywords

MSC 2020

Search

Navigation