Skip to main content
SpringerLink
Log in

Blowup for Projected \(\mathbf {2}\)-Dimensional Rotational \({\mathbf {C}}^{2}\) Solutions of Compressible Euler Equations

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

The compressible Euler equations are the classical model in fluid dynamics. In this study, we investigate the life span of the projected 2-dimensional rotational \(C^{2}\) non-vacuum solutions of the Euler equations. By examining the corresponding projected 2-dimensional solutions,

$$\begin{aligned} (\rho (t,x_{1},x_{2}),u_{1}(t,x_{1},x_{2}),u_{2}(t,x_{1},x_{2}),0), \end{aligned}$$

in \(\mathbf {R}^{3}\), we prove that there exist the corresponding blowup results for the rotational \(C^{2}\) solutions with a sufficiently large initial functional

$$\begin{aligned} H(0)= {\displaystyle \int _{\mathbf {R}^{3}}} \vec {x}\cdot \vec {u}_{0}dV. \end{aligned}$$

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Billingham, J., King, A.C.: Wave Motion. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  2. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106. Springer, New York (1994)

    Book  Google Scholar 

  3. Chae, D.H., Ha, S.Y.: On the formation of shocks to the compressible Euler equations. Commun. Math. Sci. 7, 627–634 (2009)

    Article  MathSciNet  Google Scholar 

  4. Chemin, J.Y.: Dynamique des Gaz à Masse Totale Finie. Asymptot. Anal. 3, 215–220 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Chen, G.Q., Wang, D.H.: The Cauchy problem for the Euler equations for compressible fluids. In: Handbook of Mathematical Fluid Dynamics, vol. I, pp. 421–543. North-Holland, Amsterdam (2002)

  6. Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics. Texts in Applied Mathematics, vol. 4, 3rd edn. Springer, New York (1993)

    Book  Google Scholar 

  7. Christodoulou, D.: The Formation of Shocks in 3-Dimensional Fluids. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2007)

  8. Christodoulou, D., Miao, S.: Compressible Flow and Euler’s Equations. Surveys of Modern Mathematics, vol. 9. International Press, Somerville (2014)

    MATH  Google Scholar 

  9. Constantin, A.: Breaking Water Waves. Encyclopedia of Mathematical Physics, pp. 383–386. Elsevier, Amsterdam (2006)

    Book  Google Scholar 

  10. Currie, I.G.: Fundamental Mechanics of Fluids, 3rd edn. Marcel Dekker, New York (2003)

    MATH  Google Scholar 

  11. Einzel, D.: Superfluids. Encyclopedia of Mathematical Physics, pp. 115–121. Elsevier, Amsterdam (2006)

    Book  Google Scholar 

  12. Engelberg, S.: Formation of singularities in the Euler and Euler-Poisson equations. Phys. D 98, 67–74 (1996)

    Article  MathSciNet  Google Scholar 

  13. Fefferman, C.L.: Existence and smoothness of the Navier–Stokes equation. In: The Millennium Prize Problems, pp. 57–67. Clay Mathematics Institute, Cambridge, MA (2006)

  14. Lei, Z., Du, Y., Zhang, Q.: Singularities of solutions to compressible Euler equations with vacuum. Math. Res. Lett. 20, 41–50 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  15. Lions, P.L.: Mathematical Topics in Fluid Mechanics, vols. 1, 2. Clarendon Press, Oxford (1998)

    Google Scholar 

  16. Liu, T.P.: Compressible flow with damping and vacuum. Jpn. J. Ind. Appl. Math. 13, 25–32 (1996)

    Article  MathSciNet  Google Scholar 

  17. Luk, J., Speck, J.: Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity. Invent. Math. 214, 1–169 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  18. Makino, T., Ukai, S., Kawashima, S.: Sur la Solution à Support Compact de l’Équations d’Euler Compressible (French) [On the compactly supported solution of the compressible Euler equations]. Jpn. J. Appl. Math. 3, 249–257 (1986)

    Article  Google Scholar 

  19. Serre, D.: Expansion of a compressible gas in vacuum. Bull. Inst. Math. Acad. Sin. (N.S.) 10, 695–716 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Sideris, T.C.: Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys. 101, 475–485 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  21. Sideris, T.C.: Spreading of the free boundary of an ideal fluid in a vacuum. J. Differ. Equ. 257, 1–14 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  22. Suzuki, T.: Irrotational blowup of the solution to compressible Euler equation. J. Math. Fluid Mech. 15, 617–633 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  23. Yuen, M.W.: Blowup for the Euler and Euler-Poisson equations with repulsive forces. Nonlinear Anal. TMA 74, 1465–1470 (2011)

    Article  MathSciNet  Google Scholar 

  24. Yuen, M.W.: Blowup for irrotational \(C^{1} \) solutions of the compressible Euler equations in \(R^{N}\). Nonlinear Anal. TMA 158, 132–141 (2017)

    Article  Google Scholar 

  25. Yuen, M.W.: Blowup for regular solutions and \(C^{1}\) solutions of Euler equations in \(R^{N}\) with a free boundary. Eur. J. Mech. B Fluids 67, 427–432 (2018)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks for the anonymous reviewers’ valuable comments for improving the quality of this article. This research was partially supported by the Dean’s Research Fund 2015-16 (FLASS/DRF/SFRS-6) from the Education University of Hong Kong.

Author information

Authors and Affiliations

  1. Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong

    Manwai Yuen

Authors
  1. Manwai Yuen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manwai Yuen.

Ethics declarations

Conflict of interest

The author declare that there is no conflict of interest.

Additional information

Communicated by A. Constantin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuen, M. Blowup for Projected \(\mathbf {2}\)-Dimensional Rotational \({\mathbf {C}}^{2}\) Solutions of Compressible Euler Equations. J. Math. Fluid Mech. 21, 54 (2019). https://doi.org/10.1007/s00021-019-0458-x

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-019-0458-x

Keywords

Mathematics Subject Classification

Search

Navigation