Skip to main content
SpringerLink
Log in

Singularity Formation for the Multi-dimensional Compressible Degenerate Navier–Stokes Equations

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, the multi-dimensional (M-D) isentropic compressible Navier–Stokes equations with degenerate viscosities (ICNS) is considered in the whole space. We show that for a certain class of initial data with local vacuum, the regular solution of the corresponding Cauchy problem will blow up in finite time, no matter how small and smooth the initial data are. It is worth pointing out that local existence of regular solution considered in this paper has been established.

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. Bresch, D., Desjardins, B.: Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238, 211–223 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bresch, D., Desjardins, B., Métivier, G.: Recent mathematical results and open problems about shallow water equations. In: Analysis and Simulation of Fluid Dynamics, Adv. Math. Fluid Mech., pp. 15-31. Birkhäuser, Basel, (2007)

  3. Chapman, S., Cowling, T.: The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  4. Gerbeau, J., Perthame, B.: Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. (B) 1, 89–102 (2001)

    MathSciNet  MATH  Google Scholar 

  5. Jiu, Q., Wang, Y., Xin, Z.: Remarks on blow-up of smooth solutions to the compressible fluid with constant and degenerate viscosities. J. Differ. Equ. 259, 2981–3003 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jüngel, A.: Global weak solutions to compressible Navier–Stokes equations for quantum fluids. SIAM. J. Math. Anal. 42, 1025–1045 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, J., Xin, Z.: Global existence of weak solutions to the barotropic compressible Navier–Stokes flows with degenerate viscosities, preprint, arXiv:1504.06826, (2016)

  8. Li, T., Qin, T.: Physics and Partial Differential Equations, Vol. II., SIAM: Philadelphia, Higher Education Press: Beijing, (2014)

  9. Li, Y., Pan, R., Zhu, S.: On classical solutions to 2D Shallow water equations with degenerate viscosities. J. Math. Fluid Mech. 19, 151–190 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, Y., Pan, R., Zhu, S.: On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum. Arch. Rational. Mech. Anal. 234, 1281–1334 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, T., Xin, Z., Yang, T.: Vacuum states for compressible flow. Discrete Contin. Dyn. Syst. 4, 1–32 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Marche, F.: Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. Euro. J. Mech. B/Fluids 26, 49–63 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mellet, A., Vasseur, A.: On the barotropic compressible Navier–Stokes equations. Commun. Part. Differ. Equ. 32, 431–452 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

  15. Vasseur, A., Yu, C.: Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations. Invent. Math. 206, 935–974 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, W., Xu, C.: The Cauchy problem for viscous shallow water equations. Rev. Mat. Iberoam 21, 1C24 (2005)

  17. Xin, Z., Zhu, S.: Global well-posedness of regular solutions to the three-dimensional isentropic compressible Navier–Stokes equations with degenerate viscosities and vacuum, submitted, arXiv:1806.02383v2, (2018)

  18. Xin Z., Zhu S.: Well-posedness of three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and far field vacuum, J. Math. Pure Appl. 152, 94–144 (2021). https://doi.org/10.1016/j.matpur.2021.05.004

  19. Xin, Z., Yan, W.: On blow-up of classical solutions to the compressible Navier–Stokes equations. Commun. Math. Phys. 321, 529–541 (2013)

    Article  MATH  Google Scholar 

  20. Yang, T., Zhu, C.: Compressible Navier–Stokes equations with degenerate viscosity coefficient and vacuum. Commun. Math. Phys. 230, 329–363 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research of Yucong Huang was supported in part by the UK Engineering and Physical Sciences Research Council Award EP/L015811/1. The research of Qin Wang was supported in part by National Natural Science Foundation of China (11761077), Key Project of Yunnan Provincial Science and Technology Department and Yunnan University [No. 2018FY001(-014)]. The research of Shengguo Zhu was supported in part by the Royal Society–Newton International Fellowships NF170015.

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Yucong Huang

  2. School of Mathematics, Yunnan University, Kunming, 650091, People’s Republic of China

    Qin Wang

  3. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Shengguo Zhu

Authors
  1. Yucong Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shengguo Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shengguo Zhu.

Additional information

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

Huang, Y., Wang, Q. & Zhu, S. Singularity Formation for the Multi-dimensional Compressible Degenerate Navier–Stokes Equations. J Dyn Diff Equat 35, 1769–1783 (2023). https://doi.org/10.1007/s10884-021-10038-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-021-10038-w

Keywords

Mathematics Subject Classification

Search

Navigation