Skip to main content
SpringerLink
Log in

Stability of Navier–Stokes System with Singular External Force in Fourier–Herz Space

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We prove the asymptotic properties of the solutions to the 3D Navier-Stokes system with singular external force, by making use of Fourier localization method, the Littlewood-Paley theory and some subtle estimates in Fourier-Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339 (2022)]. We deal either with the non-stationary problem or with the stationary problem where solution may be singular due to singular external force. In this paper, the Fourier-Herz space includes the function space of pseudomeasure type used in Cannone et al. [J. Differential Equations, 314, 316–339 (2022)]

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. Bahouri, H., Chemin, J. Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren er Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), Vol. 343, Springer, Heidelberg, 2011

    Book  MATH  Google Scholar 

  2. Cannone, M.: A generalization of a theorem by Kato on Navier-Stokes equations. Rev. Mat. Iberoam, 13(3), 515–541 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cannone, M.: Harmonic analysis tools for solving the incompressible Navier-Stokes equations. Handbook of Mathematical Fluid Dynamics, III, 161–244 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Cannone, M., Karch, G.: Smooth or singular solutions to the Navier-Stokes system? J. Differential Equations, 197, 247–274 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cannone, M., Karch, G., Pilarczyk, D., et al.: Stability of singular solutions to the Navier-Stokes system. Journal of Differential Equations, 314, 316–339 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cannone, M., Wu, G.: Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces. Nonlinear Analysis: Theory, Methods Applications, 75(9), 3754–3760 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Iwabuchi, T., Takada, R.: Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function space of Besov type. J. Funct. Anal, 267, 1321–1337 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Karch, G., Schonbek, M. E., Schonbek, T. P.: Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete Contin. Dyn. Syst., 40, 189–206 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Karch, G., Zheng, X.: Time-dependent singularities in the Navier-Stokes system. Discrete Contin. Dyn. Syst., 35, 3039–3057 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kato, T.: Strong Lp-solutions of Navier-Stokes equations in Rm, with applications to weak solutions. Math. Z., 187, 471–480 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Konieczny, P., Yoneda, T.: On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations. J. Differential Equations, 250, 3859–3873 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, L., Li, Y., Yan, X.: Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. I. One singularity. Arch. Ration. Mech. Anal., 227, 1091–1163 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, L., Li, Y., Yan, X.: Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. II. Classification of axisymmetric no-swirl solutions. J. Differential Equations, 264, 6082–6108 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, L., Li, Y., Yan, X.: Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. III. Two singularity. Discrete Contin. Dyn. Syst., 39, 7163–7211 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, L., Li, Y., Yan, X.: Asympototic stability of homogeneous solutions of incompressible stationary Navier-Stokes equations. J. Differential Equations, 297, 226–245 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Landau, L. D.: A new exact solution of Navier-Stokes equations. C. R. (Doklady) Acad. Sci. URSS (N.S.), 43, 286–288 (1944)

    MathSciNet  MATH  Google Scholar 

  17. Li, J., Zheng, X.: The well-posedness of the incompressible magnetohydro dynamic equations in the framework of Fourier-Herz space. J. Differential Equations, 263, 3419–3459 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nan, Z., Zheng, X.: Existence and uniqueness of solutions for Navier-Stokes equations with hyperdissipation in a large space. J. Differential Equations, 261, 3670–3703 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Šverák, V.: On Landau’s solutions of the Navier-Stokes equations. J. Math. Sci. (N.Y.), 179, 208–228 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yamazaki, M.: The Navier-Stokes equations in the weak-Ln space with time-dependent external force. Math. Ann., 317, 635–675 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their time and comments.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, P. R. China

    De Zai Min, Qing Kai Wang, Gang Wu & Zhuo Ya Yao

Authors
  1. De Zai Min
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qing Kai Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhuo Ya Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhuo Ya Yao.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 11771423)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Min, D.Z., Wang, Q.K., Wu, G. et al. Stability of Navier–Stokes System with Singular External Force in Fourier–Herz Space. Acta. Math. Sin.-English Ser. 39, 1203–1218 (2023). https://doi.org/10.1007/s10114-023-1617-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-023-1617-9

Keywords

MR(2010) Subject Classification

Search

Navigation