Skip to main content
SpringerLink
Log in

Well-Posedness of Mild Solutions for the Fractional Navier–Stokes Equations in Besov Spaces

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper addresses the existence and uniqueness of mild solution for fractional Navier–Stokes equation in the homogeneous Besov spaces. The existence and uniqueness of the global mild solution with small initial data is established. The results on existence and uniqueness of the local mild solution for arbitrary initial data are also studied. Meanwhile, we obtain properties about the regularity and decay of mild solutions. These conclusions are mainly based on the fixed point theorems, the implicit function theorem on Banach spaces and the properties of Mittag–Leffler functions.

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

Data Availability

Not applicable.

References

  1. Varnhorn, W.: The Stokes Equations. Akademie Verlag, Berlin (1994)

    Google Scholar 

  2. Cannone, M.: Ondelettes. Paraproduits et Navier–Stokes, Diderot Editeur (1995)

  3. Kato, T.: Strong \(L^{p}\)-solutions of the Navier–Stokes equation in \(\mathbb{R} ^{m}\), with applications to weak solutions. Math. Z. 187, 471–480 (1984)

    Article  MathSciNet  Google Scholar 

  4. Taylor, M.E.: Analysis on Morrey spaces and applications to Navier–Stokes equation. Commun. Partial Differ. Equ. 17, 1407–1456 (1992)

    Article  MathSciNet  Google Scholar 

  5. Cannone, M., Planchon, F.: Self-similar solutions for Navier–Stokes equations in \(\mathbb{R} ^{3}\). Commun. Partial Differ. Equ. 21, 179–194 (1996)

    Article  Google Scholar 

  6. Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  7. Weissler, F.B.: The Navier–Stokes initial value problem in \(L^{p}\). Arch. Ration. Mech. Anal. 74, 219–230 (1980)

    Article  Google Scholar 

  8. Carpio, A.: Large-time behavior in incompressible Navier–Stokes equations. SIAM J. Math. Anal. 27, 449–475 (1996)

    Article  MathSciNet  Google Scholar 

  9. Bae, H.-O., Jin, B.J.: Upper and lower bounds of temporal and spatial decays for the Navier–Stokes equations. J. Differ. Equ. 209, 365–391 (2005)

    Article  MathSciNet  Google Scholar 

  10. Kozono, H., Shimizu, S.: Strong solutions of the Navier–Stokes equations based on the maximal Lorentz regularity theorem in Besov spaces. J. Funct. Anal. 276, 896–931 (2019)

    Article  MathSciNet  Google Scholar 

  11. Kozono, H., Okada, A., Shimizu, S.: Characterization of initial data in the homogeneous Besov space for solutions in the Serrin class of the Navier–Stokes equations. J. Funct. Anal. 278, 108390 (2020)

    Article  MathSciNet  Google Scholar 

  12. Knightly, G.H.: A Cauchy Problem for the Navier–Stokes equations in \(R^{n}\). SIAM J. Math. Anal. 3, 506–511 (1972)

    Article  MathSciNet  Google Scholar 

  13. Heck, H., Kim, H., Kozono, H.: Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle. Math. Ann. 356, 653–681 (2013)

    Article  MathSciNet  Google Scholar 

  14. Zhai, X.P., Li, Y.S., Zhou, F.J.: Global large solutions to the three dimensional compressible Navier–Stokes equations. SIAM J. Math. Anal. 52, 1806–1843 (2020)

    Article  MathSciNet  Google Scholar 

  15. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing Co., Inc, River Edge, NJ (2000)

  16. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V, Amsterdam (2006)

  17. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Book  Google Scholar 

  18. Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Academic Press (2016)

  19. Hornung, G., Berkowitz, B., Barkai, N.: Morphogen gradient formation in a complex environment: an anomalous diffusion model. Phys. Rev. E 72, 041916 (2005)

    Article  MathSciNet  Google Scholar 

  20. de Carvalho-Neto, P.M., Planas, G.: Mild solutions to the time fractional Navier–Stokes equations in \(\mathbb{R} ^{N}\). J. Differ. Equ. 259, 2948–2980 (2015)

    Article  Google Scholar 

  21. Zhou, Y., Peng, L.: On the time-fractional Navier–Stokes equations. Comput. Math. Appl. 73, 874–891 (2017)

    Article  MathSciNet  Google Scholar 

  22. Zhou, Y., Peng, L.: Weak solutions of the time-fractional Navier–Stokes equations and optimal control. Comput. Math. Appl. 73, 1016–1027 (2017)

    Article  MathSciNet  Google Scholar 

  23. Zhou, Y., Peng, L., Huang, Y.Q.: Existence and Hölder continuity of solutions for time-fractional Navier–Stokes equations. Math. Methods Appl. Sci. 41, 7830–7838 (2018)

    Article  MathSciNet  Google Scholar 

  24. Wang, Y.J., Liang, T.T.: Mild solutions to the time fractional Navier–Stokes delay differential inclusions. Discrete Contin. Dyn. Syst. Ser. B 24, 3713–3740 (2019)

    MathSciNet  Google Scholar 

  25. Zheng, R.M., Jiang, X.Y.: Spectral methods for the time-fractional Navier–Stokes equation. Appl. Math. Lett. 91, 194–200 (2019)

    Article  MathSciNet  Google Scholar 

  26. Xi, X.-X., Hou, M.M., Zhou, X.-F., Wen, Y.H.: Approximate controllability for mild solution of time-fractional Navier–Stokes equations with delay. Z. Angew. Math. Phys. 72, 113 (2021)

    Article  MathSciNet  Google Scholar 

  27. Kirane, M., Aimene, D., Seba, D.: Local and global existence of mild solutions of time-fractional Navier–Stokes system posed on the Heisenberg group. Z. Angew. Math. Phys. 72, 116 (2021)

    Article  MathSciNet  Google Scholar 

  28. Kozono, H., Shimizu, S.: Navier–Stokes equations with external forces in time-weighted Besov spaces. Math. Nachr. 291, 1781–1800 (2018)

    Article  MathSciNet  Google Scholar 

  29. Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2011)

  30. Sawano, Y.: Theory of Besov Spaces, Developments in Mathematics. Springer, Singapore (2018)

    Book  Google Scholar 

  31. Bergh, J., Löfström, J.: Interpolation Spaces an Introduction, Grundlehren der Mathematische Wisssenshaften. Springer, Berlin (1976)

    Book  Google Scholar 

  32. Kaneko, K., Kozono, H., Shimizu, S.: Stationary solution to the Navier–Stokes equations in the scaling invariant Besov space and its regularity. Indiana Univ. Math. J. 68, 857–880 (2019)

    Article  MathSciNet  Google Scholar 

  33. Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252, 202–235 (2012)

    Article  MathSciNet  Google Scholar 

  34. Kozono, H., Okada, A., Shimizu, S.: Necessary and sufficient condition on initial data in the Besov space for solutions in the Serrin class of the Navier–Stokes equations. J. Evol. Equ. 21, 3015–3033 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by National Natural Science Foundation of China (12071396).

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Xuan-Xuan Xi & Yong Zhou

  2. School of Computer Science and Engineering, Macau University of Science and Technology, Macau, 999078, People’s Republic of China

    Yong Zhou

  3. School of Mathematical Sciences, Huaibei Normal University, Anhui, 235000, People’s Republic of China

    Mimi Hou

Authors
  1. Xuan-Xuan Xi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mimi Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

X.X. Xi wrote the main manuscript text after discussing main technique with Y. Zhou and M. Hou. All authors reviewed the manuscript.

Corresponding author

Correspondence to Yong Zhou.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Ethical approval

Not applicable.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xi, XX., Zhou, Y. & Hou, M. Well-Posedness of Mild Solutions for the Fractional Navier–Stokes Equations in Besov Spaces. Qual. Theory Dyn. Syst. 23, 15 (2024). https://doi.org/10.1007/s12346-023-00867-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00867-z

Keywords

Mathematics Subject Classification

Search

Navigation