Skip to main content

Advertisement

Log in

Existence and regularity of solutions for semilinear fractional Rayleigh–Stokes equations

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper deals with the semilinear Rayleigh–Stokes equation with the fractional derivative in time of order \(\alpha \in (0,1)\), which can be used to model anomalous diffusion in viscoelastic fluids. An operator family related to this problem is defined, and its regularity properties are investigated. We firstly give the concept of the mild solutions in terms of the operator family and then obtain the existence of global mild solutions by means of fixed point technique. Moreover, the existence and regularity of classical solutions are given.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

Data availability

No data, models, or code were generated or used during this study.

References

  1. Allen, M., Caffarelli, L., Vasseur, A.: A parabolic problem with a fractional time derivative. Arch. Ration. Mech. Anal. 221(2), 603–630 (2016)

    Article  MathSciNet  Google Scholar 

  2. Bazhlekova, E., Jin, B., Lazarov, R., Zhou, Z.: An analysis of the Rayleigh–Stokes problem for a generalized second-grade fluid. Numer. Math. 131(1), 1–31 (2015)

    Article  MathSciNet  Google Scholar 

  3. Dong, H., Kim, D.: \(L_{p}\)-estimates for time fractional parabolic equations with coefficients measurable in time. Adv. Math. 345, 289–345 (2019)

    Article  MathSciNet  Google Scholar 

  4. Fetecau, C., Jamil, M., Fetecau, C., Vieru, D.: The Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid. Z. Angew. Math. Phys. 60(5), 921–933 (2009)

    Article  MathSciNet  Google Scholar 

  5. Fetecau, C., Zierep, J.: The Rayleigh–Stokes problem for a Maxwell fluid. Z. Angew. Math. Phys. 54(6), 1086–1093 (2003)

    Article  MathSciNet  Google Scholar 

  6. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equation. Elsevier, Amsterdam (2006)

    Google Scholar 

  7. Kim, I., Kim, K.-H., Lim, S.: An \(L_{q}(L_{p})\)-theory for the time fractional evolution equations with variable coefficients. Adv. Math. 306, 123–176 (2017)

    Article  MathSciNet  Google Scholar 

  8. Nguyen, H.L., Nguyen, H.T., Kirane, M., Duong, D.X.T.: Identifying initial condition of the Rayleigh–Stokes problem with random noise. Math. Methods Appl. Sci. 42, 1561–1571 (2019)

    Article  MathSciNet  Google Scholar 

  9. Nguyen, H.T., Nguyen, H.L., Tuan, A.N.: Some well-posed results on the time-fractional Rayleigh–Stokes problem with polynomial and gradient nonlinearities. Math. Methods Appl. Sci. 45(1), 500–514 (2022)

    Article  MathSciNet  Google Scholar 

  10. Nguyen, H.L., Nguyen, H.T., Zhou, Y.: Regularity of the solution for a final value problem for the Rayleigh–Stokes equation. Math. Methods Appl. Sci. 42, 3481–3495 (2019)

    Article  MathSciNet  Google Scholar 

  11. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)

    Article  MathSciNet  Google Scholar 

  12. Shen, F., Tan, W., Zhao, Y., Masuoka, T.: The Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative model. Nonlinear Anal. Real World Appl. 7(5), 1072–1080 (2006)

    Article  MathSciNet  Google Scholar 

  13. Tuan, N.H., Zhou, Y., Thach, T.N., Can, N.H.: Initial verse problem for the nonlinear fractional Rayleigh–Stokes equation with random discrete date. Commun. Nonlinear Sci. Numer. Simul. 78, 104873 (2019)

    Article  MathSciNet  Google Scholar 

  14. Tuan, P.T., Ke, T.D., Thang, N.N.: Final value problem for Rayleigh–Stokes type equations involving weak-valued nonlinearities. Fract. Calc. Appl. Anal. 26(2), 694–717 (2023)

    Article  MathSciNet  Google Scholar 

  15. Wang, J.N., Alsaedi, A., Ahmad, B., Zhou, Y.: Well-posedness and blow-up results for a class of nonlinear fractional Rayleigh–Stokes problem. Adv. Nonlinear Anal. 11, 1579–1597 (2022)

    Article  MathSciNet  Google Scholar 

  16. Wang, J.N., Zhou, Y., Alsaedi, A., Ahmad, B.: Well-posedness and regularity of fractional Rayleigh–Stokes problem. Z. Angew. Math. Phys. 73, 161 (2022)

    Article  MathSciNet  Google Scholar 

  17. Zacher, R.: A De Giorgi–Nash type theorem for time fractional diffusion equations. Math. Ann. 356(1), 99–146 (2013)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Zhou, Y., Wang, J.N.: The nonlinear Rayleigh-Stokes problem with Riemann–Liouville fractional derivative. Math. Methods Appl. Sci. 44, 2431–2438 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the NSFC under the Grants 12271269 and 12326318 and the Fundamental Research Funds for the Central Universities. The author is grateful to the referees for their careful reading and valuable comments.

Author information

Authors and Affiliations

Authors

Contributions

All authors are ordered alphabetically, and all authors contributed equally to this work.

Corresponding author

Correspondence to Jingchuang Ren.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Jiang, Y., Ren, J. & Wei, Y. Existence and regularity of solutions for semilinear fractional Rayleigh–Stokes equations. Z. Angew. Math. Phys. 75, 100 (2024). https://doi.org/10.1007/s00033-024-02251-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-024-02251-6

Keywords

Mathematics Subject Classification

Navigation