Skip to main content
SpringerLink
Log in

Approximate controllability for mild solution of time-fractional Navier–Stokes equations with delay

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

Abstract

This paper addresses some results about mild solution to time-fractional Navier–Stokes equations with bounded delay existed in the convective term and the external force. Based on the Schauder’s fixed point theorem, we establish the sufficient conditions for the existence and uniqueness of the global mild solution in Banach spaces. Moreover, the polynomial decay estimate of the mild solution is given. Furthermore, the approximate controllability for the mild solution is obtained by constructing a Cauchy sequence.

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Yang, T.Y., Zhang, Z.Y.: Large time behaviour of solutions to the 3D Navier-Stokes equations with damping. Z. Angew. Math. Phys. (2020). https://doi.org/10.1007/s00033-020-01404-7

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  6. 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 

  7. Chemin, J., Gallagher, I.: On the global wellposedness of the 3-D Navier–Stokes equations with large initial data. Ann. Sci. de l’\(\acute{E}\)cole Norm. Sup\(\acute{e}\)rieure 39, 679–698 (2006)

  8. Chemin, J., Gallagher, I.: Well-posedness and stability results for the Navier–Stokes equations in \({\mathbb{R}}^{3}\). Ann. Inst. H. Poincar\(\acute{e}\) Anal. Non Lin\(\acute{e}\)aie 26, 599–624 (2009)

  9. Girault, V., Rivi\(\grave{e}\)re, B.: DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition, SIAM J. Numer. Anal. 47,2052-2089(2009)

  10. Zhou, Y., Peng, L., Ahmad, B., Alsaedi, A.: Energy methods for fractional Navier–Stokes equations. Chaos Solitons Fractals 102, 78–85 (2017)

    Article  MathSciNet  Google Scholar 

  11. Zhang, Q., Zhang, Y.H.: Global well-posedness for the 3D incompressible Keller–Segel–Navier–Stokes equations. Z. Angew. Math. Phys. (2019). https://doi.org/10.1007/s00033-019-1185-0

    Article  MathSciNet  MATH  Google Scholar 

  12. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1995)

    MATH  Google Scholar 

  13. Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45, 1561–1585 (2006)

    Article  MathSciNet  Google Scholar 

  14. Caraballo, T., Real, J.: Navier–Stokes equations with delays. Proc. R. Soc. Lond. A 457, 2441–2453 (2001)

    Article  MathSciNet  Google Scholar 

  15. Garrido-Atienza, M.J., Marin-Rubio, P.: Navier–Stokes equations with delays on unbounded domains. Nonlinear Anal. 64, 1100–1118 (2006)

    Article  MathSciNet  Google Scholar 

  16. Caraballo, T., Márquez-Durán, A.M., Real, J.: Asymptotic behaviour of the three-dimensional \(\alpha \)-Navier–Stokes model with locally Lipschitz delay forcing terms. Nonlinear Anal. 71, e227–e282 (2009)

    Article  MathSciNet  Google Scholar 

  17. Niche, C.J., Planas, G.: Existence and decay of solutions in full space to Navier–Stokes equations with delays. Nonlinear Anal. 74, 244–256 (2011)

    Article  MathSciNet  Google Scholar 

  18. Liu, L.F., Caraballo, T., Marin-Rubio, P.: Stability results for 2D Navier–Stokes equations with unbounded delay. J. Differ. Equ. 265, 5685–5708 (2018)

    Article  MathSciNet  Google Scholar 

  19. Liu, W.: Asymptotic behavior of solutions of time-delayed Burgers’ equation. Discrete Continuous Dyn. Syst. Ser. B 2, 47–56 (2002)

    Article  MathSciNet  Google Scholar 

  20. Tang, Y., Wan, M.: A remark on exponential stability of time-delayed Burgers equation. Discrete Continuous Dyn. Syst. Ser. B 12, 219–225 (2009)

    Article  MathSciNet  Google Scholar 

  21. Planas, G., Hernández, E.: Asymptotic behaviour of two-dimensional time-delayed Navier–Stokes equations. Discrete Continuous Dyn. Syst. Ser. 21, 1245–1258 (2008)

    Article  MathSciNet  Google Scholar 

  22. Guzzo, S.M., Planas, G.: On a class of three dimensional Navier–Stokes equations with bounded delay. Discrete Continuous Dyn. Syst. Ser. B 16, 225–238 (2011)

    Article  MathSciNet  Google Scholar 

  23. Bessaih, H., Garrido-Atienza, M.J., Schmalfub, B.: On 3D Navier–Stokes equations: regularization and uniqueness by delays. Physica D 376, 228–237 (2018)

    Article  MathSciNet  Google Scholar 

  24. Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011)

    Book  Google Scholar 

  25. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  Google Scholar 

  26. Wang, W.H.: Global well-posedness and analyticity for the 3D fractional magnetohydrodynamics equations in variable Fourier–Besov spaces. Z. Angew. Math. Phys. (2019). https://doi.org/10.1007/s00033-019-1210-3

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. 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 

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

    Article  MathSciNet  Google Scholar 

  30. Liu, Z.H., Li, X.W.: Approximate controllability of fractional evolution systems with Riemann–Liouville fractional derivatives. SIAM J. Control Optim. 53, 1920–1933 (2015)

    Article  MathSciNet  Google Scholar 

  31. Mahmudov, N.I.: Finite-approximate controllability of fractional evolution equations: variational approach. Fract. Calc. Appl. Anal. 21, 919–936 (2018)

    Article  MathSciNet  Google Scholar 

  32. Mahmudov, N.I.: Partial-approximate controllability of nonlocal fractional evolution equations via approximating method. Appl. Math. Comput. 334, 227–238 (2018)

    MathSciNet  MATH  Google Scholar 

  33. Li, X.W., Liu, Z.H., Li, J., Tisdell, C.: Existence and controllability for nonlinear fractional control systems with damping in Hilbert spaces. Acta Math. Sci. 39, 229–242 (2019)

    Article  MathSciNet  Google Scholar 

  34. Zhou, Y., He, J.W.: New results on controllability of fractional evolution systems with order \(\alpha \in (1,2)\). Evol. Equ. Control. Theory 9, 1–19 (2020)

    Article  MathSciNet  Google Scholar 

  35. Vijayakumar, V., Selvakumar, A., Murugesu, R.: Controllability for a class of fractional neutral integro-differential equations with unbounded delay. Appl. Math. Comput. 232, 303–312 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of semilinear fractional control systems of order \(\alpha \in (1,2]\) with infinite delay. Mediterr. J. Math. 13, 2539–2550 (2016)

    Article  MathSciNet  Google Scholar 

  37. Zhou, Y., Suganya, S., Arjunan, M.M., Ahmad, B.: Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces. IMA J. Math. Control. Inf. 36, 603–622 (2019)

    Article  MathSciNet  Google Scholar 

  38. You, Z.L., Feckan, M., Wang, J.R.: Relative controllability of fractional delay differential equations via delayed perturbation of Mittag–Leffler functions. J. Comput. Appl. Math. (2020). https://doi.org/10.1016/j.cam.2020.112939

    Article  MathSciNet  MATH  Google Scholar 

  39. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, in: North-Holland Math. Stud., vol. 204, Elsevier Science B.V., Amsterdam (2006)

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

  42. Sell, G.R., You, Y.C.: Dynamics of Evolutionary Equations. Springer, New York (2002)

    Book  Google Scholar 

  43. Mainardi, F.: On the initial value problem for the fractional diffusion-wave equation. Ser. Adv. Math. Appl. Sci. 23, 246–251 (1994)

    MathSciNet  Google Scholar 

  44. 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 

  45. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, 230039, China

    Xuan-Xuan Xi, Mimi Hou & Xian-Feng Zhou

  2. School of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230601, China

    Yanhua Wen

Authors
  1. Xuan-Xuan Xi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mimi Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xian-Feng Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yanhua Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian-Feng Zhou.

Additional information

Publisher's Note

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

This work is supported by the National Natural Science Foundation of China (11471015 and 11601003); Natural Science Foundation of Anhui Province (1508085MA01, 1608085MA12, 1708085MA15 and 2008085QA19) and Program of Natural Science Research for Universities of Anhui Province (KJ2016A023 and K1930096)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xi, XX., Hou, M., Zhou, XF. et al. Approximate controllability for mild solution of time-fractional Navier–Stokes equations with delay. Z. Angew. Math. Phys. 72, 113 (2021). https://doi.org/10.1007/s00033-021-01542-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01542-6

Keywords

Mathematics Subject Classification

Search

Navigation