Skip to main content
SpringerLink
Log in

On the global well-posedness for the 3D axisymmetric incompressible Keller–Segel–Navier–Stokes equations

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

Abstract

In this paper, we study the Cauchy problem for the three-dimensional incompressible Keller–Segel–Navier–Stokes equations. By taking advantage of the geometry of axisymmetric flow without swirl, we obtain the global well-posedness for the system.

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. Ahn, J., Kang, K., Kim, J., Lee, J.: Lower bound of mass in a chemotactic model with advection and absorbing reaction. SIAM J. Math. Anal. 49, 723–755 (2017)

    Article  MathSciNet  Google Scholar 

  2. Coll, J.C., et al.: Chemical aspects of mass spawning in corals. I. Sperm-attractant molecules in the eggs of the scleractinian coral Montipora digitata. Mar. Biol. 118, 177–182 (1994)

    Article  Google Scholar 

  3. Coll, J.C., et al.: Chemical aspects of mass spawning in corals. II. (-)-Epi-thunbergol, the sperm attractant in the eggs of the soft coral Lobophytum crassum (Cnidaria: Octocorallia). Mar. Biol. 123, 137–143 (1995)

    Article  Google Scholar 

  4. Di Francesco, M., Lorz, A., Markowich, P.: Chemotaxis fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior. Discrete Contin. Dyn. Syst. 28, 1437–1453 (2010)

    Article  MathSciNet  Google Scholar 

  5. Espejo, E., Suzuki, T.: Reaction terms avoiding aggregation in slow fluids. Nonlinear Anal. Real World Appl. 21, 110–126 (2015)

    Article  MathSciNet  Google Scholar 

  6. Jin, C.: Large time periodic solutions to coupled chemotaxis-fluid models. Z. Angew. Math. Phys. 68, 137 (2017)

    Article  MathSciNet  Google Scholar 

  7. Kiselev, A., Ryzhik, L.: Biomixing by chemotaxis and enhancement of biological reactions. Commun. Partial Differ. Equ. 37, 298–312 (2012)

    Article  MathSciNet  Google Scholar 

  8. Kiselev, A., Ryzhik, L.: Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case. J. Math. Phys. 53, 115609 (2012)

    Article  MathSciNet  Google Scholar 

  9. Liu, J.G., Lorz, A.: A coupled chemotaxis-fluid model: global existence. Ann. Inst. H. Poincaré. Anal. Non linéare 28, 643–652 (2011)

    Article  MathSciNet  Google Scholar 

  10. Lorz, A.: A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay. Commun. Math. Sci. 10, 555–574 (2012)

    Article  MathSciNet  Google Scholar 

  11. Majda, A., Bertozzi, A.L.: Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  12. Miao, C., Wu, J., Zhang, Z.: Littlewood-Paley Theory and Applications to Fluid Dynamics Equations. Monographs on Modern Pure Mathematics, vol. 142. Science Press, Beijing (2012)

  13. Miller, R.L.: Sperm chemotaxis in hydromedusae. I. Species specifity and sperm behavior. Mar. Biol. 53, 99–114 (1979)

    Article  Google Scholar 

  14. Miller, R.L.: Demonstration of sperm chemotaxis in Echinodermata: Asteroidea, Holothuroidea, Ophiuroidea. J. Exp. Zool. 234, 383–414 (1985)

    Article  Google Scholar 

  15. Tao, Y., Winkler, M.: Global existence and boundedness in a Keller–Segel–Stokes model with arbitrary porous medium diffusion. Discrete Contin. Dyn. Syst. 32, 1901–1914 (2012)

    Article  MathSciNet  Google Scholar 

  16. Tao, Y., Winkler, M.: Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system. Z. Angew. Math. Phys. 66, 2555–2573 (2015)

    Article  MathSciNet  Google Scholar 

  17. Tao, Y., Winkler, M.: Blow-up prevention by quadratic degradation in a two-dimensional Keller–Segel–Navier–Stokes system. Z. Angew. Math. Phys. 67, 138 (2016)

    Article  MathSciNet  Google Scholar 

  18. Winkler, M.: Global large-data solutions in a chemotaxis-(Navier–)Stokes system modeling cellular swimming in fluid drops. Commun. Partial Differ. Equ. 37, 319–351 (2012)

    Article  MathSciNet  Google Scholar 

  19. Winkler, M.: Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Arch. Ration. Mech. Anal. 211, 455–487 (2014)

    Article  MathSciNet  Google Scholar 

  20. Winkler, M.: Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 1329–1352 (2016)

    Article  MathSciNet  Google Scholar 

  21. Zhang, Q., Zheng, X.: Global well-posedness for the two-dimensional incompressible chemotaxis-Navier–Stokes equations. SIAM J. Math. Anal. 46, 3078–3105 (2014)

    Article  MathSciNet  Google Scholar 

  22. Zhang, Q., Wang, P.: Global well-posedness for the 2D incompressible four-component chemotaxis-Navier–Stokes equations. J. Differ. Equ. 269, 1656–1692 (2020)

    Article  MathSciNet  Google Scholar 

  23. Zhang, Q., Zheng, X.: Global well-posedness of axisymmetric solution to the 3D axisymmetric chemotaxis-Navier–Stokes equations with logistic source. J. Differ. Equ. 274, 576–612 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Q. Hua was partially supported by the Key Science and Technology Foundation of the Education Department of Hebei Province [Grant Number ZD2019021]. Q. Zhang was partially supported by the National Natural Science Foundation of China [Grant Numbers 11501160 and 11771423]; Natural Science Foundation of Hebei Province [Grant Numbers A2017201144 and A2020201014]; Young Talents Foundation of Hebei Education Department [Grant Number BJ2017058]; the Second Batch of Young Talents of Hebei Province; Nonlinear Analysis Innovation Team of Hebei University.

Author information

Authors and Affiliations

  1. Hebei Key Laboratory of Machine Learning and Computational Intelligence, School of Mathematics and Information Science, Hebei University, Baoding, 071002, People’s Republic of China

    Qiang Hua & Qian Zhang

Authors
  1. Qiang Hua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qian Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qian Zhang.

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

Hua, Q., Zhang, Q. On the global well-posedness for the 3D axisymmetric incompressible Keller–Segel–Navier–Stokes equations. Z. Angew. Math. Phys. 72, 179 (2021). https://doi.org/10.1007/s00033-021-01609-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01609-4

Keywords

Mathematics Subject Classification

Search

Navigation