Skip to main content
Log in

Global existence of weak solutions for the 3D chemotaxis–Euler equations

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we consider the Cauchy problem for the three dimensional chemotaxis–Euler equations. By exploring the new a priori estimates, we prove the global existence of weak solutions for the 3D chemotaxis–Euler equations.

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

References

  1. Biler, P., Karch, G.: Blowup of solutions to generalized Keller–Segel model. J. Evol. Equ. 10, 247–262 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blanchet, A., Dolbeault, J., Perthame, B.: Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. 44, 1–33 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Calvez, V., Corrias, L.: The parabolic–parabolic Keller–Segel model in \({\mathbb{R}}^2\). Commun. Math. Sci. 6, 417–447 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chae, M., Kang, K., Lee, J.: Existence of smooth solutions to coupled chemotaxis–fluid equations. Discrete Contin. Dynam. Syst. 33, 2271–2297 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chertock, A., Fellner, K., Kurganov, A., Lorz, A., Markowich, P.A.: Sinking, merging and stationary plumes in a coupled chemotaxis–fluid model: a high-resolution numerical approach. J. Fluid Mech. 694, 155–190 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duan, R.J., Lorz, A., Markowich, P.A.: Global solutions to the coupled chemotaxis–fluid equations. Comm. Partial Differ. Equ. 35, 1635–1673 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)

    Article  MATH  Google Scholar 

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

  10. Lorz, A.: Coupled chemotaxis fluid model. Math. Models Methods Appl. Sci. 20, 987–1004 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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  MATH  Google Scholar 

  12. Miao, C., Wu, J., Zhang, Z.: Littlewood–Paley Theory and Applications to Fluid Dynamics Equations. Science Press, Beijing (2012)

    Google Scholar 

  13. Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C.W., Kessler, J.O., Goldstein, R.E.: Bacterial swimming and oxygen transport near constant lines. Proc. Natl. Acad. Sci. USA 102, 2277–2282 (2005)

    Article  MATH  Google Scholar 

  14. Winkler, M.: Global large-data solutions in a chemotaxis–(Navier–)Stokes system modeling cellular swimming in fluid drops. Comm. Partial Differ. Equ. 37, 319–352 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Winkler, M.: Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system. Ann. I. H. Poincaré AN (2015). doi:10.1016/j.anihpc.2015.05.002

  17. Zhang, Q.: Local well-posedness for the chemotaxis–Navier–Stokes equations in Besov spaces. Nonlinear Anal. Real World Appl. 17, 89–100 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, Q.: On the inviscid limit of the three dimensional incompressible chemotaxis–Navier–Stokes equations. Nonlinear Anal. Real World Appl. 27, 70–79 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, Q.: Blowup criterion of smooth solutions for the incompressible chemotaxis–Euler equations. Z. Angew. Math. Mech. 96(4), 466–476 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Q. Zhang was partially supported by the National Natural Science Foundation of China (11501160 and 61572011), the Second Batch of Young Talents of Hebei Province and Outstanding Youth Foundation of Hebei University (2015JQ01).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Qian Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, G., Zhang, Q. Global existence of weak solutions for the 3D chemotaxis–Euler equations. RACSAM 112, 195–207 (2018). https://doi.org/10.1007/s13398-016-0374-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-016-0374-3

Keywords

Mathematics Subject Classification

Navigation