Skip to main content
SpringerLink
Log in

Well-Posedness and Singularity Formation for Inviscid Keller–Segel–Fluid System of Consumption Type

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the Keller–Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its \(C^{2}\)-norm with certain initial data.

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

Notes

  1. The case of rotation with angle \(\pi /2\) results in a very interesting system, for which the question of local well/ill-posedness seems delicate.

References

  1. Ahn, J., Kang, K.: On a Keller–Segel system with logarithmic sensitivity and non-diffusive chemical. Discrete Contin. Dyn. Syst. 34(12), 5165–5179 (2014)

    Article  MathSciNet  Google Scholar 

  2. Ahn, J., Kang, K., Yoon, C.: Global classical solutions for chemotaxis-fluid systems in two dimensions. Math. Methods Appl. Sci. 44(2), 2254–2264 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  3. Bae, H., Granero-Belinchón, R.: Singularity formation for the Serre–Green–Naghdi equations and applications to abcd-Boussinesq systems. arXiv:2001.11937

  4. Chae, M., Choi, K., Kang, K., Lee, J.: Stability of planar traveling waves in a Keller–Segel equation on an infinite strip domain. J. Differ. Equ. 265(1), 237–279 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  5. Chae, M., Kang, K., Lee, J.: Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete Contin. Dyn. Syst. 33(6), 2271–2297 (2013)

    Article  MathSciNet  Google Scholar 

  6. Chae, M., Kang, K., Lee, J.: Global existence and temporal decay in Keller–Segel models coupled to fluid equations. Commun. Partial Differ. Equ. 39(7), 1205–1235 (2014)

    Article  MathSciNet  Google Scholar 

  7. Chae, M., Kang, K., Lee, J.: Asymptotic behaviors of solutions for an aerotaxis model coupled to fluid equations. J. Korean Math. Soc. 53(1), 127–146 (2016)

    Article  MathSciNet  Google Scholar 

  8. Chae, M., Kang, K., Lee, J., Lee, K.-A.: A regularity condition and temporal asymptotics for chemotaxis-fluid equations. Nonlinearity 31(2), 351–387 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  9. 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  ADS  MathSciNet  Google Scholar 

  10. Constantin, P.: Note on loss of regularity for solutions of the \(3\)-D incompressible Euler and related equations. Commun. Math. Phys. 104(2), 311–326 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  11. Corrias, L., Perthame, B., Zaag, H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–28 (2004)

    Article  MathSciNet  Google Scholar 

  12. Duan, R., Lorz, A., Markowich, P.: Global solutions to the coupled chemotaxis-fluid equations. Commun. Partial Differ. Equ. 35(9), 1635–1673 (2010)

    Article  MathSciNet  Google Scholar 

  13. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224. Springer, Berlin (1983)

  14. Herrero, M.A., Velázquez, J.L.: A blow-up mechanism for a chemotaxis model. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24(4), 633–683 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Itô, S.: A boundary value problem of partial differential equations of parabolic type. Duke Math. J. 24, 299–312 (1957)

    Article  MathSciNet  Google Scholar 

  16. Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329(2), 819–824 (1992)

    Article  MathSciNet  Google Scholar 

  17. Jeong, I.-J., Yoneda, T.: Enstrophy dissipation and vortex thinning for the incompressible 2D Navier–Stokes equations. Nonlinearity 34(4), 1837–1853 (2021)

    Article  MathSciNet  Google Scholar 

  18. Jeong, I.-J., Yoneda, T.: Vortex stretching and enhanced dissipation for the incompressible 3D Navier–Stokes equations. Math. Ann. 380(3–4), 2041–2072 (2021)

    Article  MathSciNet  Google Scholar 

  19. Kang, K., Stevens, A.: Blowup and global solutions in a chemotaxis-growth system. Nonlinear Anal. 135, 57–72 (2016)

    Article  MathSciNet  Google Scholar 

  20. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26(3), 399–415 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  21. Lorz, A.: Coupled chemotaxis fluid model. Math. Models Methods Appl. Sci. 20(6), 987–1004 (2010)

    Article  MathSciNet  Google Scholar 

  22. Patlak, C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)

    Article  MathSciNet  Google Scholar 

  23. Tao, Y., Winkler, M.: Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equ. 252(3), 2520–2543 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  24. Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C.W., Kessler, J.O., Goldstein, R.E.: Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. 102(7), 2277–2282 (2005)

    Article  ADS  Google Scholar 

  25. Winkler, M.: Global large-data solutions in a chemotaxis-(Navier–)Stokes system modeling cellular swimming in fluid drops. Commun. Partial Differ. Equ. 37(2), 319–351 (2012)

    Article  MathSciNet  Google Scholar 

  26. Winkler, M.: Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J. Math. Pures Appl. (9) 100(5), 748–767 (2013)

    Article  MathSciNet  Google Scholar 

  27. Winkler, M.: Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Arch. Ration. Mech. Anal. 211(2), 455–487 (2014)

    Article  MathSciNet  Google Scholar 

  28. Winkler, M.: Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(5), 1329–1352 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  29. Winkler, M.: How far do chemotaxis-driven forces influence regularity in the Navier–Stokes system? Trans. Am. Math. Soc. 369(5), 3067–3125 (2017)

    Article  MathSciNet  Google Scholar 

  30. Winkler, M.: Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient Taxis(-Stokes) systems? Int. Math. Res. Not.: IMRN 11, 8106–8152 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

IJ has been supported by the New Faculty Startup Fund from Seoul National University, the Science Fellowship of POSCO TJ Park Foundation, and the National Research Foundation of Korea grant No. 2019R1F1A1058486. KK has been supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. We are grateful to the anonymous referees for various comments and suggestions, which have significantly improved the manuscript. Especially, we are grateful for pointing to us the possibility of the interesting generalization given in Remark 3.4.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences and RIM, Seoul National University, Seoul, Korea

    In-Jee Jeong

  2. Department of Mathematics, Yonsei University, Seoul, Korea

    Kyungkeun Kang

Authors
  1. In-Jee Jeong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kyungkeun Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to In-Jee Jeong.

Additional information

Communicated by A. Ionescu.

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

Jeong, IJ., Kang, K. Well-Posedness and Singularity Formation for Inviscid Keller–Segel–Fluid System of Consumption Type. Commun. Math. Phys. 390, 1175–1217 (2022). https://doi.org/10.1007/s00220-021-04292-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-021-04292-8

Search

Navigation