Skip to main content
SpringerLink
Log in

Suppressing Chemotactic Blow-Up Through a Fast Splitting Scenario on the Plane

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We revisit the question of global regularity for the Patlak–Keller–Segel (PKS) chemotaxis model. The classical 2D parabolic-elliptic model blows up for initial mass \({M > 8\pi}\). We consider a more realistic scenario which takes into account the flow of the ambient environment induced by harmonic potentials, and thus retain the identical elliptic structure as in the original PKS. Surprisingly, we find that already the simplest case of linear stationary vector field, \({Ax^\top}\), with large enough amplitude \({A}\), prevents chemotactic blow-up. Specifically, the presence of such an ambient fluid transport creates what we call a ‘fast splitting scenario’, which competes with the focusing effect of aggregation so that ‘enough mass’ is pushed away from concentration along the \({x_1}\)-axis, thus avoiding a finite time blow-up, at least for \({M < 16\pi}\). Thus, the enhanced ambient flow doubles the amount of allowable mass which evolve to global smooth solutions.

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. Bedrossian, J., Coti Zelati, M.: Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. Arch. Rat. Mech. Anal. 224(3), 1161–1204 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bedrossian, J., He, S.: Suppression of blow-up in Patlak-Keller-Segel via shear flows. SIAM J. Math. Anal. 49(6), 4722–4766 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bedrossian, J., Kim, I.: Global existence and finite time blow-up for critical Patlak-Keller-Segel models with inhomogeneous diffusion. SIAM J. Math. Anal. 45(3), 934–964 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bedrossian, J., Masmoudi, N.: Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in \(\mathbb{R}^2\) with Measure-valued initial data. Arch. Rat. Mech. Anal. 214(3), 717–801 (2014)

    Article  MATH  Google Scholar 

  5. Biler, P.: The Cauchy problem and self-similar solutions for a nonlinear parabolic equation. Studia Math. 114(2), 181–192 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blanchet, A., Calvez, V., Carrillo, J.A.: Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model. SIAM J. Numer. Anal. 46, 691–721 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blanchet, A., Dolbeault, J., Perthame, B.: Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions. EJDE 44, 1–32 (2006)

    MATH  Google Scholar 

  8. Blanchet, A., Carrillo, J.A., Masmoudi, N.: Infinite time aggregation for the critical Parlak-Keller-Segel model in \(\mathbb{R}^2\). Commun. Pure Appl. Math. 61, 1449–1481 (2008)

    Article  MATH  Google Scholar 

  9. Carlen, E., Loss, M.: Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on \(\mathbb{S}^n\). Geom. Func. Anal. 2(1), 90–104 (1992)

    Article  MATH  Google Scholar 

  10. Calvez, V., Carrillo, J.A.: Volume effects in the Keller-Segel model: energy estimates preventing blow-up. JMPA 86, 155–175 (2006)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Carrillo, J.A., Chen, L., Liu, J.-G., Wang, J.: A note on the subcritical two dimensional Keller-Segel system. Acta Appl. Math. 119(1), 43–55 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Carrillo, J.A., Rosado, J.: Uniqueness of bounded solutions to Aggregation equations by optimal transport methods. In: Proceedings of 5th European Congress of Mathmatics. European Mathematical Society, Amsterdam (2008)

  14. Delort, J.M.: Existence de nappes de tourbillon en dimension deux. J. Am. Math. Soc. 4, 386–553 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dolbeault, J., Schmeiser, C.: The two-dimensional Keller-Segel model after blow-up. Discrete Contin. Dyn. Syst. 25, 109–121 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Duan, R.-J., Lorz, A., Markowich, P.: Global solutions to the coupled chemotaxisfluid equations. Commun. Partial Differ. Equ. 35, 1635–1673 (2010)

    Article  MATH  Google Scholar 

  17. Egaa, G., Mischler, S.: Uniqueness and long time asymptotic for the Keller-Segel equation: the parabolic-elliptic case. Arch. Ration. Mech. Anal. 220(3), 1159–1194 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  19. Hortsmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein 105(3), 103–165 (2003)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  22. Khan, S., Johnson, J., Cartee, E., Yao, Y.: Global regularity of chemotaxis equations with advection. Involve 9(1), 119–131 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kiselev, A., Xu, X.: Suppression of chemotactic explosion by mixing. Arch. Ration. Mech. Anal. 222, 1077–1112 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, J.-G., Lorz, A.: A coupled chemotaxis-fluid model: global existence. Annales de lInstitut Henri Poincaré. Analyse Non Linéaire 28, 643–652 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Poupaud, F.: Diagonal defect measures, adhesion dynamics and Euler equations. Meth. Appl. Anal. 9, 533–561 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Senba, T., Suzuki, T.: Weak solutions to a parabolic-elliptic system of chemotaxis. J. Function. Anal. 47, 17–51 (2001)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Research was supported by NSF Grants DMS16-13911, RNMS11-07444 (KI-Net), ONR Grant N00014-1812465 and the Sloan research fellowship FG-2015-66019. SH thanks the Ann G.Wylie Dissertation Fellowship and Jacob Bedrossian and Scott Smith for many fruitful discussions. We thank the ETH Institute for Theoretical Studies (ETH-ITS) for its support and hospitality.

Author information

Author notes

  1. Siming He

    Present address: Department of Mathematics, Duke University, Durham, NC, 27708, USA

Authors and Affiliations

  1. Department of Mathematics and Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD, 20742, USA

    Siming He

  2. Department of Mathematics, Center for Scientific Computation and Mathematical Modeling (CSCAMM), and Institute for Physical Sciences & Technology (IPST), University of Maryland, College Park, MD, 20742, USA

    Eitan Tadmor

Authors
  1. Siming He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eitan Tadmor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eitan Tadmor.

Additional information

Communicated by A. Figalli

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, S., Tadmor, E. Suppressing Chemotactic Blow-Up Through a Fast Splitting Scenario on the Plane. Arch Rational Mech Anal 232, 951–986 (2019). https://doi.org/10.1007/s00205-018-01336-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-018-01336-7

Search

Navigation