Archive for Rational Mechanics and Analysis

, Volume 222, Issue 2, pp 1077–1112 | Cite as

Suppression of Chemotactic Explosion by Mixing

  • Alexander Kiselev
  • Xiaoqian XuEmail author


Chemotaxis plays a crucial role in a variety of processes in biology and ecology. In many instances, processes involving chemical attraction take place in fluids. One of the most studied PDE models of chemotaxis is given by the Keller–Segel equation, which describes a population density of bacteria or mold which is attracted chemically to substance they secrete. Solutions of the Keller–Segel equation can exhibit dramatic collapsing behavior, where density concentrates positive mass in a measure zero region. A natural question is whether the presence of fluid flow can affect singularity formation by mixing the bacteria thus making concentration harder to achieve. In this paper, we consider the parabolic-elliptic Keller–Segel equation in two and three dimensions with an additional advection term modeling ambient fluid flow. We prove that for any initial data, there exist incompressible fluid flows such that the solution to the equation stays globally regular. On the other hand, it is well known that when the fluid flow is absent, there exists initial data leading to finite time blow up. Thus the presence of fluid flow can prevent the singularity formation. We discuss two classes of flows that have the explosion arresting property. Both classes are known as very efficient mixers. The first class are the relaxation enhancing (RE) flows of (Ann Math:643–674, 2008). These flows are stationary. The second class of flows are the Yao–Zlatos near-optimal mixing flows (Mixing and un-mixing by incompressible flows. arXiv:1407.4163, 2014), which are time dependent. The proof is based on the nonlinear version of the relaxation enhancement construction of (Ann Math:643–674, 2008), and on some variations of the global regularity estimate for the Keller–Segel model.


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  1. 1.
    Alberti G., Crippa G., Mazzucato A. L.: Exponential self-similar mixing and loss of regularity for continuity equations. Comp. Rendus Math. 352(11), 901–906 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berestycki H., Kiselev A., Novikov A., Ryzhik L.: The explosion problem in a flow. J. d’Anal. Math. 110(1), 31–65 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Coll, J., Bowden, B., Meehan, G., Konig, G., Carroll, A., Tapiolas, D., Alino, P., Heaton, A., De Nys, R., Leone, P., 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(2), 177–182 1994Google Scholar
  4. 4.
    Constantin P.: Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations. Commun. Math. Phys. 104, 311–326 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Constantin, P., Kiselev, A., Ryzhik, L., Zlatoš, A.: Diffusion and mixing in fluid flow. Ann. Math., 643–674 2008Google Scholar
  6. 6.
    Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic theory, vol. 245. Springer Science & Business Media, New York 2012Google Scholar
  7. 7.
    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators: with application to quantum mechanics and global geometry. Springer, New York 2009Google Scholar
  8. 8.
    Deshmane S.L., Kremlev S., Amini S., Sawaya B.E.: Monocyte chemoattractant protein-1 (mcp-1): an overview. J. Interferon Cytokine Res. 29(6), 313–326 (2009)CrossRefGoogle Scholar
  9. 9.
    DiFrancesco M., Lorz A., Markowich P.: Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior. Discret. Contin. Dyn. Syst. 28(4), 1437–1453 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duan R., Lorz A., Markowich P.: Global solutions to the coupled chemotaxis-fluid equations. Communi. Partial Differ. Equations 35(9), 1635–1673 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Horstmann D.: From 1970 until present: the keller-segel model in chemotaxis and its consequences i. Jahresberichte der DMV 105, 103–165 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Horstmann D.: From 1970 until present: the keller-segel model in chemotaxis and its consequences ii. Jahresberichte der DMV 106, 51–69 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hou T.Y., Lei Z.: On the stabilizing effect of convection in three-dimensional incompressible flows. Commun. Pure Appl. Math. 62(4), 501–564 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hou T.Y., Lei Z., Luo G., Wang S., Zou C.: On finite time singularity and global regularity of an axisymmetric model for the three dimensional euler equations. Arch. Ration. Mech. Anal. 212(2), 683–706 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hou T.Y., Li C., Shi Z., Wang S., Yu X.: On singularity formation of a nonlinear nonlocal system. Arch. Ration. Mech. Anal. 199(1), 117–144 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hou T.Y., Shi Z., Wang S.: On singularity formation of a three dimensional model for incompressible navier–stokes equations. Adv. Math. 230(2), 607–641 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Iyer G., Kiselev A., Xu X.: Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows. Nonlinearity 27(5), 973 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26(3), 399–415 (1970)CrossRefzbMATHGoogle Scholar
  20. 20.
    Keller E.F., Segel L.A.: Model for chemotaxis. J. Theor. Biol. 30(2), 225–234 (1971)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kiselev A., Ryzhik L.: Biomixing by chemotaxis and enhancement of biological reactions. Commun. PDE 37, 298–318 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kiselev A., Ryzhik L.: Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case. J. Math. Phys. 53(11), 115609 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kolmogorov, A., On dynamical systems with an integral invariant on the torus (russian). In: Dokl. Akad. Nauk SSSR, vol. 93, pp. 763–766 1953Google Scholar
  24. 24.
    Larios, A., Titi, E.S.: Global regularity vs. finite-time singularities: some paradigms on the effect of boundary conditions and certain perturbations. In: Robinson, J.C., Rodigo, J.L., Sadowski, W., Vidal-López, A. (eds.) “Topics in the Theory of the Navier-Stokes Equations”. London Mathematical Society. Cambridge University Press, Cambridge 2015. arXiv:1401.1534
  25. 25.
    Lin Z., Thiffeault J.-L., Doering C. R.: Optimal stirring strategies for passive scalar mixing. J. Fluid Mech. 675, 465–476 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu J.-G., Lorz A., A coupled chemotaxis-fluid model: global existence. In: Annales de l’Institut Henri Poincare (C) Non Linear Analysis, vol. 28, pp. 643–652. Elsevier 2011Google Scholar
  27. 27.
    Lorz A.: Coupled chemotaxis fluid model. Math. Models Methods Appl. Sci. 20(06), 987–1004 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lorz, A.: Coupled Keller-Segel-Stokes model: global existence for small initial data and blow-up delay. Commun. Math. Sci. 10(2), 2012 2012Google Scholar
  29. 29.
    Lunasin E., Lin Z., Novikov A., Mazzucato A., Doering C. R.: Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows. J. Math. Phys. 53(11), 115611 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Marchioro, C., Pulvirenti, M.: Mathematical theory of incompressible nonviscous fluids, vol. 96. Springer Science & Business Media, New York 2012Google Scholar
  31. 31.
    Maz’ya, V.: Sobolev spaces. Springer, New York 2013Google Scholar
  32. 32.
    Miller R.L.: Demonstration of sperm chemotaxis in echinodermata: Asteroidea, holothuroidea, ophiuroidea. J. Exp. Zool. 234(3), 383–414 (1985)ADSCrossRefGoogle Scholar
  33. 33.
    Nagai T.: Blowup ofnonradial solutions to parabolic-elliptic systems modeling chemotaxis intwo-dimensional domains. J. Inequal. 6, 37–55 (2001)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Patlak C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15(3), 311–338 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Perthame, B.: Transport equations in biology. Springer Science & Business Media, New York 2006Google Scholar
  36. 36.
    Seis C.: Maximal mixing by incompressible fluid flows. Nonlinearity 26(12), 3279 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Taub D., Proost P., Murphy W., Anver M., Longo D., Van Damme J., Oppenheim J.: Monocyte chemotactic protein-1 (mcp-1),-2, and-3 are chemotactic for human t lymphocytes. J. Clin. Investig. 95(3), 1370 (1995)CrossRefGoogle Scholar
  38. 38.
    von Neumann J.: Zur operatorenmethode in der klassischen mechanik. Ann. Math. 33, 587–642 (1932)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Yao, Y., Zlatos, A.: Mixing and un-mixing by incompressible flows 2014. arXiv:1407.4163 (arXiv preprint)
  40. 40.
    Ziemer W.P.: Weakly differentiable functions. Spriger, New York (1989)CrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsRice UniversityHoustonUSA
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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