Suppression of Chemotactic Explosion by Mixing
- 243 Downloads
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.
Unable to display preview. Download preview PDF.
- 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
- 5.Constantin, P., Kiselev, A., Ryzhik, L., Zlatoš, A.: Diffusion and mixing in fluid flow. Ann. Math., 643–674 2008Google Scholar
- 6.Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic theory, vol. 245. Springer Science & Business Media, New York 2012Google Scholar
- 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
- 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.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
- 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
- 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
- 30.Marchioro, C., Pulvirenti, M.: Mathematical theory of incompressible nonviscous fluids, vol. 96. Springer Science & Business Media, New York 2012Google Scholar
- 31.Maz’ya, V.: Sobolev spaces. Springer, New York 2013Google Scholar
- 35.Perthame, B.: Transport equations in biology. Springer Science & Business Media, New York 2006Google Scholar
- 39.Yao, Y., Zlatos, A.: Mixing and un-mixing by incompressible flows 2014. arXiv:1407.4163 (arXiv preprint)