Suppression of Chemotactic Explosion by Mixing

By Alexander V. Kiselev, Xiaoqian Xu. 2015

137 citations recorded by the source corpus.

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.

Subfields: Modeling and Simulation, Molecular Biology, Oncology

View Paper: https://www.semanticscholar.org/paper/83b793c4829388cbe6a48112b35d146ab29d40eb

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Citations in the corpus

A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization (2019, background): “As recent analytical findings show, already cell transport through a given fluid may substantially influence the solution behavior in certain Keller-Segel-type chemotaxis systems ([28], [29]); in fact, even complete suppression of blow-up may occur ([30]).”
Linear inviscid damping and enhanced dissipation for the Kolmogorov flow (2017, background): “See [3, 13, 20, 8, 9] and references therein for more relevant works.”