  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
Keller-Segel equation, which describes a population density of bacteria or mold
which attract chemically to substance they secrete. Solution of Keller-Segel
equation can exhibit dramatic collapsing behavior, where the density
concentrates positive mass in a measure zero region. A natural question is
whether presence of fluid flow can affect singularity formation by mixing the
density thus making concentration harder to achieve.
  In this paper, we consider parabolic-elliptic Keller-Segel equation in two
and three dimensions with 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 exist
initial data leading to finite time blow up. Thus presence of fluid 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 relaxation
enhancing (RE) flow. These flows are stationary. The second class of flows are
Yao-Zlatos optimal mixing flows, which are time dependent. The proof is based
on the nonlinear version of the relaxation enhancement construction of the
paper by Peter Constantin, Alexander Kiselev, Lenya Ryzhik, Andrej Zlatos, and
on some variations of global regularity estimate for Keller-Segel model.
