  We study two types of divergence-free fluid flows on unbounded domains in two
and three dimensions -- hyperbolic and shear flows -- and their influence on
chemotaxis and combustion. We show that fast spreading by these flows, when
they are strong enough, can suppress growth of solutions to PDE modeling these
phenomena. This includes prevention of singularity formation and global
regularity of solutions to advective Patlak-Keller-Segel equations on ${\mathbb
R}^2$ and ${\mathbb R}^3$, confirming numerical observations in [Khan,
Johnson,Cartee and Yao, Global regularity of chemotaxis equations with
advection, Involve, 9(1) 2016], as well as quenching in
advection-reaction-diffusion equations.
