  We consider a simple model describing premixed combustion in the presence of
fluid flow: reaction diffusion equation with passive advection and ignition
type nonlinearity. Strong advection can suppress flames - a process we call
quenching. A flow is called quenching if any compactly supported initial data
will become extinct provided that the amplitude of the flow is chosen
sufficiently large. In this paper, we provide a sharp characterization of
quenching shear flows.The efficiency of quenching depends strongly on the
geometry and scaling of the flow. We discuss the cases of slowly and quickly
varying flows, proving analytically behavior that has been observed earlier in
numerical experiments. The technique involves probabilistic and PDE estimates,
in particular applications of Malliavin calculus and central limit theorem for
martingales.
