  We consider a simple scalar reaction-advection-diffusion equation with
ignition-type nonlinearity and discuss the following question: What kinds of
velocity profiles are capable of quenching any given flame, provided the
velocity's amplitude is adequately large? Even for shear flows, the answer
turns out to be surprisingly subtle. If the velocity profile changes in space
so that it is nowhere identically constant, (or if it is identically constant
only in a region of small measure) then the flow can quench any initial data.
But if the velocity profile is identically constant in a sizable region, then
the ensuing flow is incapable of quenching large enough flames, no matter how
much larger is the amplitude of this velocity. The constancy region must be
wider across than a couple of laminar propagating front-widths. The proof uses
a linear PDE associated to the nonlinear problem and quenching follows when the
PDE is hypoelliptic. The techniques used allow the derivation of new, nearly
optimal bounds on the speed of traveling wave solutions.
