Quenching of flames by fluid advection

We consider a simple scalar reaction-advection-diffusion equation with igni tion-type nonlinearity and discuss the following question: What kinds of ve locity profiles are capable of quenching any given flame, provided the velo city's amplitude is adequately large? Even for shear flows, the answer turn s 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 measur e), 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 incap able of quenching large enough flames, no matter how much larger the amplit ude of this velocity is. The constancy region must be wider across than a c ouple of laminar propagating front widths. The proof uses a linear PDE associated to the nonlinear problem, and quench ing follows when the PDE is hypoelliptic. The techniques used allow the der ivation of new, nearly optimal bounds on the speed of traveling-wave soluti ons. (C) 2001 John Wiley & Sons, Inc.