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 the amplitude of this velocity is. 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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics