Bulk burning rate in passive-reactive diffusion

We consider a passive scalar that is advected by a prescribed mean zero divergence-free velocity field, diffuses, and reacts according to a KPP-type nonlinear reaction. We introduce a quantity, the bulk burning rate, that makes both mathematical and physical sense in general situations and extends the often iii-defined notion of front speed. We establish rigorous lower bounds for the bulk burning rate that ale linear in the amplitude of the advecting velocity for a large class of flows. These "percolating" flows ale characterized by the presence of tubes of streamlines connecting distant legions of burned and unburned material and generalize shear flows. The bound contains geometric information on the velocity streamlines and degenerates when these oscillate on scales that are finer than the width of the laminar burning region. We give also examples of very different kind of flows, cellular flows with closed streamlines, and rigorously prove that those can produce only sub-linear enhancement of the bulk burning rate.