Bulk burning rate in passive-reactive diffusion

We consider a passive scalar that is advected by a prescribed mean zero div ergence-free velocity field, diffuses, and reacts according to a KPP-type n onlinear reaction. We introduce a quantity, the bulk burning rate, that mak es both mathematical and physical sense in general situations and extends t he often iii-defined notion of front speed. We establish rigorous lower bou nds for the bulk burning rate that ale linear in the amplitude of the advec ting velocity for a large class of flows. These "percolating" flows ale cha racterized by the presence of tubes of streamlines connecting distant legio ns of burned and unburned material and generalize shear flows. The bound co ntains geometric information on the velocity streamlines and degenerates wh en these oscillate on scales that are finer than the width of the laminar b urning region. We give also examples of very different kind of flows, cellu lar flows with closed streamlines, and rigorously prove that those can prod uce only sub-linear enhancement of the bulk burning rate.