Bulk Burning Rate in Passive - Reactive Di usion

We consider a passive scalar that is advected by a prescribed mean zero divergence-free velocity eld, diiuses, 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 ill-deened notion of front speed. We establish rigorous lower bounds for the bulk burning rate that are linear in the amplitude of the advecting velocity for a large class of ows. These "percolating" ows are characterized by the presence of tubes of streamlines connecting distant regions of burned and unburned material and generalize shear ows. The bound contains geometric information on the velocity streamlines and degenerates when these oscillate on scales that are ner than the width of the laminar burning region. We give also examples of very diierent kind of ows, cellular ows with closed streamlines, and rigorously prove that these can produce only sub-linear enhancement of the bulk burning rate.