We study flow-induced enhancement of the speed of pulsating traveling fronts for reaction–diffusion equations, and quenching of reaction by fluid flows. We prove, for periodic flows in two dimensions and any combustion-type reaction, that the front speed is proportional to the square root of the (homogenized) effective diffusivity of the flow. We show that this result does not hold in three and more dimensions. We also prove conjectures from Audoly, Berestycki and Pomeau (2000) , Berestycki (2003) , Fannjiang, Kiselev and Ryzhik (2006)  for cellular flows, concerning the rate of speed-up of fronts and the minimal flow amplitude necessary to quench solutions with initial data of a fixed (large) size.
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Andrej Zlatoš, Reaction–diffusion front speed enhancement by flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 5, pp. 711–726DOI 10.1016/J.ANIHPC.2011.05.004