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We prove the existence of reaction-diffusion traveling fronts in mean zerospace-time periodic shear flows for nonnegative reactions including theclassical KPP (Kolmogorov-Petrovsky-Piskunov) nonlinearity. For the KPPnonlinearity, the minimal front speed is characterized by a variationalprinciple involving the principal eigenvalue of a space-time periodic parabolicoperator. Analysis of the variational principle shows that adding a mean-zerospace time periodic shear flow to an existing mean zero space-periodic shearflow leads to speed enhancement. Computation of KPP minimal speeds is performedbased on the variational principle and a spectrally accurate discretization ofthe principal eigenvalue problem. It shows that the enhancement is monotonedecreasing in temporal shear frequency, and that the total enhancement frompure reaction-diffusion obeys quadratic and linear laws at small and largeshear amplitudes.

Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle