Reaction-diffusion front speeds in spatially-temporally periodic shear flows

We study the asymptotics of two space dimensional reaction-diffusion front speeds through mean zero space-time periodic shears using both analytical and numerical methods. The analysis hinges on traveling fronts and their estimates based on qualitative properties such as monotonicity and a priori integral inequalities. The computation uses an explicit second order upwind finite difference method to provide more quantitative information. At small shear amplitudes, front speeds are enhanced by an amount proportional to shear amplitude squared. The proportionality constant has a closed form expression. It decreases with increasing shear temporal frequency and is independent of the form of the known reaction nonlinearities. At large shear amplitudes and for all reaction nonlinearities, the enhanced speeds grow proportional to shear amplitude and are again decreasing with increasing shear temporal frequencies. The results extend previous ones in the literature on front speeds through spatially periodic shears and show front speed slowdown due to shear direction switching in time.