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We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type travelingfronts in space-time periodic and mean zero incompressible advection, andestablish a variational (minimization) formula for the minimal speeds. Weapproach the existence by considering limit of a sequence of front solutions toa regularized traveling front equation where the nonlinearity is combustiontype with ignition cut-off. The limiting front equation is degenerate parabolicand does not permit strong solutions, however, the necessary compactnessfollows from monotonicity of fronts and degenerate regularity. We apply adynamic argument to justify that the constructed KPP traveling fronts propagateat minimal speeds, and derive the speed variational formula. The dynamic methodavoids the degeneracy in traveling front equations, and utilizes the parabolicmaximum principle of the governing reaction-diffusion-advection equation. Thedynamic method does not rely on existence of traveling fronts.

Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds