2005/02/21 by James Nolen, Nolen, James, Matthew Rudd +3 · 1 citation
Computer Science · Earth and Planetary Sciences · Environmental Science · Mathematics · #35K57 #Analysis of PDEs (math.AP) #Climate variability and models #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Tropical and Extratropical Cyclones Research #math.AP #msc:35K57
paper · pdf · doi:10.48550/arxiv.math/0502436
34 pages
arxiv created 2005/02/21 · openalex publication_date 2005/02/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We apply a dynamic argument to justify that the constructed KPP traveling fronts propagate at minimal speeds, and derive the speed variational formula. The dynamic method avoids the degeneracy in traveling front equations, and utilizes the parabolic maximum principle of the governing reaction-diffusion-advection equation. The dynamic method does not rely on existence of traveling fronts.