Regularity, profiles of solutions and evolution of supports for modeling a flame propagation in a porous medium

Abstract

This presentation explores and extends the classical model of flame propagation in porous media, focusing on high-pressure, slowly spreading thermo-diffusive waves as described by Williams, Zeldovich, Barenblatt, Mallordy, and Roquejoffre. It aims to utilize a "non-homogeneous" spatial operator, specifically the p-Laplacian, to model diffusion, emphasizing finite propagation and energy preservation in high-resistance scenarios typical of porous domains. The study covers the Cauchy problem, introduces upper bounds for flame increase, and examines travelling waves and geometric perturbation theory. It highlights the concept of propagating supports crucial for modeling flame behavior. The conclusions suggest the need for numerical assessments and propose the introduction of an advection term to model forced convection in the porous medium. A future hypothesis explores whether a specific advection velocity could cause the compact support of a flame to vanish during propagation. Key novelties include the application of the p-Laplacian, consideration of finite propagation and energy preservation, the potential for forced convection, and the advection velocity hypothesis. This work lays a theoretical foundation for further numerical and theoretical investigations into flame propagation in porous media.