Well-posed problem for a combustion model in a multilayer porous medium

Abstract

Combustion occurring in porous media has various practical applications, such as in in-situ combustion processes in oil reservoirs, the combustion of biogas in sanitary landfills, and many others. A porous medium where combustion takes place can consist of layers with different physical properties. This study demonstrates that the initial value problem for a combustion model in a multi-layer porous medium has a unique solution, which is continuous with respect to the initial data and parameters in . In summary, it establishes that the initial value problem is well-posed in . The model is governed by a one-dimensional reaction–diffusion–convection nonlinear system, where the unknowns are the temperatures in the layers. Previous studies have addressed the same problem in . However, in this study, we solve the problem in a less restrictive space, namely . The proof employs a novel approach to combustion problems in porous media, utilizing an evolution operator defined from the theory of semigroups in Hilbert space and Kato’s theory for a well-posed associated initial value problem. Finally, we give numerical simulations of the nonlinear problem, where we observe that the characteristics of the solution profiles we find are in accordance with what is physically expected.