Application of the semigroup theory to a combustion problem in a multi-layer porous medium

TL;DR

This paper establishes the well-posedness of a reaction-diffusion-convection system modeling gasless combustion in multi-layer porous media, incorporating spatially variable parameters and using semigroup theory for a novel analytical approach.

Contribution

It introduces a new model with spatially variable physical parameters and proves well-posedness using semigroup and Kato's theories, addressing previous limitations.

Findings

Proved local and global well-posedness of the system

Extended the model to include spatially variable parameters

Ensured solution continuity with respect to initial data

Abstract

This study proved that the Cauchy problem for a one-dimensional reaction-diffusion-convection system is locally and globally well-posed in $\mathtt{H}^2(\mathbb{R})$. The system modeled a gasless combustion front through a multi-layer porous medium when the fuel concentration in each layer was a known function. Combustion has critical practical porous media applications, such as in in-situ combustion processes in oil reservoirs and several other areas. Earlier studies considered physical parameters (e.g., porosity, thermal conductivity, heat capacity, and initial fuel concentration ) constant. Here, we consider a more realistic model where these parameters are functions of the spatial variable rather than constants. Furthermore, in previous studies, we did not consider the continuity of the solution regarding the initial data and parameters, unlike the current study. This proof uses a novel approach to combustion problems in porous media. We follow the abstract semigroups theory of operators in the Hilbert space and the well-known Kato's theory for a well-posed associated initial value problem.