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

TL;DR

This paper proves the well-posedness of a combustion model in multilayer porous media within an L^2 space, using semigroup theory and Kato's approach, extending previous results from H^2 spaces.

Contribution

It establishes the existence, uniqueness, and continuous dependence of solutions for a combustion model in L^2 space, employing a novel semigroup-based method.

Findings

Proves the initial value problem is well-posed in L^2 space.

Introduces a new approach using evolution operators and Kato's theory.

Extends previous results from H^2 to L^2 spaces.

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 $\mathtt{L}^2(\mathbb{R})^n$. In summary, it establishes that the initial value problem is well-posed in $\mathtt{L}^2(\mathbb{R})^n$. The model is governed by a one-dimensional reaction-diffusion-convection system, where the unknowns are the temperatures in the layers. Previous studies have addressed the same problem in $\mathtt{H}^2(\mathbb{R})^n$. However, in this study, we solve the problem in a less restrictive space, namely $\mathtt{L}^2(\mathbb{R})^n$. 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.