Combustion waves in hydraulically resistant porous media in a special parameter regime

TL;DR

This paper analyzes the stability of combustion fronts in a specialized PDE model of hydraulically resistant porous media, using energy estimates, numerical Evans function computations, and nonlinear analysis, revealing their instability types.

Contribution

It introduces a reduction of the PDE system under specific parameters, extending stability results from the reduced to the full system, and classifies the instability of combustion fronts.

Findings

Combustion fronts are either absolutely or convectively unstable.

Stability results from the reduced system extend to the full system.

The analysis combines energy estimates, numerical Evans functions, and nonlinear methods.

Abstract

In this paper we study the stability of fronts in a reduction of a well-known PDE system that is used to model the combustion in hydraulically resistant porous media. More precisely, we consider the original PDE system under the assumption that one of the parameters of the model, the Lewis number, is chosen in a specific way and with initial conditions of a specific form. {For a class of initial conditions, then the number of unknown functions is reduced from three to two. For the reduced system, the existence of combustion fronts follows from the existence results for the original system. The stability of these fronts is studied here by a combination of energy estimates and numerical Evans function computations and nonlinear analysis when applicable. We then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the Lewis number. The fronts that we {investigate} are proved to be either absolutely unstable or convectively unstable on the nonlinear level.