Characterisations for the depletion of reactant in a one-dimensional dynamic combustion model

TL;DR

This paper investigates the depletion behavior of reactants in a one-dimensional combustion model, establishing bounds on the reactant's gradient and entropy, and introducing a novel Fisher information-based estimate.

Contribution

It provides new mathematical estimates for reactant depletion in a combustion PDE model, including a weighted gradient bound and entropy control, using Fisher information techniques.

Findings

Reactant mass fraction gradient is bounded in a weighted L^2 space.

The graph of the reactant cannot develop cusps or corners during depletion.

The entropy of the reactant remains bounded from above.

Abstract

In this paper, a novel observation is made on a one-dimensional compressible Navier--Stokes model for the dynamic combustion of a reacting mixture of $\gamma$-law gases ($\gamma>1$) with discontinuous Arrhenius reaction rate function, on both bounded and unbounded domains. We show that the mass fraction of the reactant (denoted as $Z$) satisfies a weighted gradient estimate $Z_y/ \sqrt{Z} \in L^\infty_t L^2_y$, provided that at time zero the density is Lipschitz continuous and bounded strictly away from zero and infinity. Consequently, the graph of $Z$ cannot form cusps or corners near the points where the reactant in the combustion process is completely depleted at any instant, and the entropy of $Z$ is bounded from above. The key ingredient of the proof is a new estimate based on the Fisher information, first exploited by [2, 7] with applications to PDEs in chemorepulsion and thermoelasticity. Along the way, we also establish a Lipschitz estimate for the density.