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In this paper, a novel observation is made on a one-dimensional compressibleNavier--Stokes model for the dynamic combustion of a reacting mixture of$\gamma$-law gases ($\gamma>1$) with discontinuous Arrhenius reaction ratefunction, on both bounded and unbounded domains. We show that the mass fractionof 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 isLipschitz continuous and bounded strictly away from zero and infinity.Consequently, the graph of $Z$ cannot form cusps or corners near the pointswhere the reactant in the combustion process is completely depleted at anyinstant, and the entropy of $Z$ is bounded from above. The key ingredient ofthe proof is a new estimate based on the Fisher information, first exploited by[2, 7] with applications to PDEs in chemorepulsion and thermoelasticity. Alongthe way, we also establish a Lipschitz estimate for the density.

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