Characterisations for the depletion of reactant in a one-dimensional dynamic combustion model
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917228702597120 |
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| author | Li, Siran Yang, Jianing |
| author_facet | Li, Siran Yang, Jianing |
| contents | 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 $γ$-law gases ($γ>1$) 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_16506 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Characterisations for the depletion of reactant in a one-dimensional dynamic combustion model Li, Siran Yang, Jianing Analysis of PDEs Fluid Dynamics 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 $γ$-law gases ($γ>1$) 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. |
| title | Characterisations for the depletion of reactant in a one-dimensional dynamic combustion model |
| topic | Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2308.16506 |