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On one-dimensional compressible Navier–Stokes equations for a reacting mixture in unbounded domains

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Abstract

In this paper we consider the one-dimensional Navier–Stokes system for a heat-conducting, compressible reacting mixture which describes the dynamic combustion of fluids of mixed kinds on unbounded domains. This model has been discussed on bounded domains by Chen (SIAM J Math Anal 23:609–634, 1992) and Chen–Hoff–Trivisa (Arch Ration Mech Anal 166:321–358, 2003), among others, in which the reaction rate function is a discontinuous function obeying the Arrhenius’ law of thermodynamics. We prove the global existence of weak solutions to this model on one-dimensional unbounded domains with large initial data in \(H^1\). Moreover, the large-time behaviour of the weak solution is identified. In particular, the uniform-in-time bounds for the temperature and specific volume have been established via energy estimates. For this purpose we utilise techniques developed by Kazhikhov–Shelukhin (cf. Kazhikhov in Siber Math J 23:44–49, 1982; Solonnikov and Kazhikhov in Annu Rev Fluid Mech 13:79–95, 1981) and refined by Jiang (Commun Math Phys 200:181–193, 1999, Proc R Soc Edinb Sect A 132:627–638, 2002), as well as a crucial estimate in the recent work by Li–Liang (Arch Ration Mech Anal 220:1195–1208, 2016). Several new estimates are also established, in order to treat the unbounded domain and the reacting terms.

Keywords

Navier–Stokes equations Compressible Reacting mixture Combustion Global existence Uniform estimates Large-time behaviour 

Mathematics Subject Classification

Primary 35Q30 35Q35 35Q79 Secondary 76N10 76N15 

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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