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

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 H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document}. 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.