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

TL;DR

This paper proves the global existence and analyzes the long-term behavior of weak solutions to the one-dimensional compressible Navier-Stokes equations modeling a reacting mixture on unbounded domains, extending previous bounded domain results.

Contribution

It establishes the global existence of weak solutions for large initial data and characterizes their large-time behavior in unbounded domains, using novel estimates and energy methods.

Findings

Global existence of weak solutions is proved.

Uniform bounds for temperature and specific volume are established.

Long-term behavior of solutions is characterized.

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 Jour. Math. Anal., 23 (1992), 609--634) and Chen-Hoff-Trivisa (Arch. Rat. Mech. Anal. 166 (2003), 321--358) among others, in which the reaction rate function is a discontinuous function obeying the Arrhenius Law. 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 and proved. 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 and coauthors ({\it cf.} Siber. Math. Jour. 23 (1982), 44--49; Jour. Appl. Math. Mech., 41 (1977), 273--282), as well as a crucial estimate in the recent work by Li-Liang (Arch. Rat. Mech. Anal. 220 (2016), 1195--1208). Several new estimates are also established, in order to treat the unbounded domain and the reacting terms.