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

Abstract

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. Several new estimates are also established, in order to treat the unbounded domain and the reacting terms.Mathematics Subject Classification. Primary 35Q30, 35Q35, 35Q79; Secondary 76N10, 76N15.

Global symmetric classical solutions for radiative compressible Navier‐Stokes equations with temperature‐dependent viscosity coefficients

“…wherēis given by (26). With (172), we can apply Poincaré inequality and (171) to derive In order to apply Grönwall's inequality, we eliminate the terms on the right side of the inequalities.…”

One-dimensional Dynamic Model and Experimental Verification of Scramjet

“…More refined decay estimates have been obtained by Wang-Wen [37] and Wang-Wu [38], among other works. The first named author in [22] generalised the aforementioned existence and asymptotic results in [3] to the unbounded domain case Ω = R. Other types of boundary conditions have also been considered in [3,22,37,38].…”

“…For Z, it has been shown by maximum principle arguments that 0 ≤ Z ≤ 1 (cf. [3, Lemma 2] and [22,Lemma 2.2]. Strictly speaking, these inequalities only hold almost everywhere; see however Remark 0.5 below), but we have had no knowledge about the set of degeneracy Z −1 {0}.…”

“…Regularised system. The proof of Proposition 1.1 in [3,22] relies on the C 1 -regularisation of the discontinuous reaction rate function ϕ(•) defined in Eq. (0.3), which shall be described as follows for convenience of the readers.…”

“…In the case Ω = [0, 1], it is established in [3] by linearisation and classical Schauder estimates that, for regular initial data, there exists a unique classical solution in parabolic Hölder spaces. The analogue on Ω = R is obtained in [22]. The crucial ingredients of the arguments therein are adapted from Jiang [15,16] (which, in turn, rely on the localisation lemma by Kazhikhov [20] and Kazhikhov-Shelukhin [18,19]) and Li-Liang [23].…”