The limits of coefficients of the species diffusion and the rate of reactant to one-dimensional compressible Navier\u2013Stokes equations for a reacting mixture

Mingyu Zhang

doi:10.1186/s13662-019-2205-2

Abstract

In this paper, we consider the initial-boundary value problem of the one-dimensional compressible viscous and heat-conductive Navier–Stokes equations with a reacting mixture. This model is used to describe the dynamic combustion. Respectively, we obtain the vanishing species diffusion limit, the rate of reactant limit, and the convergence rates.

Highlights

The equations of one-dimensional compressible viscous and heat-conductive Navier–Stokes equations for a reacting mixture in the Lagrange coordinates are of the following form:
Our third and fourth results of this paper are concerned with the vanishing species diffusion and rate of reactant limit, which are shown by making a full use of some strong condition of the heat-conductive Navier–Stokes equations for a reacting mixture
We use the following four lemmas to show species diffusion limit and convergence rates with L2-norm and H1-norm, respectively, and this can be illustrated by Theorem 1.3

Summary

Introduction

The equations of one-dimensional compressible viscous and heat-conductive Navier–. Stokes equations for a reacting mixture in the Lagrange coordinates are of the following form (see [1,2,3]):. In [1], when φ(θ ) is discontinuous, existence theorems are established for global generalized solutions to the compressible Navier–Stokes equations for a reacting mixture. The study of the vanishing species diffusion and rate of reactant limits relies on the global uniform-in-λ estimates and the global uniform-in-λ, k estimates of the solutions respectively of problem (1.1)–(1.4), which are more difficult to achieve than those for problem (1.5)–(1.8) and (1.9), (1.10), (1.11), and (1.12) due to the presence of reacting-diffusion equation. Our third and fourth results of this paper are concerned with the vanishing species diffusion and rate of reactant limit, which are shown by making a full use of some strong condition of the heat-conductive Navier–Stokes equations for a reacting mixture. With the help of global (uniform) estimates, we justify the vanishing species diffusion and rate of reactant limit and obtain the convergence rates

The vanishing species diffusion limit

L2 θ xx

The species diffusion and the rate of reactant limits