Asymptotic Stability of a Viscous Contact Wave for the One-Dimensional Compressible Navier-Stokes Equations for a Reacti
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
ASYMPTOTIC STABILITY OF A VISCOUS CONTACT WAVE FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE∗
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Lishuang PENG (
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China E-mail : [email protected] Abstract We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact discontinuity and the initial perturbation are suitably small. We apply the approach introduced in Huang, Li and Matsumura (2010) [1] and the elementary L2 -energy methods. Key words
reacting mixture; viscous contact wave; asymptotic stability; energy estimates
2010 MR Subject Classification
1
35Q30; 35D30; 76N10
Introduction
In this article, we consider the one-dimensional compressible Navier-Stokes equations for a reacting mixture in Lagrangian coordinates: vt − ux = 0, µux ut + p(v, θ)x = ( v )x , (1.1) µu2x κθx + ( ) + λφz, e + p(v, θ)u = t x x v v dzx zt = ( 2 )x − φz, v where x ∈ R is the Lagrangian space variable and t ∈ R+ is the time variable. Here, the primary dependent variables v = v (t, x) > 0, u = u (t, x) and θ = θ (t, x) > 0 denote the specific volume, fluid velocity and absolute temperature, respectively. z = z (t, x) is the mass fraction of the reactant, while the positive constants d and λ are the specific diffusion coefficients
∗ Received October 17, 2019; revised May 20, 2020. This work was supported by the National Natural Science Foundation of China (11871341).
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
and the difference in heat between the reactant and the product, respectively. The reaction rate function φ = φ (θ) is defined, from the Arrhenius law [2], by A β φ (θ) = Kθ exp − , θ where positive constants K and A are the coefficients of the rate of the reactant and the activation energy, respectively. β is a non-negative number. The positive constants µ and κ denote the viscosity coefficient and the heat conduction coefficient, respectively. The pressure p and the internal energy e are given by the state equation Rθ , e (v, θ) = Cv θ, v where R and Cv are the positive constants. We impose the following initial and far field conditions as: (v, u, θ, z)(0, x) = (v , u , θ , z )(x) x ∈ R, 0 0 0 0 (1.2) (v, u, θ, z)(0, ±∞) = (v± , u± , θ± , 0)(x) t > 0, p (v, θ) =
where v± > 0, u± and θ± > 0 are given constants, and where we assume inf R v0 > 0, inf R θ0 > 0 and (v0 , u0 , θ0 , z0 ) (±∞) = (v± , u± , θ± , 0) as compatibility conditions. It is known that the large-time behavior of solutions of the Cauchy problem (1.1)–(1.2) is closely related to the
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