Zero dissipation limit to rarefaction wave with vacuum for the one-dimensional non-isentropic micropolar equations

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Abstract

The zero dissipation limit of the one-dimensional non-isentropic micropolar equations is studied in this paper. If the given rarefaction wave which connects to vacuum at one side, a sequence of solution to the micropolar equations can be constructed which converge to the above rarefaction wave with vacuum as the viscosity and the heat conduction coefficient tend to zero. Moreover, the uniform convergence rate is obtained. The key point in our analysis is how to control the degeneracies in the vacuum region in the zero dissipation limit process.

Section snippets

Introduction and main result

In the present paper, we consider the zero dissipation limit of the one-dimensional non-isentropic micropolar equations ρt+(ρu)x=0,(ρu)t+(ρu2+p)x=(ϵux)x,(ρw)t+(ρuw)x+ϵw=(ϵwx)x,(ρE)t+(ρuE+up)x=(κθx)x+(ϵuux)x+ϵ(wx)2+ϵw2,here, xR,t>0, the unknowns ρ(x,t)0, u(x,t), w(x,t) and θ(x,t)0 are the mass density, fluid velocity, microrotational velocity and absolute temperature respectively. Besides, p=p(ρ,θ) is pressure and E=e+u22 represents the specific total energy, where e=e(ρ,θ) is the specific

Rarefaction wave

As we all know, there is no exact rarefaction wave profile for the micropolar equations (1.1), and the following approximate rarefaction wave profile satisfying the Euler equations was motivated by Matsumura–Nishihara [28] and Xin [29].

Consider the Riemann problem for the inviscid Burgers equation: Wt+WWx=0,W(x,0)=W,x<0,W+,x>0.If W<W+, the Riemann problem (2.1) admits a rarefaction wave solution Wr(x,t)=Wr(xt) given by Wrxt=W,xtW,xt,WxtW+,W+,xtW+.However, Wrxt is only Lipschitz

Reformulation of the problem

To prove Theorem 1.1, we consider the Cauchy problem (1.1) with the smooth initial data (ρϵ,uϵ,θϵ,wϵ)(x,t=0)=(ρ̄,ū,θ̄,0)(x,0),and treat the global smooth solution (ρϵ,uϵ,θϵ,wϵ) of system (1.1), (3.1) as the perturbation around the approximate rarefaction wave (ρ̄,ū,θ̄,0). For convenience, we reformulate the system by introducing a scaling for the independent variables y=xϵ,τ=tϵ.Then we denote the perturbation (ϕ,ψ,χ,w)(y,τ)=(ρϵρ̄,uϵū,θϵθ̄,wϵ0)(x,t).For simplicity of notation, without

Proof of Theorem 1.1

In this section, we prove Theorem 1.1. Actually, there only left (1.15) with a,b,c,d given in (1.16)–(1.18) need to be proved. By Lemmas 2.2, 2.3, Proposition 3.1 and μ=ϵa|lnϵ| and δ=ϵa, it holds that for any given positive constant l there exists a positive constant Cl which is independent of ϵ such that suptlρ(,t)ρr3tLsupτ[0,+)ϕ(,t)L+suptlρ̄(,t)ρμr3tL+suptlρμr3tρr3tLClϵ12(γ+2)|lnϵ|14γ2+δ|lnδ|+μ,(1<γ2),Clϵ18|lnϵ|3(1γ)4+δ|lnδ|+μ,(γ>2),Clϵa|lnϵ|,similarly we have

Acknowledgments

The work was supported by the grants from the National Natural Science Foundation of China under contracts 11731008, 11671309 and 11971359. We would like to express our thanks to the anonymous referees for their valuable comments, which lead to substantial improvements of the original manuscript. Last but not least, the author would like to thank Professor Huijiang Zhao for his support and encouragement.

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