Asymptotic Stability of a Boundary Layer and Rarefaction Wave for the Outflow Problem of the Heat-Conductive Ideal Gas w
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
ASYMPTOTIC STABILITY OF A BOUNDARY LAYER AND RAREFACTION WAVE FOR THE OUTFLOW PROBLEM OF THE HEAT-CONDUCTIVE IDEAL GAS WITHOUT VISCOSITY∗
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Lili FAN (
School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China E-mail : [email protected]
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Meichen HOU (
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China E-mail : [email protected] Abstract This article is devoted to studying the initial-boundary value problem for an ideal polytropic model of non-viscous and compressible gas. We focus our attention on the outflow problem when the flow velocity on the boundary is negative and give a rigorous proof of the asymptotic stability of both the degenerate boundary layer and its superposition with the 3-rarefaction wave under some smallness conditions. New weighted energy estimates are introduced, and the trace of the density and velocity on the boundary are handled by some subtle analysis. The decay properties of the boundary layer and the smooth rarefaction wave also play an important role. Key words
non-viscous; degenerate boundary layer; rarefaction wave; outflow problem
2010 MR Subject Classification
1
35B35; 35B40; 35M20; 35Q35; 76N10; 76N15
Introduction
Mathematically, the motion of heat-conductive ideal gas without viscosity in the Eulerian coordinates is a hyperbolic-parabolic coupled system of the following form: ρt + (ρu)x = 0, (ρu)t + (ρu2 + p)x = 0, (1.1) n o n o 2 2 ρ(e + u ) + ρu(e + u ) + pu = κθxx , 2 t 2 x where x ∈ R+ , t > 0 and ρ(x, t) > 0, u(x, t), θ(x, t) > 0, e(x, t) > 0 and p(x, t) represent density, fluid velocity, absolute temperature, internal energy and pressure respectively. This ∗ Received
October 7, 2019; revised July 31, 2020. This work was supported by the Fundamental Research grants from the Science Foundation of Hubei Province (2018CFB693). The research of L.L.Fan was supported by the Natural Science Foundation of China (11871388) and in part by the Natural Science Foundation of China (11701439).
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
is a fundamental system to describe the motion of a compressible gas without the viscosity phenomenon, and it has many applications. Here the constant κ > 0 is the coefficient of the heat conduction. Throughout this article, we will concentrate on the ideal polytropic gas p = Rρθ = Aργ exp(
γ−1 s), R
e = Cv θ =
R θ, γ−1
(1.2)
where s is the entropy, γ > 1 is the adiabatic exponent and A, R are both positive constants. As far as we know, most of the existing results in this field concern the analysis of the global-in-time existence and stability of the elementary wave for the heat-conductive ideal gas with viscosity; that is, there are many works on the large-time behavior of solutions to the compressible Navier-Stokes equations towards the viscous versions of the three basic wave patterns: rarefaction wave, contact wave, and sh
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