Nonlinear Stability of Rarefaction Waves for a Compressible Micropolar Fluid Model with Zero Heat Conductivity
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
NONLINEAR STABILITY OF RAREFACTION WAVES FOR A COMPRESSIBLE MICROPOLAR FLUID MODEL WITH ZERO HEAT CONDUCTIVITY∗
7¬)
Jing JIN (
1†
Noor REHMAN2
)
Qin JIANG (
1
1. School of Mathematics and Statistics, Huanggang Normal University, Huanggang 438000, China 2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China E-mail : [email protected]; [email protected]; [email protected] Abstract In 2018, Duan [1] studied the case of zero heat conductivity for a one-dimensional compressible micropolar fluid model. Due to the absence of heat conductivity, it is quite difficult to close the energy estimates. He considered the far-field states of the initial data to be constants; that is, lim (v0 , u0 , ω0 , θ0 )(x) = (1, 0, 0, 1). He proved that the solution x→±∞
tends asymptotically to those constants. In this article, under the same hypothesis that the heat conductivity is zero, we consider the far-field states of the initial data to be different constants – that is, lim (v0 , u0 , ω0 , θ0 )(x) = (v± , u± , 0, θ± )-and we prove that if both the x→±∞
initial perturbation and the strength of the rarefaction waves are assumed to be suitably small, the Cauchy problem admits a unique global solution that tends time – asymptotically toward the combination of two rarefaction waves from different families. Key words
micropolar fluids; rarefaction wave; zero-heat conductivity
2010 MR Subject Classification
1
35Q35; 76D03
Introduction
This article is concerned with the large time behavior of global solutions of the Cauchy problem for compressible micropolar fluid with zero heat conductivity, which can be expressed as follows: vt − ux = 0, x ∈ R, t > 0, µu x u + p = , x t v x (1.1) µuu 1 2 ωx2 x 2 (e + u ) + (pu) = + + vω , t x 2 v v x h i ω x ωt = − vω , v x where v = 1ρ , u, p, e and ω represent the specific volume, velocity, pressure, internal energy density and microrotation velocity, respectively. µ and κ stand for viscosity coefficient and ∗ Received
December 11, 2018; revised May 12, 2020. The first author was supported by Hubei Natural Science (2019CFB834 ). The second author was supported by the NSFC (11971193 ). † Corresponding author: Jing JIN.
No.5
1353
J. Jin et al: RAREFACTION WAVE STABILITY
heat conductivity. For the deduction of this model please refer to [2–5]; for a more physical explanation, please refer to [6, 7]. We assume, as is usual in thermodynamics, that by using any given two of the five thermodynamical variables – v, p, e, θ, and s – the remaining three variables are their functions. Furthermore, their relation is implied by the second law of thermodynamics: θds = de + pdv. According to this, if we choose (v, θ) as independent variables and write (p, e, s) = (p(v, θ), e(v, θ), s(v, θ)), then we can deduce that sv (v, θ) = pθ (v, θ), eθ (v, θ) (1.2) sθ (v, θ) = , θ ev (v,
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