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arXiv 2005.00276math.AP
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大初始扰动粘性辐射和反应气体稀疏波的非线性稳定性

Nonlinear stability of rarefaction waves for a viscous radiative and reactive gas with large initial perturbation

Guiqiong Gong, Lin He, Yongkai Liao

备注：
This paper has been accepted for publication in SCIENCE CHINA Mathematics
发表信息：
SCIENCE CHINA Mathematics (2020)
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摘要：

本文研究了一维可压缩Navier-Stokes型系统的Cauchy问题的稀疏波的时间渐近非线性稳定性，其中压力$p$、比内能$e$、比体积$v$、绝对温度$θ$和比熵$s$的本构关系由$p=Rθ/v +aθ^4/3$给出，$e=C_vθ+avθ^4$和$s=C_v\ln θ+ 4avθ^3/3+R\ln v$，其中$R>0$、$C_{v}>0$和$a>0$分别是理想气体常数、比热和辐射常数。对于这样一个特定的气体运动，一个有些令人惊讶的事实是，一般来说，压力$\widetilde{p}(v,s)$不是比容$v$和比熵$s$的凸函数。即便如此，我们在本文中表明，稀疏波的时间渐近稳定的大的初始扰动提供的辐射常数$a$和稀疏波的强度是足够小的。分析的关键是推导出比容和绝对温度的正的上下界，它们与空间和时间变量一致，但与辐射常数$a$无关。

英文摘要：

We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of an one-dimensional compressible Navier-Stokes type system for a viscous, compressible, radiative and reactive gas, where the constitutive relations for the pressure $p$, the specific internal energy $e$, the specific volume $v$, the absolute temperature $θ$, and the specific entropy $s$ are given by $p=Rθ/v +aθ^4/3$, $e=C_vθ+avθ^4$, and $s=C_v\ln θ+ 4avθ^3/3+R\ln v$ with $R>0$, $C_{v}>0$, and $a>0$ being the perfect gas constant, the specific heat and the radiation constant, respectively. For such a specific gas motion, a somewhat surprising fact is that, general speaking, the pressure $\widetilde{p}(v,s)$ is not a convex function of the specific volume $v$ and the specific entropy $s$. Even so, we show in this paper that the rarefaction waves are time-asymptotically stable for large initial perturbation provided that the radiation constant $a$ and the strength of the rarefaction waves are sufficiently small. The key point in our analysis is to deduce the positive lower and upper bounds on the specific volume and the absolute temperature, which are uniform with respect to the space and the time variables, but are independent of the radiation constant $a$.

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