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具有线性状态方程 $p=Kρ$（其中 $1/3<K<1$）的相对论性完美流体的稳定性

On the stability of relativistic perfect fluids with linear equations of state $p=Kρ$ where $1/3<K<1$

Elliot Marshall, Todd A. Oliynyk

发表信息：
Lett. Math. Phys. 113 (2023), 102
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AI总结：

本文研究相对论性欧拉方程在指数膨胀时空中的稳定性，证明非各向同性解的稳定性并数值发现各向同性解的密度对比度在未来无穷远处无界。

摘要：

对于 $1/3<K<1$，我们考虑在指数膨胀的 FLRW 时空上，具有线性状态方程 $p=Kρ$ 的相对论性欧拉方程的两类不同的空间齐次解的稳定性。这两类解的区别在于一类是空间各向同性的，而另一类则不是。我们建立了非各向同性类的非线性扰动的未来稳定性，适用于全部参数范围 $1/3<K<1$，这改进了第二作者先前建立的稳定性结果，该结果要求 $K$ 位于受限范围 $(1/3,1/2)$ 内。作为理解各向同性类的非线性扰动行为的第一步，我们在 $\mathbb{T}^2$-对称性假设下构造了相对论性欧拉方程的数值解。这些解由固定时刻的初始数据生成，该数据被选择为与各向同性解的初始数据足够接近。我们的数值结果表明，对于全部参数范围 $1/3<K<1$，与各向同性解的非线性扰动相关的密度对比度 $\frac{\partial_{x}ρ}ρ$ 在有限个空间点附近发展出陡峭的梯度，并在未来类时无穷远处变得无界。这种行为由 Rendall 在 \cite{Rendall:2004} 中预期，具有特别的意义，因为它与宇宙学中暴胀的标准图景不一致。

英文摘要：

For $1/3<K<1$, we consider the stability of two distinct families of spatially homogeneous solutions to the relativistic Euler equations with a linear equation of state $p=Kρ$ on exponentially expanding FLRW spacetimes. The two families are distinguished by one being spatially isotropic while the other is not. We establish the future stability of nonlinear perturbations of the non-isotropic family for the full range of parameter values $1/3<K<1$, which improves a previous stability result established by the second author that required $K$ to lie in the restricted range $(1/3,1/2)$. As a first step towards understanding the behaviour of nonlinear perturbations of the isotropic family, we construct numerical solutions to the relativistic Euler equations under a $\mathbb{T}^2$-symmetry assumption. These solutions are generated from initial data at a fixed time that is chosen to be suitably close to the initial data of an isotropic solution. Our numerical results reveal that, for the full parameter range $1/3<K<1$, the density contrast $\frac{\partial_{x}ρ}ρ$ associated to a nonlinear perturbation of an isotropic solution develops steep gradients near a finite number of spatial points where it becomes unbounded at future timelike infinity. This behaviour, anticipated by Rendall in \cite{Rendall:2004}, is of particular interest since it is not consistent with the standard picture for inflation in cosmology.

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