论文标题
爱因斯坦 - 欧拉特系统在负爱因斯坦空间上的线性稳定性
The linear stability of the Einstein-Euler system on negative Einstein spaces
论文作者
论文摘要
在这里，我们证明了“ $ n+1 $'家族的线性稳定性 - 尺寸弗里德曼·莱玛特·罗伯逊·沃克（FLRW）宇宙相对论的宇宙学模型。我们表明，由于常规初始数据的常规初始数据，围绕具有紧凑型空间拓扑的一类线性欧元欧特场方程（$ n = 3 $）均具有均匀界限和衰减的恒定负空间尺度曲率的家族。在CMCSH仪表中，利用液体的$ n- $速度1形式的霍奇分解，将einstein-euler系统在CMCSH仪表中变成了椭圆形的 - 纤维性（和非自治），从而促进了能量类型参数的应用。利用从关联的椭圆方程得出的估计值，我们首先证明了在扩展方向上Lyapunov功能（控制数据的适当规范）的统一界限。利用统一的界限，我们后来获得了一个清晰的衰减估计值，这表明该特定宇宙模型的扩展可能足以控制爱因斯坦欧拉群岛系统的非线性（包括可能的冲击形成），以实现完全非线性稳定性的潜在证明。此外，流体的$ n- $速度场夫妇的旋转和谐波部分在较高阶的剩余自由度上，这再次表明，在足够小的数据限制中，当前分析直接扩展到完全非线性的设置。此外，我们的结果需要在扩展因子和适当范围的绝热索引$γ_{a} $（$（1，\ frac {n+1} {n}）$ I.E.，$（1，\ frac {4} {3}）$中$ p =（γ_{a} -1）ρ$。
Here we prove the linear stability of a family of `$n+1$'-dimensional Friedmann Lemaître Robertson Walker (FLRW) cosmological models of general relativity. We show that the solutions to the linearized Einstein-Euler field equations around a class of FLRW metrics with compact spatial topology (negative Einstein spaces and in particular hyperbolic for $n=3$) arising from regular initial data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. Utilizing a Hodge decomposition of the fluid's $n-$velocity 1-form, the linearized Einstein-Euler system becomes elliptic-hyperbolic (and non-autonomous) in the CMCSH gauge facilitating an application of an energy type argument. Utilizing the estimates derived from the associated elliptic equations, we first prove the uniform boundedness of a Lyapunov functional (controlling appropriate norm of the data) in the expanding direction. Utilizing the uniform boundedness, we later obtain a sharp decay estimate which suggests that expansion of this particular universe model may be sufficient to control the non-linearities (including possible shock formation) of the Einstein-Euler system in a potential future proof of the fully non-linear stability. In addition, the rotational and harmonic parts of the fluid's $n-$velocity field couple to the remaining degrees of freedom in higher orders, which once again indicates a straightforward extension of current analysis to the fully non-linear setting in the sufficiently small data limit. In addition, our results require a certain integrability condition on the expansion factor and a suitable range of the adiabatic index $γ_{a}$ ($(1,\frac{n+1}{n})$ i.e., $(1,\frac{4}{3})$ in the physically relevant `$3+1$' universe) if the equation of state $p=(γ_{a}-1)ρ$ is chosen.
