标题： 时空膨胀对相对论流体的稳定作用 对于辐射状态方程的精确结果 显示英文标题

标题： The Stabilizing Effect of Spacetime Expansion on Relativistic Fluids With Sharp Results for the Radiation Equation of State

摘要： 在本文中，我们研究在一个预设的共形平坦膨胀时空背景上的1 + 3维相对论欧拉方程，其空间片微分同构于$\mathbb{R}^3.$。我们假设流体满足状态方程$p = c_s^2 \rho,$，其中$0 \leq c_s \leq \sqrt{1/3}$是声速。 我们还假设与膨胀时空度量相关的尺度因子的倒数满足一个与$c_s-$相关的时间可积条件。 在这些假设下，我们使用向量场能量方法证明，一个显式族物理上合理的、空间均匀且空间各向同性的流体解在初始条件的小扰动下是全局未来稳定的。 每个尺度因子对应的显式解是众所周知的空间平坦弗里德曼-勒梅特-罗伯逊-沃尔克族的类似物。 我们的非线性分析利用了由膨胀产生的耗散项，表明扰动解在所有未来时间都存在，并且保持接近显式解。 这项工作是对先前结果的扩展，这些结果表明当时空呈指数膨胀时，类似的稳定性结果成立。 在辐射状态方程$p = (1/3)\rho,$的情况下，我们还表明，如果尺度因子倒数的时间可积条件不成立，则显式流体解是不稳定的。 更准确地说，我们表明显式解初始条件的任意小光滑扰动可以引发在有限时间内形成激波的扰动解。 激波形成的证明基于相对论欧拉方程在$c_s^2 = 1/3,$时的共形不变性，这使得能够简化为克里斯托多卢的已知结果。 显示英文摘要

摘要： In this article, we study the 1 + 3 dimensional relativistic Euler equations on a pre-specified conformally flat expanding spacetime background with spatial slices that are diffeomorphic to $\mathbb{R}^3.$ We assume that the fluid verifies the equation of state $p = c_s^2 \rho,$ where $0 \leq c_s \leq \sqrt{1/3}$ is the speed of sound. We also assume that the inverse of the scale factor associated to the expanding spacetime metric verifies a $c_s-$dependent time-integrability condition. Under these assumptions, we use the vectorfield energy method to prove that an explicit family of physically motivated, spatially homogeneous, and spatially isotropic fluid solutions is globally future-stable under small perturbations of their initial conditions. The explicit solutions corresponding to each scale factor are analogs of the well-known spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker family. Our nonlinear analysis, which exploits dissipative terms generated by the expansion, shows that the perturbed solutions exist for all future times and remain close to the explicit solutions. This work is an extension of previous results, which showed that an analogous stability result holds when the spacetime is exponentially expanding. In the case of the radiation equation of state $p = (1/3)\rho,$ we also show that if the time-integrability condition for the inverse of the scale factor fails to hold, then the explicit fluid solutions are unstable. More precisely, we show that arbitrarily small smooth perturbations of the explicit solutions' initial conditions can launch perturbed solutions that form shocks in finite time. The shock formation proof is based on the conformal invariance of the relativistic Euler equations when $c_s^2 = 1/3,$ which allows for a reduction to a well-known result of Christodoulou.