The Stabilizing Effect of Spacetime Expansion on Relativistic Fluids With Sharp Results for the Radiation Equation of State
The Stabilizing Effect of Spacetime Expansion on Relativistic Fluids With Sharp Results for the Radiation Equation of State
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DOI：
10.1007/s00205-013-0655-3
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发表时间：
2012-01
影响因子：
2.5
通讯作者：
Jared Speck
中科院分区：
文献类型：
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作者：
Jared Speck
在本文中，我们研究了1 + 3维相对论性欧拉方程在一个预先确定的共形平坦膨胀时空背景上的解，其中空间切片是双态的。我们假设流体验证状态方程，其中声速是。我们还假设与膨胀时空度规相关的标度因子的倒数验证了acs依赖的时间可积性条件。在这些假设下，我们使用的矢量场能量方法来证明，一个明确的家庭的物理动机，空间均匀，空间各向同性的流体解决方案是全球未来稳定的小扰动下，他们的初始条件。对应于每个比例因子的显式解类似于著名的空间平坦的Friedmann-Lemaintre-Robertson-步行者族。我们的非线性分析，利用耗散项产生的扩展，表明扰动的解决方案存在的所有未来的时间，并保持接近显式解决方案。这一工作是以前结果的推广，表明当时空指数膨胀时，类似的稳定性结果成立。对于p =（1/3）ρ的辐射方程，我们还证明了如果标度因子的倒数的时间可积性条件不成立，则显式流体解是不稳定的.更精确地说，我们证明了一个开放的初始数据族的存在性，使得（i）它包含显式解的数据的任意小的光滑扰动和（ii）相应的扰动解必然在有限时间内形成冲击。激波形成的证明是基于相对论性欧拉方程的共形不变性，当允许减少到一个众所周知的结果Christodoulou。
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 toWe assume that the fluid verifies the equation of statewhereis the speed of sound. We also assume that the reciprocal of the scale factor associated with the expanding spacetime metric verifies acs−dependent time-integrability condition. Under these assumptions, we use the vector field energy method to prove that an explicit family of physically motivated, spatially homogeneous, and spatially isotropic fluid solutions are 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–Lemaître–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 statep= (1/3)ρ, we also show that if the time-integrability condition for the reciprocal of the scale factor fails to hold, then the explicit fluid solutions are unstable. More precisely, we show the existence of an open family of initial data such that (i) it contains arbitrarily small smooth perturbations of the explicit solutions’ data and (ii) the corresponding perturbed solutions necessarily form shocks in finite time. The shock formation proof is based on the conformal invariance of the relativistic Euler equations whenwhich allows for a reduction to a well-known result of Christodoulou.