Well-posedness of the Einstein-Euler system in asymptotically flat spacetimes: The constraint equations
Well-posedness of the Einstein-Euler system in asymptotically flat spacetimes: The constraint equations
复制标题
DOI：
10.1016/j.jde.2011.05.037
复制
发表时间：
2011-09-15
影响因子：
2.4
通讯作者：
Karp, Lavi
中科院分区：
文献类型：
--
作者：
Brauer, Uwe;Karp, Lavi
本文讨论了耦合Einstein-Euler方程组的初值构造问题。我们考虑的条件下，能量密度可能会消失或趋于零，在无穷大，压力是一个分数的能量密度的幂。为了达到我们的目的，我们使用了一种分数阶的加权Sobolev空间，但不能直接使用常用的求解约束方程的Lichnerowicz-约克标度法（Choquet-Bruhat and约克，1980 [9]; Cantor，1979 [7]）。基本问题是物质源被共形地标度，流体变量必须从共形变换的物质源中恢复。Dain和Nagy（2002）[11]已经解决了这个问题，尽管是在不同的背景下。我们证明了当物质变量被限制在一定的区域内时，Einstein约束方程在分数阶加权Sobolev空间中存在唯一解。规律性取决于状态方程的分数次方。(C)2011 Elsevier Inc. All rights reserved.
This paper deals with the construction of initial data for the coupled Einstein-Euler system. We consider the condition where the energy density might vanish or tend to zero at infinity, and where the pressure is a fractional power of the energy density. In order to achieve our goals we use a type of weighted Sobolev space of fractional order.The common Lichnerowicz-York scaling method (Choquet-Bruhat and York, 1980 [9]; Cantor, 1979 [7]) for solving the constraint equations cannot be applied here directly. The basic problem is that the matter sources are scaled conformally and the fluid variables have to be recovered from the conformally transformed matter sources. This problem has been addressed, although in a different context, by Dain and Nagy (2002) [11]. We show that if the matter variables are restricted to a certain region, then the Einstein constraint equations have a unique solution in the weighted Sobolev spaces of fractional order. The regularity depends upon the fractional power of the equation of state. (C) 2011 Elsevier Inc. All rights reserved.