论文标题
渐近地类似Kasner的奇异性
Asymptotically Kasner-like singularities
论文作者
论文摘要
我们证明了对爱因斯坦真空方程的解决方案的存在，独特性和规律性，采用表格$ {^{（4）} g} = -dt^2 + sum_ {i，j = 1}^3 a_ {ij {ij} X^i \，\ Mathrm {d} X^J $$在$（0，t] _t \ times \ Mathbb t^3_x $中，其中$ a_ {ij}（t，x）$和$ p_i（x）$和$ p_i（x）$是无需对称或分析性假设的常规函数。这些$ $ singulic和usymptions $ usympt insympt kas+kasnemptime kas+\ the+\ the+\ the+\ the+\ the+\ casnece+\ the+。预计解决方案将是高度非生成的，我们的构造可以被视为通过fuchsian型分析解决一个单一的初始值问题，在“奇异性超表面”上，数据构成了$ \ {t = 0 \} $。 为了进行分析，我们研究了同步坐标系中的问题。特别是，我们基于估计常数$ t $ hypersurfaces的第二种基本形式，在此类坐标系中介绍了一种新颖的方法来执行（加权）能量估计。
We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form $${^{(4)}g} = -dt^2 + \sum_{i,j = 1}^3 a_{ij}t^{2p_{\max\{i,j\}}}\, \mathrm{d} x^i\, \mathrm{d} x^j$$ on $(0,T]_t \times \mathbb T^3_x$, where $a_{ij}(t,x)$ and $p_i(x)$ are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as $t\to 0^+$. These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the "singular hypersurface" $\{ t = 0\}$. This is the first such result without imposing symmetry or analyticity. To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-$t$ hypersurfaces.
