发表机构
Sorbonne Université; Centre National de la Recherche Scientifique（索邦大学; 法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对T³上具有T²对称性的真空和物质时空，在满足特定条件的本构方程下，通过JKL形式等方法建立爱因斯坦-欧拉和爱因斯坦-纳维-斯托克斯面积流的有限能量柯西稳定性，结果覆盖多种情况且与相关机制无关。
AI 中文摘要
我们针对T³上具有T²对称性的真空和物质时空，在满足双曲性以及真空和大质量-能量密度处温和渐近条件的一般本构方程下，建立了有限能量柯西稳定性。首先，我们利用对称轨道的面积作为时间函数，引入了爱因斯坦面积流的JKL形式，这是一个一阶演化-约束系统，将几何的波映射结构与物质的双曲平衡定律以及用于扭转和对称轨道切向动量的加权输运方程耦合在一起。其次，我们还定义了两类双曲爱因斯坦-纳维-斯托克斯模型，由我们所称的纳维-斯托克斯势和弛豫率映射构建。所提出的粒子产生模型具有无散度物质应力和非负粒子数产生，而所提出的耗散能量模型守恒粒子数并耗散流体质量-能量，同时辅助应力恢复了收缩比安基恒等式所需的总应力-能量守恒。对于正则有限能量（爱因斯坦、欧拉、纳维-斯托克斯）流，我们建立了最大值和不变域原理、几何和能量单调性公式、类空和类时能量估计以及总变差估计。第三，这些估计共同在未来膨胀区域以及收缩区域（直至类空体积坍缩前）得到有限能量柯西稳定性。它们与粘性（或弛豫）机制无关，覆盖真空、无界质量-能量密度和任意有限快度，还可扩展至满足参考数学熵不等式的弱正则爱因斯坦-欧拉面积流。
英文摘要
We establish finite-energy Cauchy stability for vacuum and matter spacetimes with $T^2$ symmetry on $T^3$, under general constitutive equations satisfying hyperbolicity and mild asymptotic conditions at vacuum and large mass-energy density. First, using the area of the symmetry orbits as a time function, we introduce the JKL formulation of Einstein areal flows}, a first-order evolution-constraint system coupling a wave-map structure for the geometry to hyperbolic matter balance laws and weighted transport equations for the twists and the momentum tangent to the symmetry orbits. Second, we also define two classes of hyperbolic Einstein-Navier-Stokes models, constructed from (as we call them) a Navier-Stokes potential and a relaxation rate map. The proposed particle-production model has divergence-free matter stress and non-negative particle-number production, while the proposed dissipated-energy model conserves particle number and dissipates fluid mass-energy, while an auxiliary stress restores the total stress-energy conservation required by the contracted Bianchi identity. For regular finite-energy (Einstein, Euler, Navier-Stokes) flows, we establish maximum and invariant-domain principles, geometric and energy monotonicity formulas, spacelike and timelike energy estimates, and total-variation estimates. Third, together, these estimates yield finite-energy Cauchy stability in the future expanding regime and, in the contracting regime, until the spacelike volume collapses. They are independent of the viscosity (or relaxation) mechanisms and cover vacuum, unbounded mass-energy density, and arbitrary finite rapidity. They also extend to weakly regular Einstein-Euler areal flows satisfying a reference mathematical entropy inequality.
CommentsThis paper includes some estimates from version 1 of arXiv:2605.31585 for completeness
