Bigorgne Léo, Fajman David, Joudioux Jérémie, Smulevici Jacques, Thaller Maximilian
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK.
Gravitational Physics, Faculty of Physics, University of Vienna, Vienna, Austria.
Arch Ration Mech Anal. 2021;242(1):1-147. doi: 10.1007/s00205-021-01639-2. Epub 2021 Jul 22.
We prove the global asymptotic stability of the Minkowski space for the massless Einstein-Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at the top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein-Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components.
我们证明了在波动坐标下无质量爱因斯坦 - 弗拉索夫系统的闵可夫斯基空间的全局渐近稳定性。与之前关于该主题的工作不同，对于弗拉索夫场在空间或动量变量方面的初始数据，不需要紧致支集假设。事实上，初始时的衰减是最优的。当前的证明基于弗拉索夫场的向量场和加权向量场技术，这是由法伊曼、朱迪厄克斯和斯穆列维奇在之前的工作中发展起来的，并且严重依赖于无质量弗拉索夫方程的几个结构性质，类似于零条件和弱零条件。为了处理度规的弱衰减率，我们传播精心选择的分层加权能量范数，这些范数反映了远离光锥的粒子密度所满足的强衰减性质。当我们无法获得某些度规分量的改进逐点衰减估计时，在最高阶会出现一个特殊的分析困难。利用无质量爱因斯坦 - 弗拉索夫系统中的一种新颖分层结构解决了这个困难，该结构利用了不同度规分量能量范数的不同增长率的传播。
