We show that any $$3+1$$ 3 + 1 -dimensional Milne model is future nonlinearly, asymptotically stable in the set of solutions to the Einstein–Vlasov system. For the analysis of the Einstein equations we use the constant-mean-curvature-spatial-harmonic gauge. For the distribution function the proof makes use of geometric $$L^2$$ L 2 -estimates based on the Sasaki-metric. The resulting estimates on the energy-momentum tensor are then upgraded by employing the natural continuity equation for the energy density. The combination of $$L^2$$ L 2 -estimates and the continuity equation reveals a powerful tool to analyze massive transport equations with potential applications beyond the result presented here.

中文翻译：

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含物质的 Milne 模型的非线性稳定性

我们证明了任何 $$3+1$$ 3 + 1 维 Milne 模型在爱因斯坦-弗拉索夫系统的一组解中都是非线性的、渐近稳定的。对于爱因斯坦方程的分析，我们使用常数平均曲率空间谐波规范。对于分布函数，证明使用基于 Sasaki 度量的几何 $$L^2$$ L 2 -估计。然后通过采用能量密度的自然连续性方程来升级对能量-动量张量的估计结果。$$L^2$$L 2 估计值和连续性方程的组合揭示了一个强大的工具来分析大规模输运方程，其潜在的应用超出了这里给出的结果。