Kasner奇点附近的相对论理想流体
Relativistic perfect fluids near Kasner singularities
AI总结：
本文证明了Kasner背景下奇点附近欧拉方程稳定解族的存在性，通过转化为对称双曲Fuchsian系统并应用新存在性理论，在声速远大于Kasner指数时成立，为大爆炸奇点附近流体行为提供理论依据。
摘要：
我们证明了Kasner背景下奇点附近欧拉方程稳定解族的存在性，该解族具有全部预期的渐近数据自由度，且无对称性或各向同性限制。通过将欧拉方程转化为对称双曲Fuchsian系统的形式，再应用奇异初值问题的新存在性理论，我们实现了存在性证明。稳定性则由arXiv:1907.04071中发展的Fuchsian系统（正则）全局初值问题的存在性理论推导得出。实际上，对于该族中的每个解，我们证明了存在具有相同定性渐近行为的邻近解的开集，并表明任何此类扰动解均与奇异初值问题的另一个解一致。我们的所有结果均在流体声速远大于所有Kasner指数的情况下成立，这被解释为Kasner大爆炸奇点附近的稳定流体渐近状态。
英文摘要：
We establish the existence of a stable family of solutions to the Euler equations on Kasner backgrounds near the singularity with the full expected asymptotic data degrees of freedom and no symmetry or isotropy restrictions. Existence is achieved through transforming the Euler equations into the form of a symmetric hyperbolic Fuchsian system followed by an application of a new existence theory for the singular initial value problem. Stability is shown to follow from the existence theory for the (regular) global initial value problem for Fuchsian systems that was developed in arXiv:1907.04071. In fact, for each solution in the family, we prove the existence of an open set of nearby solutions with the same qualitative asymptotics and show that any such perturbed solution agrees again with another solution of the singular initial value problem. All our results hold in the regime where the speed of sound of the fluid is large in comparison to all Kasner exponents. This is interpreted as the regime of stable fluid asymptotics near Kasner big bang singularities.
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