Well-posedness and stability for relativistic Euler equations with free boundaries

对星星的严格数学描述需要两个著名的偏微分方程系统之间的耦合：流体力学的欧拉方程和广义相对论的爱因斯坦方程。这些系统具有丰富的数学结构，描述了基本的物理过程，在数学和物理学中起着重要的作用。它们的耦合给出了所谓的欧拉-爱因斯坦系统，这是分析与重力耦合的流体的基本模型。最重要的例子之一是恒星，理想化为流体或气体云，有一个移动的边界界面将它们与真空隔开。关于自由边界欧拉-爱因斯坦系统，人们可以问的第一个也是基本的数学问题是：在给定星星的某些初始构型的情况下，人们能否建立一个严格的数学框架来建立这个系统解的存在性和唯一性？我们能否同样地追踪运动真空边界的行为和规律性？初始配置的更改如何影响解决方案？这类问题在技术上被称为适定性问题，该方案的主要目的是为运动真空边界的Euler-Einstein系统建立一个严格的适定性框架。在数学上，这个问题交织着与自由边界流体和爱因斯坦方程相关的各种困难。由于其高度非线性的性质，它是一个先验不清楚的自由真空边界是否会导致退化的模型，导致潜在的崩溃的解决方案，即使在很短的时间。即使在牛顿的背景下，约翰·冯·诺依曼早在1949年就暗示了问题的退化性质。然而，在过去的几十年里，一方面在牛顿自由边界欧拉方程的严格研究方面，另一方面在数学广义相对论方面，都有了惊人的发展。在过去的三年中，牛顿自由边界可压缩流体的适定性理论得到了很好的发展。同样，对相对论性流体进行严格的数学研究是一个丰富而广泛的主题，在过去的十年中产生了大量的数学研究。2007年Christodoulou在相对论性流体稳定激波形成的研究中取得了重大突破，这为这一理论的提出提供了重要的动力，但自由边界几何与相对论几何之间复杂而美丽的相互作用，给现有文献带来了新的数学结构和额外的挑战。该提案详细探讨了这些结构，并提出了新的想法来证明1）闵可夫斯基时空上的自由真空边界欧拉方程和2）自由真空边界欧拉-爱因斯坦系统的适定性。然后，它使用这样建立的框架来解决著名的弗里德曼-勒梅特-罗伯逊-沃克解决方案的稳定性，描述了一个加速膨胀的宇宙。

A rigorous mathematical description of a star requires a coupling between two famous systems of partial differential equations: the Euler equations of fluid mechanics and the Einstein equations of general relativity. These systems possess a rich mathematical structure and describe fundamental physical processes, playing an important role in both mathematics and physics. Their coupling gives the so-called Euler-Einstein system, a fundamental model in the analysis of fluid bodies coupled to gravity. One of the most important examples are the stars, idealised as fluid or gas clouds with a moving boundary interface separating them from the vacuum.The first and basic mathematical question that one can ask about the free boundary Euler-Einstein system is the following: can one develop a rigorous mathematical framework that establishes the existence and uniqueness of solutions to this system given some initial configuration of the star? Can we similarly track down the beahviour and the regularity of the moving vacuum boundary? How do the changes in initial configurations affect the solutions? Such questions are technically termed as problems of well-posedness, and the principal aim of the proposal is to develop a rigorous well-posedness framework for the moving vacuum boundary Euler-Einstein system.Mathematically, this problem intertwines various difficulties associated with both the free boundary fluids and the Einstein equations. Due to its highly nonlinear nature, it is a priori unclear whether the free vacuum boundary can cause a degeneracy in the model, leading to a potential breakdown of the solutions, even after a very short time. Even in the Newtonian setting, the degenerate nature of the problem was hinted at by John Von Neumann as early as 1949. However, the past few decades have seen striking developments in the rigorous study of the Newtonian free boundary Euler equations on one hand, and in the mathematical general relativity on the other. A satisfactory well-posedness theory for the Newtonian free boundary compressible fluids has been developed in the past 3 years. Similarly, a rigorous mathematical study of relativistic fluids is a rich and broad topic, that has generated a lot of mathematical research over the past decade. As an example, a momentous breakthrough in the study of stable shock formation for relativistic fluids was accomplished by Christodoulou in 2007.While such works provide an important impetus for this proposal, the complicated, but beautiful interaction between the free boundary geometry and the relativistic geometry, gives rise to new mathematical structures and additional challenges with respect to the existing literature. The proposal explores these structures in detail and develops novel ideas to show the well-posedness of 1) the free vacuum boundary Euler equations on the Minkowski spacetime and 2) the free vacuum boundary Euler-Einstein system. It then uses the thus established framework to address the stability of the well-known Friedmann-Lemaitre- Robertson-Walker solutions, describing an accelerating expanding universe.