The Global Analysis of Fluids in General Relativity

本文的重点是研究相对论性欧拉方程解的全局非线性行为，这是相对论性流体力学的基本偏微分方程。特别是，首席研究员将研究时空膨胀对流体整体行为的影响。主要目标是了解能够抑制由小而平滑的初始数据发起的溶液中冲击形成的最小膨胀率。一个相关的次要目标是在平滑初始数据上找到保证冲击演化形成的条件。在更广泛的背景下，这项工作将促进我们对扩展洛伦兹流形上拟线性双曲偏微分方程解的整体行为的理解。解决这些问题需要在几何、分析和流体力学的界面上发展和应用数学技术。具体来说，首席研究员将使用几何能量方法来推导双曲偏微分方程解的耗散和色散估计。该提案中的项目有望帮助为一些最基本的宇宙学预测提供严格的数学证明，其中许多预测是基于相对论性欧拉方程显式解的行为；在宇宙学中，宇宙中的大部分“正常物质”都被认为是由流体有效地模拟出来的。了解时空膨胀对流体行为的影响尤为重要，因为实验观测表明，我们自己的宇宙正在经历加速膨胀。尽管标准宇宙学模型所描述的流体行为的物理图景现在已经建立起来，但其预测的完整数学证明尚未出现。特别是，为了使显式流体解具有预测价值，必须证明它们在初始条件的小扰动下是全局稳定的；这是提案的主要目标之一。另一方面，首席研究员对激波形成的研究可能会暴露出宇宙学中流体模型的一些局限性。除了进行研究外，首席研究员还将监督与本提案主题相关的本科生研究项目。

The focus of this proposal is the study of the global nonlinear behavior of solutions to the relativistic Euler equations, which are the fundamental PDEs of relativistic fluid mechanics. In particular, the principal investigator will study the effect that spacetime expansion has on the global behavior of fluids. A primary goal is to understand the minimal expansion rate that is capable of suppressing the formation of shocks in solutions that are launched by small, smooth initial data. A related secondary goal is to find conditions on smooth initial data that guarantee the evolutionary formation of shocks. In a broader context, this work will advance our understanding of the global behavior of solutions to quasilinear hyperbolic PDEs on expanding Lorentzian manifolds. Attacking these problems requires the development and application of mathematical techniques lying at the interface of geometry, analysis, and fluid mechanics. Specifically, the principal investigator will use geometric energy methods to derive dissipative and dispersive estimates for solutions to hyperbolic PDEs.The projects in this proposal are expected to help provide rigorous mathematical justification for some of the most fundamental predictions of cosmology, many of which are based on the behavior of explicit solutions to the relativistic Euler equations; in cosmology, much of the "normal matter" content of the universe is assumed to be effectively modeled by a fluid. It is especially important to understand the effect that spacetime expansion has on the behavior of fluids, for experimental observations indicate that our own universe is undergoing accelerated expansion. Although the physical picture of the fluid behavior set forth by the standard cosmological model is by now well-established, a full mathematical justification of its predictions has yet to emerge. In particular, in order for the explicit fluid solutions to have predictive value, it is essential to show that they are globally stable under small perturbations of their initial conditions; this is one of the major proposal goals. On the other hand, the principal investigator's investigation of shock formation may expose some limitations of the fluid model in cosmology. In addition to conducting research, the principal investigator will also supervise undergraduate research projects connected to the main themes of this proposal.