Qualitative dynamics of self-gravitating Newtonian liquids

本博士项目的主要目标是系统而严谨地研究由引力欧拉-泊松系统描述的自引力牛顿液体的整体动力学特性。在这里，液体是被真空包围的紧密支撑的流体团，其密度在流体-真空边界处不连续。这应该与气体形成对比，在气体中，密度是假定在流体-真空界面连续的。这个方程组来自天体物理学，被用作孤立恒星的基本数学模型。该项目的第一个目标是确定导致恒星在有限时间内坍缩的初始数据。用专业术语来说，目标是研究上述方程组的奇点形成。这些问题是非线性偏微分方程领域的几个现代发展的中心，需要分析、动力系统和数学物理的工具。在这个过程中，需要开发一些新的成分，包括彻底重新审视现有的局部适定性理论。该项目的第二个目标是研究液体欧拉-泊松系统的平稳（即时间无关）解的存在性和稳定性。虽然人们对欧拉-泊松气体（与液体相反）了解很多，但有必要系统地探讨在液体情况下静止恒星的存在问题并研究它们的稳定性。在此过程中，将详细研究问题的尺度不变性，并探讨临界性的概念，以促进静止液体恒星的稳定性分析。虽然该项目的核心工具牢牢扎根于数学分析，但它与其他领域的许多联系——最著名的是几何、数学物理和动力系统——将在该项目中发挥重要作用。

The main goal of this PhD project is to initiate a systematic and rigorous study of global dynamic properties of self gravitating Newtonian liquids described by the gravitational Euler-Poisson system. Here liquids are compactly supported blobs of fluid surrounded by vacuum, whose density is discontinuous at the fluid-vacuum boundary. This should be contrasted to gases, where the density is assumed continuous at the fluid-vacuum interface. This system of equations comes from astrophysics and is used as a basic mathematical model of an isolated star. The first goal of the project is to identify initial data that lead to stellar collapse in finite time. In technical terms, the goal is to study singularity formation for the above mentioned system of equations. Such questions lie at the centre of several modern developments in the field of nonlinear partial differential equations and necessitate tools from analysis, dynamical systems, and mathematical physics. In the process, several new ingredients need to be developed, including a thorough revisiting of the existing local well-posedness theory. The second goal of the project, is the study of the existence and stability properties of stationary (i.e. time-independent) solutions to the liquid Euler-Poisson system. While much is known for the Euler-Poisson gases (as opposed to liquids), it is necessary to systematically approach the question of existence of stationary stars in the liquid case and study their stability. In the process, the scaling invariances of the problem will be studied in detail and a notion of criticality will be investigated with the aim of facilitating the stability analysis for stationary liquid stars. While the core tools for the project are firmly rooted in mathematical analysis, there are numerous connections to other fields - most notably geometry, mathematical physics, and dynamical systems - that will play an important role in the project.