Mathematical Questions in Relativistic Fluids

相对论流体中的数学问题

基本信息

批准号：
2107701
负责人：
Marcelo Disconzi
金额：
$ 18万
依托单位：
Vanderbilt University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-15 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2107701&HistoricalAwards=false
关键词：
Mathematical Questions Relativistic Fluids

项目摘要

The general theory of relativity is currently the best description of gravitational phenomena and their interactions with matter. Proposed by Einstein in 1915, it has been overwhelmingly confirmed by experiments and is today an essential part of the toolbox employed by physicists studying astrophysics and cosmology. Mathematically, the theory is rich, and the topic of mathematical general relativity is now an exciting and active field of research among mathematicians. This project deals with mathematical properties of relativistic fluids, and in particular with the mathematical structure of theories describing the motion of fluids under conditions in which the laws of relativity cannot be neglected. Examples include accretion disks near black holes, the dynamics of the quark-gluon plasma that forms in collisions of heavy-ions in particle accelerators, and the study of fine properties of star evolution. The mathematical advances and techniques brought about by this project will be important for applications, particularly for the study of viscous effects on mergers of neutron stars. In terms of broader impact, the research has a strong interdisciplinary component and will continue a fruitful interaction between mathematics and physics. This project will also contribute to the education of young scientists, continuing the PI's efforts to disseminate some of the most recent findings in relativistic fluid dynamics to graduate students and post-docs.Specifically, this project will investigate: (a) local well-posedness of the Einstein-Euler system with a physical vacuum boundary; (b) shock formation for the relativistic Euler equations; (c) causality and local well-posedness of theories of relativistic viscous fluids; and (d) formulations of the equations of relativistic viscous fluids coupled to the Einstein equations that are suitable for the numerical simulations of neutron star mergers. A unifying feature of all the partial differential equations (PDEs) studied in this project is that they form a system with multiple characteristic speeds. Understanding systems of this type in more than one spatial dimension is currently one of the main challenges in hyperbolic PDEs. All problems will be studied under realistic physical assumptions. This involves, in particular, considering relativistic fluids in three spatial dimensions, with vorticity, and without symmetry assumptions. Not only is such treatment essential for applications, but it also involves a great deal of rich mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

相对论的一般理论目前是重力现象及其与物质的相互作用的最佳描述。爱因斯坦（Einstein）在1915年提出的，这是通过实验绝大多数证实的，如今已成为研究天体物理学和宇宙学的物理学家使用的工具箱的重要组成部分。从数学上讲，该理论是丰富的，数学一般相对论的话题现在已成为数学家的激动人心和积极的研究领域。该项目涉及相对论流体的数学特性，尤其是在无法忽略相对论规律的条件下描述流体运动的理论的数学结构。例子包括黑洞附近的积聚磁盘，在粒子加速器中重型离子碰撞中形成的夸克 - 胶子等离子体的动力学以及对星星进化的细性的研究。该项目带来的数学进步和技术对于应用将很重要，特别是对于研究中子恒星合并的粘性影响。在更广泛的影响方面，该研究具有强大的跨学科组成部分，并将继续在数学和物理学之间进行富有成果的互动。该项目还将为年轻科学家的教育做出贡献，继续PI向研究生和培训后的相对论流体动力学的一些最新发现和事后。特别是，该项目将调查：（a）Einstein-Euler System的本地良好性，具有物理真空边界；（b）相对论欧拉方程的冲击形成；（c）相对论粘性流体理论的因果关系和局部良好的特性；（d）相对论粘性流体的方程式与爱因斯坦方程相连的配方，该方程适用于中子星合并的数值模拟。该项目中研究的所有部分微分方程（PDE）的统一特征是它们形成具有多个特征速度的系统。当前，在多个空间维度中了解这种类型的系统是双曲线PDE的主要挑战之一。所有问题将在现实的物理假设下进行研究。特别是，这涉及考虑三个空间维度的相对论流体，具有涡度，没有对称性假设。这种治疗不仅对应用必不可少，而且还涉及大量丰富的数学。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准通过评估来支持的。