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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.

广义相对论是目前对引力现象及其与物质相互作用的最佳描述。它由爱因斯坦于1915年提出，已被实验压倒性地证实，今天是研究天体物理学和宇宙学的物理学家所使用的工具箱的重要组成部分。在数学上，理论是丰富的，数学广义相对论的主题现在是数学家中令人兴奋和活跃的研究领域。该项目涉及相对论流体的数学性质，特别是描述在相对论定律不能被忽略的条件下流体运动的理论的数学结构。例子包括黑洞附近的吸积盘，粒子加速器中重离子碰撞形成的夸克胶子等离子体的动力学，以及星星演化的精细性质的研究。该项目所带来的数学进步和技术将是重要的应用，特别是对中子星合并的粘性效应的研究。就更广泛的影响而言，该研究具有很强的跨学科成分，并将继续数学和物理之间富有成效的互动。该项目还将有助于青年科学家的教育，继续PI向研究生和博士后传播相对论流体动力学的一些最新发现的努力，具体而言，该项目将研究：（a）具有物理真空边界的Einstein-Euler系统的局部适定性;（B）相对论Euler方程的激波形成;（c）相对论粘性流体理论的因果性和局部适定性;（d）与爱因斯坦方程耦合的相对论粘性流体方程的公式，适用于中子星星合并的数值模拟。在这个项目中研究的所有偏微分方程（PDE）的一个统一的特点是，他们形成一个系统的多个特征速度。在一个以上的空间维度上理解这种类型的系统是目前双曲偏微分方程的主要挑战之一。所有问题都将在现实的物理假设下进行研究。这涉及到，特别是，考虑相对论性流体在三维空间，涡量，没有对称性假设。这一奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。