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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。它由爱因斯坦在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，广义相对论的理论是丰富的，数学广义相对论的主题现在是数学家们兴奋和活跃的研究领域。这个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可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，以及恒星演化的精细性质的研究。该项目带来的数学进步和技术将对应用，特别是对中子星合并的粘性效应的研究具有重要意义。就更广泛的影响而言，这项研究具有很强的跨学科成分，并将继续在数学和物理之间进行卓有成效的互动。这个项目还将有助于对年轻科学家的教育，继续PI向研究生和博士后传播相对论流体力学的一些最新成果的努力。具体地说，这个项目将调查：(A)具有物理真空边界的爱因斯坦-欧拉系统的局部适定性；(B)相对论欧拉方程的激波形成；(C)相对论粘性流体理论的因果关系和局部适定性；以及(D)适用于中子星合并的数值模拟的爱因斯坦方程耦合的相对论粘性流体方程的公式。本课题所研究的偏微分方程组有一个共同的特点，那就是它们形成了一个具有多个特征速度的系统。在一个以上的空间维度上理解这类系统是目前双曲偏微分方程组的主要挑战之一。所有问题都将在现实的物理假设下进行研究。这尤其涉及到考虑三个空间维度的相对论流体，有涡度，没有对称性假设。这样的待遇不仅对申请至关重要，而且还涉及大量丰富的数学知识。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。