Analytic Problems in the Physics of Fluids, Gravitation and Conformal Geometry

该奖项将解决有关物理理论的数学基础的重要问题：流体动力学中欧拉方程的解；相对论耗散流体的柯西问题；广义相对论中的彭罗斯不等式；弦理论中的有效势。它还解决了具有非脐带边界的流形上的Yamabe问题的紧凑性问题。理解自由边界欧拉方程的解在固定域内收敛于标准欧拉方程的解，将为应用科学中使用的几种近似方案提供数学依据。它也可能为如何改进此类计划提供有用的提示。天体物理学和宇宙学在研究相对论性粘性流体方面取得了重要进展。因此，对描述这些情况的柯西问题给出一个恰当的数学处理是至关重要的。彭罗斯不等式是引力物理学中一个长期存在的开放性问题。在不同的情况下证明它是建立宇宙审查猜想的重要一步，而宇宙审查猜想反过来又可以被视为对广义相对论一致性的检验。有效势是弦理论中最有希望构建现实模型的方法之一。最后，对非脐形流形上Yamabe问题解的紧性的研究是对已有结果的重要推广。所有这些问题都是PI和合作者之前所做工作的延伸。更广泛的影响：本项目中描述的所有问题肯定会导致物理和数学之间的新互动，以及新技术的发展，这些新技术无疑会应用于物理、分析和几何中的其他问题。研究相对论耗散流体中的柯西问题的技术可能适用于偏微分方程领域的其他问题。对于具有非脐带边界的流形，Yamabe问题解的紧性证明将需要一个类似于脐带张量的Weyl消失定理。PI和M. A. Khuri最近给出的带电彭罗斯不等式的证明依赖于引入一个新的准局部质量，该质量是为带电初始数据集量身定制的。它的推广可能会为广义相对论中更广泛的质量问题带来新的见解。在过去的几十年里，几何与弦理论之间进行了卓有成效的交流，有效势的研究必将为这种相互作用提供新的途径。这个项目的想法也将通过主题课程、研讨会等方式传播给PI直接专业领域以外的受众。最后，我们希望这个项目的一些成果最终将有助于建立一个科学教育程度更高的社会。在大型强子对撞机的时代，流行书籍和电视节目向公众展示了黑洞、巨星和额外维度等概念，跟踪我们物理理论的数学和逻辑可靠性可以帮助公民决定哪些是已经确立的科学，哪些是尚未达到学术严格标准的想法。

This award will address important questions regarding the mathematical foundations of physical theories: solutions to the Euler equations in Fluid Dynamics; the Cauchy problem for relativistic dissipative fluids; the Penrose inequality in General Relativity; and effective potentials in String Theory. It also addresses the issue of compactness of solutions to the Yamabe problem on manifolds with non-umbilic boundary. Understanding the convergence of solutions of the free boundary Euler equations to solutions of the standard Euler equations in a fixed domain will provide mathematical justification to several approximating schemes used in the Applied Sciences. It may also give useful hints on how to improve such schemes. There have been important developments in Astrophysics and Cosmology which deal with relativistic viscous fluids. It is therefore paramount to give a proper mathematical treatment of the Cauchy problem describing these situations. The Penrose inequality is a longstanding open problem in the physics of gravitation. Proving it in different situations is an important step towards establishing the Cosmic Censorship Conjecture, which in turn can be viewed as a test for the consistency of General Relativity. Effective potentials are among the most promising approaches to construct realistic models in String Theory. Finally, the study of compactness of solutions to the Yamabe problem on manifolds with non-umbilic boundary is an important extension of the results known so far. All these problems are extensions of previous work done by the PI and collaborators. Broader impact: All problems described in this project will certainly lead to new interactions between Physics and Mathematics, as well as the development of new techniques which will undoubtedly have applications to other problems in Physics, Analysis and Geometry. The techniques developed to study the Cauchy problem in relativistic dissipative fluids will likely be applicable to other problems in the field of Partial Differential Equations. The proof of compactness of solutions to the Yamabe problem for manifolds with non-umbilic boundary will require an analogue of the Weyl Vanishing Theorem for the umbilicity tensor. A recent proof of the charged Penrose inequality given by the PI and M. A. Khuri relies on the introduction of a new quasi-local mass tailored to electrically charged initial data sets. Its generalization can potentially bring new insights to the broader issue of mass in General Relativity. The last decades have seen an extremely fruitful exchange between Geometry and String Theory, and the study of effective potentials is certain to provide new avenues for this interaction. The ideas of this project will also reach an audience outside the PI immediate field of expertise through their dissemination via topics courses, seminars etc. Finally, one hopes that some of the outcomes of this project will eventually help the task of building an ever more scientifically educated society. In the age of the Large Hadron Collider, where popular books and TV shows present the general public with concepts like black holes, giant stars and extra dimensions, keeping track of the mathematical and logical solidity of our physical theories can help citizens to decide what to take as well-established science versus ideas which have yet to meet the standards of academic rigor.