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

Vanderbilt University, Nashville TN

Investigators

Abstract

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.