Well-posedness and stability for relativistic Euler equations with free boundaries

A rigorous mathematical description of a star requires a coupling between two famous systems of partial differential equations: the Euler equations of fluid mechanics and the Einstein equations of general relativity. These systems possess a rich mathematical structure and describe fundamental physical processes, playing an important role in both mathematics and physics. Their coupling gives the so-called Euler-Einstein system, a fundamental model in the analysis of fluid bodies coupled to gravity. One of the most important examples are the stars, idealised as fluid or gas clouds with a moving boundary interface separating them from the vacuum.

The first and basic mathematical question that one can ask about the free boundary Euler-Einstein system is the following: can one develop a rigorous mathematical framework that establishes the existence and uniqueness of solutions to this system given some initial configuration of the star? Can we similarly track down the beahviour and the regularity of the moving vacuum boundary? How do the changes in initial configurations affect the solutions? Such questions are technically termed as problems of well-posedness, and the principal aim of the proposal is to develop a rigorous well-posedness framework for the moving vacuum boundary Euler-Einstein system.

Mathematically, this problem intertwines various difficulties associated with both the free boundary fluids and the Einstein equations. Due to its highly nonlinear nature, it is a priori unclear whether the free vacuum boundary can cause a degeneracy in the model, leading to a potential breakdown of the solutions, even after a very short time. Even in the Newtonian setting, the degenerate nature of the problem was hinted at by John Von Neumann as early as 1949. However, the past few decades have seen striking developments in the rigorous study of the Newtonian free boundary Euler equations on one hand, and in the mathematical general relativity on the other. A satisfactory well-posedness theory for the Newtonian free boundary compressible fluids has been developed in the past 3 years. Similarly, a rigorous mathematical study of relativistic fluids is a rich and broad topic, that has generated a lot of mathematical research over the past decade. As an example, a momentous breakthrough in the study of stable shock formation for relativistic fluids was accomplished by Christodoulou in 2007.

While such works provide an important impetus for this proposal, the complicated, but beautiful interaction between the free boundary geometry and the relativistic geometry, gives rise to new mathematical structures and additional challenges with respect to the existing literature. The proposal explores these structures in detail and develops novel ideas to show the well-posedness of 1) the free vacuum boundary Euler equations on the Minkowski spacetime and 2) the free vacuum boundary Euler-Einstein system. It then uses the thus established framework to address the stability of the well-known Friedmann-Lemaitre- Robertson-Walker solutions, describing an accelerating expanding universe.

Besides the academic impact that has been described in the "Academic Beneficiaries" section, this multifaceted proposal connects to the broader public in several important ways.

From its conception in the 18th century, the Euler equations of fluid mechanics have represented one of the focal points in the study of partial differential equations. They led to the study of a variety of important phenomena such as the turbulence, formation of shocks, boundary layers etc. Another beautiful theory, the general theory of relativity, offered us a mathematical tool to study phenomena on large scales under the influence of gravity. Not long after its formulation in 1915, equations for fluids interacting with gravity as described by Einstein were formulated and led to an important system of equations that we call the Euler-Einstein system. It embodies the complexities of both the Euler equation and the Einstein equations individually but at the same time their coupling gives rise to exciting new mathematical phenomena. When the fluid that we are describing is contained in a finite bounded region, which is allowed to change its shape in the course of time, then this system becomes a basic mathematical idealisation of a star.

One of my aims is to explain some of the intricate mathematical structures underpinning the free vacuum boundary Euler equations when relativistic effects are present. Notions such as black holes, white dwarves, gravitational collapse etc. easily capture the imagination of the public, yet there exists a significant gap in understanding the mathematical mechanisms behind such phenomena. Many of the necessary tools come from the field of nonlinear PDEs and geometric analysis and my aim is to take steps in bringing some of these ideas closer to the general nonexpert audiences. I will therefore give presentations on the role of mathematical analysis in our description of stars through the outreach programs at King's. Most importantly, I will speak at conferences and events designed for the UK secondary school maths teachers. I will separately give talks to secondary school students with the aim of popularising mathematical analysis and motivating some of the basic analytical tools in a clear and engaging way. I will also present the outlines of the proposed research at international summer schools for undergraduate students with encouraged participation of the UK maths undergraduates. This is part of my long term goal to foster the development of nonlinear mathematical analysis in the UK. I also plan to create lecture notes that will come out of this series of lectures, as a stepping stone for my future supervision of Masters and Ph.D students.

The UK has a longstanding tradition of world leading research in the field of mathematical analysis. Nevertheless, the relative amount of research in the field of nonlinear partial differential equations is small in comparison to the USA and Europe. This proposal will help shrink that gap and contribute to the development of the field in the UK. In line with this expectation, I plan to organise a workshop at King's College London that will bring world leading experts in nonlinear PDEs together. Special resources will be allocated to encourage the participation of Ph.D.-students and junior researchers from the UK.