EPSRC Reference: 
EP/N016777/1 
Title: 
Wellposedness and stability for relativistic Euler equations with free boundaries 
Principal Investigator: 
Hadzic, Dr M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
Kings College London 
Scheme: 
First Grant  Revised 2009 
Starts: 
01 March 2016 
Ends: 
28 February 2018 
Value (£): 
93,937

EPSRC Research Topic Classifications: 
Mathematical Analysis 
Mathematical Physics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
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 socalled EulerEinstein 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 EulerEinstein 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 wellposedness, and the principal aim of the proposal is to develop a rigorous wellposedness framework for the moving vacuum boundary EulerEinstein 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 wellposedness 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 wellposedness of 1) the free vacuum boundary Euler equations on the Minkowski spacetime and 2) the free vacuum boundary EulerEinstein system. It then uses the thus established framework to address the stability of the wellknown FriedmannLemaitre RobertsonWalker solutions, describing an accelerating expanding universe.

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