EPSRC Reference: 
EP/S02218X/1 
Title: 
Singularities and mixing in Euler flows 
Principal Investigator: 
Hadzic, Dr M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
UCL 
Scheme: 
EPSRC Fellowship 
Starts: 
01 April 2019 
Ends: 
31 March 2024 
Value (£): 
957,306

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


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Summary on Grant Application Form 
One of the oldest and most intriguing puzzles of modern science is to unravel and understand the true nature of instabilities and turbulence that we associate with the behaviour of fluids. Thereby the word fluid can refer to a mixing between between water and oil in a cup, or much more violent processes such as the collapse of a star.
A common mathematical thread between these two seemingly unrelated physical situations is the famous Euler equation, written down by Leonhard Euler as early as 1755. This is among the earliest examples of partial differential equations, that to this day, remains a focal point of mathematical research.
The proposal aims to rigorously examine solutions of Euler equations that describe singular processes within which a moving fluid region can shrink to a point, expand to infinity, or exhibit some other form of, what mathematicians like to call, a topological singularity.
Even though we want to describe physical processes that happen at vastly different spatial scales, there is a precise mathematical sense in which the solutions of the Euler equation can be "scaleindependent", a notion aptly termed selfsimilarity. Streams A and B of the proposed research focus precisely on finding and studying such (approximately) selfsimilar solutions of Euler equation as they are mathematical building blocks for the singularities mentioned above. In stream A we focus on the celebrated problem of describing the gravitational collapse of a shrinking star, while in stream B we examine the nature of collapsing cavities surrounded by a fluid. In both cases, surprising links between mathematical analysis, geometry, and physics play a key role.
In stream C we turn our attention to 2D (or planar) fluid flows, with the aim of deepening our knowledge of mixing properties of fluids. By stirring milk into the coffee, one may wonder how far must we zoom into the mixture before we see the milk particles separated from the coffee particles. This simple question, it turns out, motivates a rigorous geometric definition of mixing, which allows mathematicians to formulate conjectures and prove theorems. Our focus here is on a famous problem known as the Bressan's mixing conjecture and a link to the longtime behaviour of solutions to the 2D Euler equation.

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