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Details of Grant 

EPSRC Reference: EP/S02218X/1
Title: Singularities and mixing in Euler flows
Principal Investigator: Hadzic, Dr M
Other Investigators:
Researcher Co-Investigators:
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Department: Mathematics
Organisation: UCL
Scheme: EPSRC Fellowship
Starts: 01 April 2019 Ends: 31 March 2024 Value (£): 957,306
EPSRC Research Topic Classifications:
Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
23 Jan 2019 EPSRC Mathematical Sciences Fellowship Interviews January 2019 Announced
28 Nov 2018 EPSRC Mathematical Sciences Prioritisation Panel November 2018 Announced
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 "scale-independent", a notion aptly termed self-similarity. Streams A and B of the proposed research focus precisely on finding and studying such (approximately) self-similar 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 2-D (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 long-time behaviour of solutions to the 2-D Euler equation.

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