EPSRC Reference: 
EP/G007470/1 
Title: 
The NavierStokes equations: functional analysis and dynamical systems 
Principal Investigator: 
Robinson, Professor JC 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Warwick 
Scheme: 
Leadership Fellowships 
Starts: 
26 November 2008 
Ends: 
30 September 2014 
Value (£): 
999,366

EPSRC Research Topic Classifications: 
Mathematical Analysis 
Nonlinear Systems Mathematics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
26 Jun 2008

Fellowship Allocation Panel Meeting

Announced

10 Jun 2008

Fellowships 2008 Interviews  Panel B

Deferred


Summary on Grant Application Form 
The NavierStokes equations are well established as the mathematical model for the flow of fluids. But while they are used extensively in both theoretical and computational analyses of every aspect of fluid flow, their mathematical foundations are still uncertain.In the year 2000, the Clay Mathematics Institute announced a list of Seven Millennium problems, solutions for each of which will attract a prize of one million dollars. Included in this list are 'classic problems' such as the Riemann Hypothesis and the Poincar conjecture (now solved by the work of Perelman); but here one can also find the question of the existence (or otherwise) of unique solutions for the threedimensional NavierStokes equations.The point of a mathematical model is that it enables prediction: if you know what happens at an initial time, you can predict what will happen in the future. However, being able to make a 'prediction' relies on the model having only one solution: two (or more) solutions starting from the same initial setup make prediction a matter of divination rather than science.This is the 'uniqueness problem' (which can be formulated precisely given the correct mathematical language) that remains unresolved for the threedimensional NavierStokes equations: although used routinely, there is no mathematical proof that they have any predictive power. Part of this proposal focuses on questions related to this fundamental difficulty, which is a fault line running through mathematical fluid dynamics. The formation of a 'singularity' is the process by which predictive power can be lost, and this project will consider how one can limit the formation of these singularities (should they actually occur). Related to this is the question of how the NavierStokes equations relate to the Euler equations, an older and some sense simpler model that neglects the effect of viscosity.The other half of the proposal considers questions that arise when one considers the twodimensional NavierStokes equations. The twodimensional model has less physical relevance, but does not suffer from the fundamental problems that bedevil its threedimensional counterpart: this makes it a useful testbed for techniques that could eventually be applied in the threedimensional case.The theory of dynamical systems (of which 'chaos theory' forms a part) can be applied to the twodimensional equations. In this context, it is possible to show that the equations have an attractor that is finitedimensional. In a very loose way this says that 'what happens in the long run should be relatively easy to describe'; in the language of physics one might express this as 'fullydeveloped twodimensional turbulence has a finite number of degrees of freedom'.Giving a rigorous (and mathematically concrete) interpretation of this idea forms the other half of this proposal.

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Summary 

Date Materialised 


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Project URL: 

Further Information: 

Organisation Website: 
http://www.warwick.ac.uk 