| EPSRC Reference: |
EP/G007470/1 |
| Title: |
The Navier-Stokes equations: functional analysis and dynamical systems |
| Principal Investigator: |
Robinson, Professor JC |
| Other Investigators: |
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| Researcher Co-Investigators: |
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| Project Partners: |
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| Department: |
Mathematics |
| Organisation: |
University of Warwick |
| Scheme: |
Leadership Fellowships |
| Starts: |
26 November 2008 |
Ends: |
30 September 2014 |
Value (£): |
999,366
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| EPSRC Research Topic Classifications: |
| Mathematical Analysis |
Non-linear Systems Mathematics |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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| Related Grants: |
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| Panel History: |
| Panel Date | Panel Name | Outcome |
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26 Jun 2008
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Fellowship Allocation Panel Meeting
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Announced
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10 Jun 2008
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Fellowships 2008 Interviews - Panel B
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Deferred
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Summary on Grant Application Form |
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The Navier-Stokes 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 three-dimensional Navier-Stokes 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 three-dimensional Navier-Stokes 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 Navier-Stokes 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 two-dimensional Navier-Stokes equations. The two-dimensional model has less physical relevance, but does not suffer from the fundamental problems that bedevil its three-dimensional counterpart: this makes it a useful testbed for techniques that could eventually be applied in the three-dimensional case.The theory of dynamical systems (of which 'chaos theory' forms a part) can be applied to the two-dimensional equations. In this context, it is possible to show that the equations have an attractor that is finite-dimensional. 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 'fully-developed two-dimensional 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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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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| Date Materialised |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.warwick.ac.uk |