Last Date: Sunday December 15, 2024
Dr Tobias Barker, Prof Manuel Del Pino
Competition Funded PhD Project (Students Worldwide)
University of Bath Department of Mathematical Sciences
About the Project
This project is one of a number that are in competition for funding from the University of Bath URSA competition, for entry in September 2025.
Overview of the Research:
This project is centred on the rigorous analysis of partial differential equations (PDEs) concerning fluids, with a particular emphasis on the three-dimensional incompressible Navier-Stokes equations (3D NSE). Despite being widely used, it is not clear if the mathematical properties of these equations accurately reflect certain ubiquitous physical phenomena. One of the most challenging mathematics questions of our time asks if solutions to the 3D NSE remain smooth for all times or develop a singularity. Resolving this is a Millenium Prize Problem, with a reward of 1 million dollars.
Recent numerical evidence [Hou23] suggests that the 3D NSE may form a singularity. One direction is to obtain a detailed mathematical understanding of potentially singular solutions in the setting explored in [Hou23]. Though the Millenium Prize Problem concerns the initial value problem of 3D NSE, certain singularity scenarios (with certain symmetry) that have been ruled out in this case remain potentially viable in the presence of a boundary with no-slip boundary conditions (see the introduction of [BP20]). An additional direction is to further investigate regularity properties of solutions to the 3D NSE in this setting with a boundary. Furthermore, it may also be possible for the candidate to interact with projects undertaken in the EPSRC grant “APP11616: Dynamics and regularity criteria for nonlinear incompressible partial differential equations”.
Progress in these projects will rely upon ideas from Functional Analysis, Harmonic Analysis and the Analysis of PDEs. Advanced training in these areas is offered through courses and seminars such as the Taught Course Centre and the weekly Analysis Seminar.
There is also the possibility of participating in the activities of the doctoral training centre in ‘Statistical and Applied Mathematics in Bath’ (SAMBa), whose remit includes Applied Analysis.
The Department of Mathematical Sciences at the University of Bath is an ideal environment to pursue such a project. It is home not only to experts in PDEs, but also to experts in pure and applied aspects of fluid mechanics.
Project keywords: partial differential equations, Navier-Stokes equations, singular set, boundary regularity, boundary effects
Candidate Requirements:
Applicants should hold, or expect to receive, a First Class or good Upper Second Class UK Honours degree (or the equivalent) in a relevant subject. A master’s level qualification would also be advantageous. The main requirements for a successful application are a sound background in Analysis and the rigorous theory of Partial Differential Equations, a high level of motivation and an ability to think and work independently.
Non-UK applicants must meet the programme’s English language requirement by the application deadline.
Enquiries and Applications:
Informal enquiries are encouraged and should be directed to Dr Tobias Barker, tb2130@bath.ac.uk.
Formal applications should be submitted via the University of Bath’s online application form for a PhD in Mathematics prior to the closing date of this advert.
IMPORTANT:
When completing the application form:
1. In the Funding your studies section, select ‘University of Bath URSA’ as the studentship for which you are applying.
2. In the Your PhD project section, quote the project title of this project and the name of the lead supervisor in the appropriate boxes.
Failure to complete these two steps will cause delays in processing your application and may cause you to miss the deadline.
More information about applying for a PhD at Bath may be found on our website.
Equality, Diversity and Inclusion:
We value a diverse research environment and aim to be an inclusive university, where difference is celebrated and respected. We welcome and encourage applications from under-represented groups.
If you have circumstances that you feel we should be aware of that have affected your educational attainment, then please feel free to tell us about it in your application form. The best way to do this is a short paragraph at the end of your personal statement.
Funding Notes
Candidates may be considered for a University of Bath studentship tenable for 3.5 years. Funding covers tuition fees, a stipend (£19,237 p/a in 2024/5) and access to a training support budget.
References
[BP20] Scale-Invariant Estimates and Vorticity Alignment for Navier–Stokes in the Half-Space with No-Slip Boundary Conditions (with Christophe Prange). Arch. Ration. Mech. Anal., 235(2), 881-926 (2020)
[Hou23] Hou, Thomas Y. “Potentially singular behavior of the 3D Navier–Stokes equations.” Foundations of Computational Mathematics 23.6 (2023): 2251-2299.
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