# David McCormick

I completed my PhD in the MASDOC Centre for Doctoral Training, having completed my MSc (with Distinction) under MASDOC in 2010/11. I previously studied my undergraduate degree at Warwick from 2005-2009, graduating with a Master of Mathematics (MMath), before taking a gap year in 2009/10. You can read about my gap year exploits and more recent travels on my blog here.

**Email: **D.S.McCormick *-obvious symbol-* warwick.ac.uk

**Office:** B3.04, desk 17 (2012-14) ~~B3.04, desk 5 (2011/12)~~ ~~D2.08, desk 9 (2010/11)~~

## Current Research

My research interests lie in the theory of PDEs, particularly in the equations of magnetohydrodynamics, and the related Navier--Stokes and Euler equations. I am currently investigating the behaviour (including existence, uniqueness and regularity) of the non-resistive magnetohydrodynamics (MHD) equations, with a view to extending the work of my MSc dissertation to a more rigorous setting in two dimensions.

## Papers and Preprints

C.L. Fefferman, D.S. McCormick, J.C. Robinson, J.L. Rodrigo (2014), *Higher order commutator estimates and local existence for the non-resistive MHD equations and related models*. Journal of Functional Analysis **267**(4):1035-1056. Available online here. Preprint: arXiv:1401.5018 [math.AP]

D.S. McCormick, J.C. Robinson, J.L. Rodrigo (2014), *Existence and uniqueness for a coupled parabolic-elliptic model with applications to magnetic relaxation. *Archive for Rational Mechanics and Analysis **214**(2):503-523. Available online here. Preprint: arXiv:1303.6352 [math.AP]

D.S. McCormick, J.C. Robinson, J.L. Rodrigo (2013), *Generalised Gagliardo-Nirenberg inequalities using weak Lebesgue spaces and BMO*. Milan Journal of Mathematics **81**(2):265-289. Available online here. Preprint: arXiv:1303.6351 [math.AP]

C.E.A. Brett, K.F. Lam, K.J.H. Law, D.S. McCormick, M.R. Scott, A.M. Stuart (2013), *Accuracy and stability of filters for dissipative PDEs*. Physica D **245** (2013), 34-45. Preprint: arXiv:1203.5845 [math.OC]; old version: arXiv:1110.2527 [math.OC].

## Talks and Posters

I have given a number of talks and poster presentations. The title should link to the slides or poster in each case.

- "Fractional derivatives, functional analysis and fluid dynamics", Postgraduate Seminar, June 4th 2014 (chalk-and-talk, so no slides).
- "A generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem", Applied PDE seminar, University of Warwick, February 26th 2013.
- "The Navier-Stokes Equations: what's the problem (and why do we care)?", Postgraduate Seminar, January 23rd 2013 (chalk-and-talk, so no slides).
- "A generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem", contributed talk at SWPDE Winter School, University of Bath, January 3rd 2013.
- "(Magnetic) Fluid Dynamics from a PDE Theoretical Point of View", Postgraduate Seminar, University of Warwick, May 9th 2012.
- "Stationary Euler flows and ideal magnetohydrodynamics", contributed talk at Young Researchers in Mathematics 2012, University of Bristol, April 3rd 2012.

## MSc Dissertation (2011)

In my MSc dissertation, I studied the relationship between the long-term behaviour of the Euler equations and the magnetohydrodynamics equations, working with James Robinson (as my supervisor) and José Rodrigo (as second supervisor / mentor).

* Abstract: *In this dissertation, we study stationary solutions of the Euler equations through the approach of magnetic relaxation, where a viscous, non-resistive, incompressible fluid is evolved under the laws of magnetohydrodynamics to a state of equilibrium, in which the limiting magnetic field ought to satisfy the stationary Euler equations. After introducing the "analogy" - first proposed by Moffatt (1985) - between stationary Euler flows and magnetostatics, we expound the rigorous analysis of Núñez (2007) in showing that, while the kinetic energy decays to zero, the magnetic field may fail to have a weak limit, unless an additional condition on the velocity holds. We also introduce a reduced model, suggested by Moffatt (2009), in which some of the terms in the equations are neglected, and present an original approach to prove that the kinetic energy also decays to zero for this model; however, we also show that the magnetic field may still fail to have a weak limit.

- Dissertation (PDF, 650KB)

## Research Study Group (2010/11)

The Research Study Group is an innovative group project forming about one sixth of the MASDOC MSc year. Students work together in groups in two phases:

- In phase 1 (November-February), the groups formulate a research proposal in an area of interest in applications to physics, chemistry, biology, engineering, and so on.
- In phase 2 (February-May), the groups implement the solution to one of the
*other groups'*research proposals.

More information is available here (Warwick login required).

### Phase 1: Biomembranes

During phase 1, I worked with Charles Brett, Michael Eyers and Michael Scott on formulating a research proposal on Biomembranes, working with Charlie Elliott and Björn Stinner.

- Research proposal (PDF, 533KB)

This proposal was then implemented by Don Praveen Amarasinghe, Andrew Aylwin, Pravin Madhavan and Christopher Pettitt, and you may read more about their research on their webpages.

### Phase 2: Data Assimilation

During phase 2, I worked with Charles Brett, Andrew Lam and Michael Scott on Data Assimilation for the 2D Navier--Stokes equations. We worked under the guidance of Andrew Stuart, with much help from Kody Law and Masoumeh Dashti, based (loosely) on the proposal written by Andrew Aylwin, Adam Hall, Christopher Pettitt and Maria Veretennikova.

- Final report (PDF, 1.2MB)
- Poster (PDF, 1.5MB)

We presented our poster at the Workshop on Inverse Problems and Optimal Control for PDEs at the University of Warwick, 23rd-27th May 2011. The research resulted in a paper, *Accuracy and stability of filters for dissipative PDEs*, published in Physica D (see above).

## Teaching

In 2013/14 I lectured MA4J1 Continuum Mechanics.

## Notes for courses

Over the years I have typed up notes for a number of courses: notes for some of them will be made available below. In addition, I wrote a number of revision guides for the Warwick Mathematics Society which are available at their website if you are a member of the society.

### MA4J0 Advanced Real Analysis (autumn 2010, lectured by José Luis Rodrigo)

The notes are still in preparation; however, a preliminary draft is available:

Version 0.1 (compiled January 17th, 2011): PDF -- please note that this is only complete up to lecture 23.