- Room No. :
**CM T.21** -
Address :
**Mathematics Department**

Heriot-Watt University

Edinburgh EH14 4AS

Scotland - Telephone No. :
**+44 131 451 3254** - Fax No. :
**+44 131 451 3249** - E-Mail Address :
**S.J.A.Malham [at] hw.ac.uk**

**The Navier-Stokes equations and turbulent flows.**Do solutions of the Navier-Stokes equations develop a singularity in finite time? Since Leray suggested these singular solutions may signify the onset of turbulence in a*real*fluid flow, it is also an interesting engineering issue. This remains an important open problem and is currently one of the millenium prize problems set by the Clay Institute and worth a million dollars (see the criteria devised by Charles Fefferman). Also see the**LMS-EPSRC Short Course: Theoretical Fluid Dynamics**held at Heriot-Watt University: August 29th - September 2nd 2011.**Reaction-diffusion phenomena:**Travelling waves; Autocatalysis; Combustion.**Computational spectral theory:**Evans function; Geometric methods; Grassmannian/Riccati methods; Multi-dimensional stability.**Stochastic differential equations:**Lie group integration; Efficient simulation; Algebraic structure; Heston model.**Nonlinear partial differential equations:**Fredholm Grassmannian flows, linearisation and solution methods; Smoluchowksi coagulation.

If you would like to know more or are interested in doing a PhD, email me (see address above).

- Bartuccelli, M.V., Gibbon, J.D. and Malham, S.J.A. 1993,
Lattice methods and the pressure field for solutions of the Navier-Stokes equations
(pdf),
*Nonlinearity*6(1), pp. 79-92. - Bartuccelli, M.V., Doering, C.R., Gibbon, J.D. and Malham, S.J.A. 1993,
Length scales in solutions of the Navier-Stokes equations
(pdf),
*Nonlinearity*6(4), pp. 549-568. - Malham, S. and Xin, J. 1998,
Global solutions to a reactive Boussinesq system with front
data on an infinite domain
(pdf),
*Comm. Math. Phys.*193(2), pp. 287-316. - Balmforth, N.J., Craster, R.V.
and Malham, S.J.A. 1999, Unsteady fronts in an autocatalytic system
(pdf),
*Proc. R. Soc. Lond.*A 455, pp. 1401-1433. - Malham, S.J.A. and Oliver, M. 2000, Accelerating fronts in autocatalysis
(pdf)
,
*Proc. R. Soc. Lond.*A 456, pp. 1609-1624. - Ford, R., Malham, S.J.A. and Oliver, M. 2002, A new model
for shallow water in the low Rossby-number limit
(pdf)
,
*JFM*450, pp. 287-296. - Aparicio, N.D., Malham, S.J.A. and Oliver, M. 2005,
Numerical evaluation of the Evans function by Magnus integration
(pdf),
*BIT*45, pp. 219-258. - Lord, G., Malham, S.J.A. and Wiese, A.,
Efficient strong integrators for linear stochastic systems
(arxiv link),
*SIAM J. on Numerical Analysis*46(6) (2008), pp. 2892-2919. - Malham, S. and Niesen, J.,
Evaluating the Evans function: order reduction in numerical methods
(arxiv link),
*Math. Comp.*77 (2008), pp. 159-179. - Malham, S.J.A. and Wiese, A.,
Stochastic Lie group integrators
(arxiv link),
*SIAM J. Sci. Comput.*30 (2) (2008), pp. 597-617. - Ledoux, V., Malham, S.J.A. and Thummler, V.,
Grassmannian spectral shooting
(arxiv link),
*Math. Comp.*79 (2010), pp. 1585-1619. - Malham, S.J.A. and Wiese, A.,
Chi-square simulation of the CIR process and the Heston model
(arxiv link),
*International Journal of Theoretical and Applied Finance*16(3) (2013). - Ledoux, V., Malham, S.J.A., Niesen, J. and Thummler, V.,
Computing stability of multi-dimensional travelling waves
(arxiv link),
*SIAM J. on Applied Dynamical Systems*8(1) (2009), pp. 480-507. - Malham, S.J.A. and Wiese, A.,
Stochastic expansions and Hopf algebras
(arxiv link),
*Proc. R. Soc. A*465 (2009), pp. 3729-3749. doi:10.1098/rspa.2009.0203 - Malham, S.J.A. and Wiese, A.,
Efficient almost-exact Levy area sampling
(arxiv link),
*Statistics and Probability Letters*88 (2014), pp. 50-55. - Ebrahimi-Fard, K., Lundervold, A., Malham, S.J.A.,
Munthe-Kaas, H. and Wiese, A.,
Algebraic structure of stochastic expansions and efficient simulation
(arxiv link),
*Proc. R. Soc. A*468 (2012), pp. 2361-2382. - Curry, C., Ebrahimi-Fard, K., Malham, S.J.A. and Wiese, A.,
Levy processes and quasi-shuffle algebras
(arxiv link),
*Stochastics*86(4) (2014) pp. 632-642. - Beck, M. and Malham, S.J.A. 2014,
Computing the Maslov index for large systems
(arxiv link),
*Proc. Am. Math. Soc.*(doi.10.1090/S0002-9939-2014-12575-5). - Karambal, I. and Malham, S.J.A.,
Evans function and Fredholm determinants,
(arxiv link),
*Proc. R. Soc. A*:2015471 20140597;DOI: 10.1098/rspa.2014.0597. Published 14 January 2015. - Ebrahimi-Fard, K., Malham, S.J.A., Patras, F. and Wiese, A.,
Flows and stochastic Taylor series in Ito calculus,
(arxiv link),
*2015 J. Phys. A: Math. Theor. 48*: 495202 doi:10.1088/1751-8113/48/49/495202 - Ebrahimi-Fard, K., Malham, S.J.A., Patras, F. and Wiese, A.,
The exponential Lie series for continuous semimartingales,
(arxiv link),
*Proc. R. Soc. A*:2015 471 20150429; DOI: 10.1098/rspa.2015.0429. Published 16th December 2015. - Curry, C., Ebrahimi-Fard, K., Malham, S.J.A. and Wiese, A.,
Algebraic structures and stochastic differential equations driven by Levy processes,
(arxiv link),
*Proceedings of the Royal Society A*2018. - Beck, M., Doikou, A., Malham, S.J.A., and Stylianidis, I., 2019,
Grassmannian flows and applications to nonlinear partial differential equations,
(arxiv link),
*Proceedings of the Abel Symposium 2016*. - Beck, M., Doikou, A., Malham, S.J.A., and Stylianidis, I., 2018,
Partial differential systems with nonlocal nonlinearities: Generation and solutions
(arxiv link),
*Phil. Trans. R. Soc. A***376:**20170195. - Doikou, A., Malham, S.J.A., and Wiese, A., 2019, Stochastic analysis and discrete quantum systems, preprint (arxiv link).
- Doikou, A., Malham, S.J.A., Stylianidis, I. and Wiese A., 2019, Applications of Grassmannian and graph flows to nonlinear systems, preprint (arxiv link).
- Malham, S.J.A., Shen, J. and Wiese A., 2019, Series expansions and direct inversion for the Heston model, submitted.

- Malham, S.J.A., Regularity assumptions and length scales for the Navier-Stokes equations, PhD Thesis (July 1993), 129 pages.
- Malham, S.J.A., An introduction to asymptotic analysis, Lecture notes (Nov 2005), 56 pages.
- Malham, S.J.A., Differential equations and linear algebra, Lecture notes (March 2008), 95 pages.
- Ledoux, V. and Malham, S.J.A., Spectral shooting is Schubert calculus, Unpublished working manuscript (Nov 2009), 14 pages.
- Ledoux, V. and Malham, S.J.A., Introductory Schubert calculus, Review notes (v1: Jan 2010; v2: Sept 2010), 18 pages.
- Malham, S.J.A. and Wiese, A., An introduction to SDE simulation, Lecture notes (Feb 2010), 27 pages.
- Karambal, I., Ledoux, V., Malham, S.J.A. and Niesen, J., Introductory Fredholm theory and computation, Review notes (v1: Nov 2010; v2: Aug 2014), 16 pages.
- Malham, S.J.A., Fundamental functional analysis, Lecture notes (Aug 2011), 7 pages.
- Malham, S.J.A., A Navier-Stokes equations regularity tutorial and solutions, Tutorial notes (Sept 2011), 4 pages.
- Malham, S.J.A., Summary: Constantin and Fefferman's regularity result, Summary lecture notes (April 2015), 6 pages.
- Malham, S.J.A., An introduction to Lagrangian and Hamiltonian mechanics, Lecture notes (Aug 2016), 87 pages. (Complete detailed solutions available on request.)
- Malham, S.J.A., Introductory fluid mechanics, Lecture notes (Sept 2014), 74 pages. (Complete detailed solutions available on request.)

Last updated: 20th August 2019. S.J.A.Malham [at] hw.ac.uk