All seminars (unless otherwise stated) will take place on Mondays at 3.00 pm in Room 505 which is located on the 5th floor of the Mathematics Department. See Where to Find Us for further details. There will be tea afterwards in room 606.

If you require any more information on the Applied seminars please contact Professor Yaroslav Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

## 09 January 2012

Yves Capdeboscq - University of Oxford

##### Regularity Estimates in High Conductivity Homogenization

Abstract:

In a recent work with Marc Briane and Luc Nguyen, we considered the case of a periodic micro-structure with highly conducting fibres (i.e. metal rods). Fenchenko and Khruslov showed 30 years ago that for a particular scaling range, the effective problem includes a non-local term. We show that

- From a homogenization corrector result, one can deduce a lower bound on all norms W^{1,p}_{loc} of the solution for p>2, and this bound blow-ups like exp(C/eps^2), a given power of the inverse of the radius of the rods.
- This is not a surface effect : the blow-up occurs also outside the fibres.
- Everywhere but at a distance less than eps^{1+\delta} from the fibres, the solution is uniformly C^{1\alpha} smooth. The measure of the forbidden domain tends to zero with a given rate in epsilon.

I will then discuss the interpretation of this result for two applications, a enhanced resolution in imaging, and meta-materials.

## 16 January 2012

Sofia Olhede - UCL Department of Statistics

##### TTheory and Estimation of Modulated Oscillations

Abstract:

Modulated oscillations are ubiquitous in applications, ranging from econometric indices to acoustic phenomena. Of particular interest is the observation using multiple sensors or measurement devices, where each device is observing parts of the same oscillation. We mention bivariate records of freely-drifting instrument position in oceanography, trivariate records of acceleration in seismology, neuroscience measurements such as EEG or MEG, bloodflow and the generic application of acoustic transducers, as examples. While the theory of the single modulated oscillation was well-developed already in the 1980s, the theory of the multivariate modulated oscillation has been proposed over the last few years. I will discuss the problem of defining a consistent theory for such objects and inherent issues in their estimation, using the analytic signal, or other methods developed for oscillatory signals. I will illustrate aspects of the theory with examples from float and drifter data (oceanography) and Electroencephalography (EEG) measurements of infants subjected to noxious stimuli.

This is joint work with Jonathan Lilly (NWRA, Seattle), Shane Elipot (POL) and Maria Fitzgerald (UCL).

## 23 January 2012

Jamal Uddin - University of Birmingham

##### Breakup of Spiralling Liquid Jets

Abstract:

The industrial prilling process is amongst the most favourite technique employed in generating monodisperse droplets. In such a process long curved jets are generated from a rotating drum which in turn breakup and from droplets. In this talk we describe the experimental set-up and the theory to model this process. We will consider the effects of changing the rheology of the fluid as well as the addition of surface agents to modify breakup characteristics. Both temporal and spatial instability will be considered as well as nonlinear numerical simulations with comparisons between experiments.

## 30 January 2012

Evgeny Lakshtanov - University of Aveiro (Portugal)

##### Scattering by obstacles

Abstract:

The first part of my talk concerns the case of the Dirichlet or Neumann boundary conditions. I will present several recent results concerning high frequency regime. In particular, we will discuss a justification of approximations of solutions in the case of obstacles with Lipshitz boundaries or/and trapping obstacles.

The second part of my talk is devoted to the Transmission Scattering Problem, namely to, so called Interior Transmission Eigenvalues. Such eigenvalues play the same role for the transmission scattering problem as the eigenvalues of the Dirichlet or Neumann Laplacian play for the scattering by an obstacle with the corresponding (Dirichlet or Neumann) boundary conditions. I will provide some recent results on this topic including Weyl formula and construction of the branching billiard which allows to construct quasimodes.

This talk is based on joined results with B.Sleeman and B.Vainberg.

## 06 February 2012

Adrian Constantin - King's College London/ University of Vienna

##### Analyticity of the streamlines for periodic traveling free surface water waves with vorticity

Abstract:

The streamlines of periodic irrotational traveling water waves are known to be real-analytic, with exception of the free surface in the case the wave of greatest height which has a corner at the wave crest (the lateral tangents being at an angle of 2\pi/3). The regularity of waves of small and moderate amplitude is, perhaps surprisingly, little affected by the presence of vorticity in the flow (in the absence of stagnation points). This is joint work with J. Escher (Hannover University).

## 13 February 2012

READING WEEK - NO SEMINAR

## 20 February 2012

Alexey Slunyaev - Keele University, Staffordshire, UK/Institute of Applied Physics, Nizhny Novgorod, Russia

##### Physical mechanisms and simulations of rogue waves in the ocean

Abstract:

A brief introduction to the actual state of the rogue (freak) wave research will be given. The own recent results will be presented highlighting the abilities and lack of abilities of approximate nonlinear theories to describe and predict these dangerous waves.

## 27 February 2012

Charles R. Doering - University of Michigan, USATBA

##### Ultimate state of two-dimensional Rayleigh-B\'enard convection

Abstract:

Rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-B\'enard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number is bounded in terms of the Rayleigh number, uniformly in the Prandtl number. The asymptotic high Rayleigh number scaling of the upper bound challenges some theoretical arguments regarding heat transport by turbulent convection. (Joint work with Jared Whitehead.).

## 12 March 2012

S. Hill - Buckingham University

##### Using computers to investigate problems in differential geometry

Abstract:

I will give a non-technical overview of the problem of finding `special' Riemannian metrics ( e.g. Einstein metrics, Ricci solitons, constant scalar curvature metrics...). In general, solving these equations involves finding solutions to non-linear PDEs. I will report on some methods of investigating these equations with numerical methods.

Some of this is joint work with Andrew Dancer and McKenzie Wang and also with Robert Haselhofer and Michael Seipmann.

## 19 March 2012

M. Heil - University of Manchester

##### The stability of the liquid lining in fluid-conveying curved tubes

Abstract:

Motivated by an interest in the fluid mechanical behaviour of the liquid film that lines the pulmonary airways, we demonstrate that the surface-tension-driven migration of fluid towards the outer wall of a curved vessel can be opposed by the azimuthal shear stresses generated by the Dean-like secondary flows that develop when air is driven along the vessel. Assuming the pressure-driven flow along the curved vessel to be fully developed, we employ a combination of numerical and asymptotic approaches to demonstrate that the competition between the two effects allows the existence of steady solutions in which the liquid lining has finite thickness along the entire perimeter of the vessel. We study the stability of these steady solutions and contrast the predictions obtained from a thin-film model that allows a detailed analysis of the system's bifurcation structure to the results of full numerical simulations based on the free-surface Navier-Stokes equations.

This is joint work with Andrew Hazel (Manchester), Sarah Waters and James Oliver (both at Oxford).