## Spring 2019

All seminars (unless otherwise stated) will take place on **Tuesdays at 3.00pm in ****Room B15 in the Darwin Building** - see the map for further details. There will be tea afterwards in **Maths Room 606** (6th Floor, 25 Gordon Street) - see map for further details. If you require any more information on the Applied seminars please contact Prof Jean-Marc Vanden-Broeck (e-mail: j.vanden-broeck AT ucl.ac.uk or tel: 020-7679-2835) or Prof Ilia Kamotski (e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937).

## 15 January 2019

### Speaker: Prof Arnaud Diego Münch (Université Clermont Auvergne; Laboratoire de mathématiques Blaise Pascal )

##### Title: About least-squares methods to solve direct and control problems

**Abstract: **

We introduce and analyze least-squares methods to solve direct and controllability problems associated to non linear PDEs, mainly the incompressible Navier-Stokes system. The methods are illustrated with some numerical experiments. This is a joint work with Jerome Lemoine (Université Clermont Auvergne, Clermont-Ferrand) and Pablo Pedregal (Universidad Castilla-La-Mancha, Ciudad Real).

## 22 January 2019

### Speaker: Dr Bin Cheng (University of Surrey)

##### Title: Analysis of nonlinear dynamics with three time scales

**Abstract: **

A PDE/ODE system can evolve in 3 time scales when the fast scales are associated with 2 small parameters that tend to zero at different rates. We investigate the limiting dynamics when the fast dynamics is generated by 2 skew-seft-adjoint operators and the initial time derivative is uniformly bounded regardless of the small parameters. To find the subspace that the limiting dynamics resides, we rely on matrix perturbation theory a la Kato (1982, Springer-Verlag).

## 29 January 2019

### Speaker: Dr Michael Dallaston (Coventry University)

##### Title: Singularities of the thin film equation

**Abstract: **

The lubrication approximation is commonly used in interfacial fluid dynamics to derive simplified evolution equations (PDEs) for the thickness of a liquid film. Many models fit into the framework of a canonical nonlinear equation, in which the thickness is stabilised by a fourth order term, and destabilised by a second order term, with coefficients as some power of the thickness. This equation exhibits a rich display of nonlinear phenomena, particularly related to the formation of singularities by either finite-time rupture (thickness going to zero at a point), or finite-time blow up (thickness going to infinity at a point). Such singularities can be examined through the assumption of self-similarity near the singularity.

We will discuss two aspects of singularity formation in different parameter regimes. First, the loss of stability of self-similar rupture solutions leading to the creation of discretely self-similar rupture profiles (resulting from periodic orbits in a scaled version of the problem); and secondly, details of the matched asymptotic analysis of blowup profiles above a critical line in parameter space.

## 05 February 2019

### Dept Colloquium Cancelled

## 12 February 2019

**NO SEMINAR - READING WEEK**

## 19 February 2019

### Speaker: Professor Tom Bridges (University of Surrey)

##### Title: Reappraisal of Whitham modulation theory

**Abstract: **

The first part of the talk will review the conventional version of Whitham modulation theory found in textbooks, including the conservation of waves, conservation of wave action, averaged Lagrangians, and the modulation equations themselves. The basic theory produces a pair of first-order nonlinear PDEs which are dispersionless and may be either hyperbolic or elliptic, and the transition between the two is the Lighthill instability transition. The second part of the talk will present a new approach to Whitham modulation theory, which recovers the classical theory, but extends to include dispersion in a natural and universal way. New scaling produces a heirachy of modulation equations. An example is the morphing of the conservation of wave action into a two-way Boussinesq equation due to coalescing characteristics at Lighthill points. The third part of the talk considers modulation of multiphase wavetrains, which were traditionally thought of as possible only in integrable systems. However Lagrangians invariant under the action of a Lie group, which are plentiful, generate robust multiphase wavetrains. Some of the generalisations of concepts from single phase modulation, such as role of coalescing characteristics, will be discussed. Examples in the talk include non-integrable coupled NLS equations and the water-wave problem in shallow water.

## 26 February 2019

### Speaker: Dr Alice Thompson (University of Manchester)

##### Title: Bubble propagation in modified Hele-Shaw channels

**Abstract: **

The propagation of a deformable air finger or bubble into a fluid-filled channel with an imposed pressure gradient was first studied by Saffman and Taylor. Assuming large aspect ratio channels, the flow can be depth-averaged and the free-boundary problem for steady propagation solved by conformal mapping. Famously, at zero surface tension, fingers of any width may exist, but the inclusion of vanishingly small surface tension selects symmetric fingers of discrete finger widths. At finite surface tension, Vanden-Broeck later showed that other families of 'exotic' states exist, but these states are all linearly unstable.

In this talk, I will discuss the related problem of air bubble propagation into rigid channels with axially-uniform, but non-rectangular, cross-sections. By including a centred constriction in the channel, multiple modes of propagation can be stabilised, including symmetric, asymmetric and oscillatory states, with a correspondingly rich bifurcation structure. These phenomena can be predicted via depth-averaged modelling, and also observed in our experiments, with quantitative agreement between the two in appropriate parameter regimes. I will also outline our efforts to understand how the system dynamics is affected by the presence of nearby unstable solution branches and how we can predict phenomena such as bubble break up from weakly nonlinear stability analysis.

## 05 March 2019 - CANCELLED

## 12 March 2019

##### See the Departmental Colloquium webpage

## 19 March 2019

### Speaker: Dr Angelika Manhart (Imperial College London)

##### Title: From filament networks to collective behavior - Using PDEs to explain biology

**Abstract: **

Partial differential equation (PDE) models can be a powerful tool for understanding emerging patterns in the life sciences. To mathematically capture these structures, one of the biggest challenges to overcome is the problem of scales: small scale events can result in large scale effects. I will present two projects, which exemplify how applied mathematics and molecular biology can gain from each other.

In the first part of the talk I will focus on a structure that lies at the heart of many types of cell movement: dynamic networks of branched actin fibres. Using experimental data to inform the models, we investigate how cofilin, a known disassembly agent, creates dynamic networks of fixed lengths. To capture the observed macroscopic fragmentation of the network, we combine PDE-based modelling of the cofilin binding dynamics with a discrete network disassembly model. This allows to predict the equilibrium network length across various control parameters.

In the second part I will discuss modelling, simulation and analysis of travelling waves in collectively moving bacterial swarms. The macroscopic PDEs describing the bacteria are derived from an individual-based description using a biology-focused coarse-graining method. Using the continuous model we can further investigate the unusual phenomena of counter-propagating travelling waves: Two families of interacting travelling waves whose profiles remain unchanged, but whose composition is modified by the oncoming wave.

## 26 March 2019

### Speaker: Thomas Richter (Magdeburg University)

##### Title: MODELS and NUMERICS FOR SEA ICE DYNAMICS

**Abstract: **

The subject of this talk is the analysis of a common model for simulating the dynamics of the ice layer on the arctic and antarctic ocean. The model under consideration goes back to Hibler ("A dynamic thermodynamic sea ice model", J. Phys. Oceanogr., Hibler 1979) and is based on a viscous-plastic description of the ice as a two-dimensional thin layer on the ocean surface.

In the first part of this talk we will derive and discuss the model in order to find a presentation that is suitable for applying modern numerical approximation tools. Then, in the second part we will focus on numerical tools for an approximation of the model with finite elements. A major numerical difficulty is due to the strong nonlinearity that is coming from the viscous-plastic rheology. We discuss a modified Newton solver that is robust in typical configurations.