All seminars (unless otherwise stated) will take place on **Tuesdays at 3.00pm in ****Room 505 in the Mathematics Department **(25 Gordon Street). See see how to find us for further details. There will be tea afterwards in Maths Room 606. If you require any more information on the Applied seminars please contact Prof Slava Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

## 13 January 2015

Prof Beth Wingate - University of Exeter

##### Title: The influence of fast waves and fluctuations on the evolution of slow solutions of the Boussinesq equations

Abstract:

I will present results from studies of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. My collaborators and I work in the framework of fast singular limits that derives from the work of Bogoliubov and Mitropolsky [1961], Klainerman and Majda [1981], Shochet [1994], Embid and Majda [1996] and others. To understand how the flow approaches and interacts with the slow dynamics we decompose the full solution into its slow component and everything else. We use this decomposition to find evolution equations for the components of the flow (and the corresponding energy) into and out of the 'slow manifold'. Numerical simulations indicate that for the geometry considered (triply periodic) and the type of forcing applied, the fast dynamics act as a conduit, moving energy onto the slow component.

I will also discuss generalizations of the method of cancellations of oscillations of Schochet for two distinct fast time scales, i.e. which fast time scale is fastest? I will give an example for the quasi-geostrophic limit of the Boussinesq equations.

The talk includes joint work with Jared Whitehead (Brigham Young University) and Terry Haut (Los Alamos National Laboratory).

## 20 January 2015

Prof Alexey Chernov - University of Reading

##### Title: Quasi-optimal stability estimates for the hp-Raviart-Thomas projection operator on the cube

Abstract:

Stability of the hp-Raviart-Thomas projection operator as a mapping H^1(K) -> H^1(K) on the unit cube K in R^3 has been addressed e.g. in [2], see also [1]. These results are suboptimal w.r.t. the polynomial degree. In this talk we present quasi-optimal stability estimates for the hp-Raviart-Thomas projection operator on the cube. The analysis involves elements of the polynomial approximation theory on an interval and the real method of Banach space interpolation.

(Joint work with Herbert Egger, TU Darmstadt)

[1] Mark Ainsworth and Katia Pinchedez. hp-approximation theory for BDFM and RT finite elements on quadrilaterals. SIAM J. Numer. Anal., 40(6):2047-2068 (electronic) (2003), 2002.

[2] Dominik Schötzau, Christoph Schwab, and Andrea Toselli. Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal., 40(6):2171-2194 (electronic) (2003), 2002.

## 27 January 2015

Prof Alexander Stegner - Ecole Polytechnique, France

##### Title: Cyclone-anticyclone asymmetries of island wake flows

Abstract:

A large number of recent studies show that even if the upstream forcing is symmetrical (uniform flow and circular island) the wake flow may exhibit a strong cyclone-anticyclone asymmetry due to the combined effects of rotation and stratification. At mesoscale, laboratory experiments and numerical simulations reveal the preferred formation of anticyclonic vortices within a shallow-water wake if the island diameter is larger than the first baroclinic radius. Once they are formed, these large-scale anticyclones tend to be more stable and robust to external strain perturbations than their cyclonic counterparts. At submesoscale, large scale experiments performed on the Coriolis plateform and 3D numerical simulations have shown that an island wake flow may exhibit a transient and three-dimensional instability in the region of intense anticyclonic vorticity. This instability is a branch of the inertial-centrifugal instability in the framework of rotating,stratified shallow-water flows. In such case, we could expect a predominance of intense cyclone in the island wake.

These various dynamical regimes are mainly controlled by three dimensionless numbers, namely the island Rossby number RoI , the Burger number Bu and the Ekman number Ek. A global diagram, which estimates the various regions of cyclonic or anticyclonic predominance is proposed in the (RoI ; Bu) parameter space.

## 3 February 2015

Dr Olga Trichtchenko - UCL

##### Title: Stability of Periodic Gravity-Capillary Water Waves

Abstract:

I will present results on the computation and stability of periodic surface gravity-capillary waves. First, I will show how we solve Euler's equations to compute these waves. Then I will present the results of the stability analysis for these solutions by making use of Hill's method. Depending on the coefficient of surface tension, we see resonant effects called Wilton's ripples. These resonant solutions for gravity-capillary waves are found to have interesting instabilities. Since this stability analysis is general to all Hamiltonian systems, we can also use it to compare and contrast the results for different models for water waves.

## 10 February 2015 - NO SEMINAR

## 17 February 2015 - NO SEMINAR (READING WEEK)

## 24 February 2015

Dr Camilo Garcia Trillos - UCL

##### Title: A probabilistic approach to the solution of second order semiliear PDEs

Abstract:

In this talk, we will discuss a probabilistic interpretation to some second order partial differential equations (PDEs). This probabilistic view can be extremely useful in the study of several properties of their solutions (existence, uniqueness, differentiability, growth control,… ) and suggests novel probabilistic approaches to their numerical approximation.

More specifically, we will focus on studying the semi-linear second order PDE case. I will recall some results linking their solutions to the so called Backward Stochastic Differential Equations (BSDEs). Then, I will present some key properties of the solution to BSDEs and their implications from the PDE point of view. Finally, we will use these concepts to introduce an algorithm to solve a system of second order semi-linear PDEs (with non-local coefficients), arising from some control problems in finance.

## 3 March 2015

Dr David Pritchard - University of Strathclyde

##### Title: Wrestling Mud

Abstract:

I will describe a number of problems in non-Newtonian fluid dynamics, loosely motivated by geophysical flows of mud. Mud has properties including shear thinning, plasticity and thixotropy: I will discuss how these are represented in rheological models and the challenges that face the fluid dynamicist faced with a bewildering variety of such models. In particular, I will present recent results for the lubrication flow of generalised Newtonian or thixotropic fluids, and relate these to the ongoing challenge of describing non-Newtonian flow in porous media.

## 10 March 2015 - COLLOQUIUM TALK

Prof Sylvia Serfaty - Université Pierre et Marie Curie

##### - please see the Departmental Colloquia webpage

## 17 March 2015

Dr. Andrés A. León Baldelli - Mathematical Institute, Oxford

##### Title: Emerging nonlinearities from linear elasticity. From elastic foundations to complex crack patterns

Abstract:

Exploring asymptotic regimes in linear elasticity, we will close a problem that has been open in mechanics for more than 150 years: the origin and calibration of linear elastic foundation models.

Often regarded as heuristic, phenomenological models, they emerge asymptotically from standard, linear, three-dimensional elasticity providing the first order linear model leading to size effects. This geometric nonlinearity entails, among other involved phenomena, the structuration of crack patterns in thin film systems under homogeneous loads.

I will finally present the associated variational fracture problem, and show some large scale numerical results in 2D involving hexagonal crack networks and parallel crack patterns.