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Centre for Nonlinear Mechanics     University of Bath

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                                                 Seminars and Meetings.

On Tuesdays we hold our formal seminars in
 Nonlinear Mechanics and Interdisciplinary Applied          mathematics

These and other events that happen throughout the year are listed below.

All seminars are at 1.15pm in 4W 1.7  (The Wolfson Room) University of Bath,
with lunch in the Claverton rooms beforehand.

                            CNM/IAM  Seminars and Workshops

                                 Spring 2014

   Tuesday 28th January                                                  Gareth Alexander (Warwick)

    Knotted Nematics

Tuesday 4th February                                                   Alexandra Tzella (Birmingham)

    Front propagation in steady cellular flows: A large deviation approach

   Tuesday 11th February                                                 Martin Paesold (Mech. Eng., Bath)

    Rock folding & Rock folding Think Tank

    Tuesday  18th February                                                Cancelled             

    Tuesday 25th February                                                 Lyubov Chumakova (Edinburgh)

Leaky modes -- discrete versus continuous spectrum in the atmosphere.

Much of our understanding of tropospheric dynamics is based on the concept of discrete internal modes. Internal gravity waves, such as those associated with convective systems, propagate at definite speeds, typically associated with the first to third baroclinic vertical modes. These waves are the dynamical backbone of the tropospheric dynamics, even though their nature and speed can be altered significantly by nonlinearity, moist convection, mean wind shear, etc. These discrete modes are a signature of systems of finite extent, and are derived in a case when the atmosphere is bounded above by a rigid lid. In reality, the atmosphere does not have a definite top, and, some argue, should be modeled as semi-infinite, leading to a continuous spectrum. Are the discrete rigid lid modes then just a fallacy of overly simplified theoretical models? In this talk I will present a correction to the rigid lid by using a boundary condition at the top of the troposphere, that allows for a fraction of waves to escape to the stratosphere. The new discrete "leaky" modes decay with characteristic time-scales, which are in the ballpark of many atmospheric phenomena. I will also address the mathematical question of why in seemingly identical physical situations of an unbounded atmosphere and its subsection with a radiation condition at the top give different spectral characteristics. This is joint work with R.R. Rosales (MIT) and E.G. Tabak (NYU).

    Tuesday 4th March

     Rogue waves Think Tank

     Tuesday 11th March                                                       Prof. Beth Wingate (Exeter)

     The influence of fast waves and fluctuations on the evolution of three slow solutions of the Boussinesq equations.

I will begin this seminar by discussing new challenges from the emergence of exascale computing and what this means for computing highly oscillatory systems of equations such as those used in climate and weather prediction. From this motivation I will present results from a study of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. We examine three known slow limits of the rotating and stratified Boussinesq equations: strongly stratified flow (Fr → 0,Ro ≈ O(1)), strongly rotating flow (Ro → 0,Fr ≈ O(1)) and Quasi-Geostrophy (Ro → 0,Fr → 0,Fr/Ro = f/N finite).
Numerical simulations indicate that for the geometry considered (triply periodic) and the type of forcing applied, the fast waves act as a conduit, moving energy onto the slow manifold. This decomposition clarifies how the energy is exchanged when either the stratification or the rotation is weak. In the quasi-geostrophic limit the energetics are less clear, however it is observed that the energy off the slow manifold equilibrates to a quasi-steady value.

    Wednesday 12th March  (NOTE DATE)                     Dr. Giacomo Canevari (Paris VI)

      Biaxiality in the Landau-de Gennes model for liquid crystals

Nematic liquid crystals are a special phase of matter, somehow intermediate between solid crystals and liquids. Several variational models have been proposed to describe their behaviour. In this talk we focus on the Landau de Gennes model, in a 2-D stationary case, with Dirichlet boundary conditions. In the low temperature range, we show that the stable equilibria are biaxial - that is, the molecules align locally along more preferred directions, at some point. Next, we discuss the asymptotic behavior of minimizers, as the elastic constant tends to zero, and prove the convergence to a locally harmonic map with singularities.

Tuesday 18th March                                                        Pete Ashwin (Exeter)

    Minimal geometric networks and dynamics of the Endoplasmic Reticulum

     Tuesday 25th March                                                      Ansgar Jungel    ( Vienna)  

    Entropy-dissipation methods for nonlinear parabolic equations and their systems

Entropy-dissipation methods have been developed recently to investigate the well-posedness and the qualitative behavior of solutions to nonlinear parabolic equations. The strength of the method lies in its flexibility and applicability to a large class of nonlinear equations. In this talk, two aspects of entropy methods will be detailed.

First, a priori estimates for higher-order parabolic equations will be derived by means of (Lyapunov) functionals which are called entropies. The estimations are usually based on skillful integration by parts. These integrations can be made systematic by formulating the task as a decision problem in real algebraic geometry, which can be solved in an algorithmic way. The method is applied to a fourth- and sixth-order quantum diffusion problem for semiconductors.

Second, we present a technique to derive a priori estimates for cross-diffusion systems whose diffusion matrix may be non-symmetric and not positive definite. The key idea is to exploit a formal gradient-flow structure. The corresponding entropy (or free energy) functional yields new variables which make the diffusion matrix positive definite. In certain cases, the new variables also allow for the proof of bounded solutions, although no classical maximum principle can be used. The method is applied to a diffusion system arising in tumor-growth modelling.

Monday 31st March                                                        Marshall Slemrod (Jerusalem)

    From Boltzmann to Euler: Hilbert’s 6th problem revisited

This talk addresses the hydrodynamic limit of the Boltzmann equation, namely the compressible Euler equations of gas dynamics. An exact summation of the Chapman–Enskog expansion originally given by Gorban and Karlin is the key to the analysis. An appraisal of the role of viscosity and capillarity in the limiting process is then given where the analogy is drawn to the limit of the Korteweg–de Vries–Burgers equations as a small parameter tends to zero.

     Tuesday 1st April                                                            Patrick Dondl (Durham)

    A phase field model for the optimization of the Willmore energy in the class of connected surfaces

 In many applications structures can be described as (local) minimizer of suitable bending energies. The most prominent example is the variational characterization of shapes of biomembranes by the use of Helfrich- or Willmore-type functionals. Whereas the restriction to topological spheres is natural in many applications, it is sometimes more reasonable to consider the class of orientable connected surfaces of arbitrary genus instead. For example the inner membrane of mitochondria cells shields the inside matrix from the outside but shows - in contrast to old textbook illustrations - a lot of of handle-like junctions and therefore represents a higher genus surface. In this example another natural constraint comes into play, given by the confinement of the inner membrane to a 'container' that is given by the outer membrane of the mitochondria.

This motivates to consider the following variational problem: Minimize Willmore's energy in the class of all compact, connected, orientable surfaces without boundary that are embedded in a bounded domain and have prescribed surface area.

We consider a phase field approximation to this problem, i.e., we approximate the surface by a level set function u admitting the value +1 on the inside of the surface and -1 on its outside. The confinement of the surface is now simply given by the domain of definition of $u$. Diffuse interface approximations for the area functional, as well as for the Willmore energy are well known. We address the main difficulty, namely the topological constraint of connectedness by a nested minimization of two phase fields, the second one being used to identify connected components of the surface. We present a proof of Gamma-convergence of our model to the sharp interface limit. This is joint work with Matthias Röger (TU Dortmund) and Luca Mugnai (MPI Leipzig).

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