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).