{\bf Keble Workshop on Partial Differential Equations
in Science and Engineering}
\\
\\Place: Roy Griffiths Room in the ARCO Building, Keble College
\\Time: 1:00pm-5:10pm, Thursday, June 10.
\\
\\
Program:\\
\\ 1:00-1:20pm: Coffee and Tea
\\
\\ 1:20-2:10pm: Prof. Walter Craig (Joint with OxPDE Lunchtime Seminar)
\\
\\ 2:20-2:40pm Prof. Mikhail Feldman
\\
\\ 2:50-3:10pm Prof. Paul Taylor
\\
\\ 3:20-3:40pm Coffee and Biscuits
\\
\\ 3:40-4:00pm: Prof. Sir John Ball
\\
\\ 4:10-4:30pm: Dr. Apala Majumdar
\\
\\ 4:40-5:00pm: Prof. Robert Pego
\\
\\ 5:10-6:00pm: Free Discussion
\\
\\{\bf Titles and Abstracts:}
\\
1.{\bf Title: On the singular set of the Navier-Stokes equations
\\ Speaker: Prof. Walter Craig, McMaster University, Canada}
\\ Abstract:\\
The Navier-Stokes equations are important in
fluid dynamics, and a famous mathematics problem is the
question as to whether solutions can form singularities.
I will describe these equations and this problem, present
three inequalities that have some implications as to the
question of singularity formation, and finally, give a
new result which is a lower bound on the size of the
singular set, if indeed singularities exist.
\\
\\{\bf 2. Title: Shock Analysis and Nonlinear Partial Differential Equations of Mixed Type.
\\ Speaker: Prof. Mikhail Feldman, University of Wisconsin-Madison, USA}
\\
\\ Abstract:\\ Shocks in gas or compressible fluid arise in various physical
situations, and often exhibit complex structures. One example is reflection
of shock by a wedge. The complexity of reflection-diffraction configurations
was first described by Ernst Mach in 1878. In later works, experimental and
computational studies and asymptotic analysis have shown that various patterns
of reflected shocks may occur, including regular and Mach reflection. However,
many fundamental issues related to shock reflection are not understood,
including transition between different reflection patterns. For this reason
it is important to establish mathematical theory of shock reflection,
in particular existence and stability of regular reflection solutions for PDEs
of gas dynamics. Some results in this direction were obtained recently.
\\
In this talk we start by discussing examples of shocks in supersonic and
transonic flows of gas. Then we introduce the basic equations of gas dynamics:
steady and self-similar compressible Euler system and potential flow equation.
These equations are of mixed elliptic-hyperbolic type. Subsonic and supersonic
regions in the flow correspond to elliptic and hyperbolic regions of solutions.
Shocks correspond to certain discontinuities in the solutions. We discuss some
results on existence and stability of steady and self-similar shock solutions,
in particular the recent work (joint with G.-Q. Chen) on global existence of
regular reflection solutions for potential flow. We also discuss open problems
in the area.
\\
\\{\bf 3. Title: Shallow water waves - a rich source of interesting solitary wave
solutions to PDEs
\\ Speaker: Prof. Paul H. Taylor, Keble College and Department of Engineering Science, Oxford}
\\
\\Abstract:\\ In shallow water, solitary waves are ubiquitous: even the wave crests
in a train of regular waves can be modelled as a succession of solitary waves.
When successive crests are of different size, they interact when the large ones
catch up with the smaller. Then what happens? John Scott Russell knew by experiment
in 1844, but answering this question mathematically took 120 years!
This talk will examine solitary wave interactions in a range of PDEs, starting
with the earliest from Korteweg and De Vries, then moving onto Peregrine's
regularized long wave equation and finally the recently introduced Camassa-Holm
equation, where solitary waves can be cartoon-like with sharp corners at the crests.
For each case the interactions can be described using the conserved quantities,
in two cases remarkably accurately and in the third exactly, without actually
solving any of the PDEs.
The methodology can be extended to other equations such as the various versions
of the Boussinesq equations popular with coastal engineers, and perhaps even
the full Euler equations.
\\
{\bf 4. Title: Austenite-Martensite interfaces
\\ Speaker: Prof. Sir John Ball, Queen's College and Mathematical Institute, Oxford}
\\
\\Abstract:\\ Many alloys undergo martensitic phase transformations
in which the underlying crystal lattice undergoes a change of shape
at a critical temperature. Usually the high temperature phase (austenite)
has higher symmetry than the low temperature phase (martensite).
In order to nucleate the martensite it has to somehow fit geometrically
to the austenite. The talk will describe different ways in which this
occurs and how they may be studied using nonlinear elasticity and
Young measures.
\\
\\{\bf 5. Title: Partial Differential Equations in Liquid Crystal Science and
Industrial Applications
\\ Speaker: Dr. Apala Majumdar, Keble College and Mathematical Institute, Oxford}
\\
\\Abstract:\\
Recent years have seen a growing demand for liquid crystals in modern
science, industry and nanotechnology. Liquid crystals are mesophases or
intermediate phases of matter between the solid and liquid phases of
matter, with very interesting physical and optical properties.
We briefly review the main mathematical theories for liquid crystals and
discuss their analogies with mathematical theories for other soft-matter
phases such as the Ginzburg-Landau theory for superconductors. The
governing equations for the static and dynamic behaviour are typically
given by systems of coupled elliptic and parabolic partial differential
equations. We then use this mathematical framework to model liquid crystal
devices and demonstrate how mathematical modelling can be used to make
qualitative and quantitative predictions for practical applications in
industry.
\\
\\{\bf 6. Title: Bubble bath, shock waves, and random walks --- Mathematical
models of clustering
\\Speaker: Prof. Robert Pego, Carnegie Mellon University, USA}
\\Abstract:\\ Mathematics is often about abstracting complicated phenomena into
simple models. This talk is about equations that model aggregation
or clustering phenomena --- think of how aerosols form soot particles
in the atmosphere, or how interplanetary dust forms comets, planets
and stars. Often in such complex systems one observes universal trend
toward self-similar growth. I'll describe an explanation for this
phenomenon in two simple models describing: (a) ``one-dimensional
bubble bath,'' and (b) the clustering of random shock waves.