Geometric Analysis, Elasticity and PDE

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Geometric Analysis, Elasticity and PDE

 23 - 27 Jun 2008
 Heriot Watt University, Edinburgh

Organiser

Name
Institution
Bona, JerryUniversity of Illinois at Chicago
Carr, JackHeriot-Watt University
James, RichardUniversity of Minnesota
Marsden, JerryCalifornia Institute of Technology
Murat, FrançoisUniversité Pierre et Marie Curie (Paris VI)
Slemrod, MarshallUniversity of Wisconsin

Scientific Organiser
Jack Carr

Scientific Committee
Professor Jerry Bona, Co-Chair (University of Illinois)
Professor Jack Carr, Heriot-Watt University
Professor Dick James (University of Minnesota)
Professor Jerry Marsden, Co-Chair (California Institute of Technology)
Professor Stefan Müller (Max Planck Institute for Mathematics, Leipzig)
Dr François Murat (Université Pierre et Marie Curie, Paris VI)
Professor Marshall Slemrod (University of Wisconsin)

Local Organising Committee
Professor Tony Carbery (University of Edinburgh)
Professor Jack Carr (Heriot-Watt University)
Professor Chris Eilbeck (Heriot-Watt University)
Professor Robin Knops (Heriot-Watt University)
Professor John Toland (University of Bath)

2008 sees the 60th birthday of Sir John Ball, FRS, FRSE, and we will be celebrating this occasion during the workshop. Lecture notes and photographs from the workshop are now available on a page maintained by Professor Eilbeck.

Nonlinear Partial Differential Equations (PDE) are of fundamental importance in mathematics and although major progress in their study has recently been achieved, many profound challenges remain. Developments are motivated not only by advances in pure and applied mathematics but also by new issues in the physical and biological sciences, mathematical finance and risk theory and many other applied disciplines.

Analytical, dynamical systems, as well as geometric methods continue to play a critical role in both fundamental investigations as well as numerical methods in solid and fluid mechanics, including complex materials, such as material microstructures and liquid crystals. It is clear that this subject is in a period of exciting growth and development. The principal aim of the workshop is to promote international exchange of recent developments in the analysis and geometry of PDE and their applications.

Specific objectives include:

  1. mapping new frontiers of research
  2. building bridges between the most recent developments in the general mathematical theory and the areas of applications that have developed most actively in the last few years,
  3. identifying new applications of PDE

Specific themes of the workshop include:

  • Nonlinear elasticity and liquid crystals
  • Material science; microstructure, minimizer in the presence of symmetry, relation with experiment
  • Calculus of variations
  • Systems of conservation laws
  • Transport equations
  • Biomaterials Dynamical systems methods and the PDEs of mechanics


Supporting Institutions
EPSRC
Maxwell Institute for Mathematical Sciences (Centre for Analysis and Nonlinear PDE)
University of Oxford
LMS
Edinburgh Mathematical Society

 

Arrangements

Participation
You may apply to attend this meeting by completing the application form which is available at the top of this page. (Please note that the application form will close on 23 May, or when we have reached the maximum number for the meeting, 150.) You will then receive an e-mail confirming your place. Speakers and committee members will receive a separate invitation via e-mail.

If you do not receive an e-mail please contact Morag Burton.

Venue
The Workshop will be held at Heriot-Watt University. The campus is on the western edge of the city, near the airport, with excellent public transport to and from Edinburgh city centre. Click here for instructions on how to reach the university by various forms of transport.

Registration
There will be a combined Registration and Welcome Buffet on Sunday 22 June, at Heriot-Watt University. Those who cannot register on Sunday evening may do so on Monday 23 June prior to the start of the lectures or during the first coffee break. There will be a Registration Fee of 60.00 GBP. For those of you who want to pay by credit card, the credit card form can be found here. You may post or fax (+44 (0)131 220 1053) this to us in advance or bring it with you, already filled in, to registration. Please note that there is a 1.75% charge for use of credit cards

Poster Sessions
Contributed presentations in poster format are invited in all areas consistent with the conference themes. If you wish to submit a poster please submit a title and abstract (maximum 75 words) to J. Carr (at) ma.hw.ac.uk and a prompt decision will be made on its acceptability. Information on the format of the poster sessions will be available in March 2008.

Audio/Visual Facilities
A data projector, overhead projector, PC, laptopconnection and blackboards will be available.

Accommodation and Facilities
Single study-bedrooms are available at Heriot-Watt University. To book contact the Edinburgh Conference Centre directly on +44 (0)131 451 3669 or e-mail info@edinburgh-conference.com .
Invited participants should refer to their personal invitation letter for accommodation arrangements.

All rooms have a private shower and toilet and a telephone. The telephone will access incoming calls. You need to buy a telephone card from Reception in order to make outgoing calls. Each room also has an internet point should you wish to bring your laptop, although there is a charge for this service. Free email and internet access will be arranged in the central computer labs, the Library and to the university wireless network.

Meals and Refreshments
Breakfast will be provided for Heriot-Watt University residents for the period of their stay in the Middle Floor Dining Room.

Refreshments will be available during workshop breaks.

There will be a buffet supper (free of charge to participants) on the evening of Sunday 22 June between 19.00 and 20.30. Participants may register for the workshop during this buffet.

The Workshop Dinner will be on Thursday 26 June. Non-invitees are welcome to come to this on payment of £30.00, which will cover the dinner and wine. Payment for the dinner may be added to the amount being paid for the Registration Fee (see above).

Programme

Provisional Timetable

Monday 23 June
09:00-09:30 John Toland: Welcome
Chair: Jack Carr
09:30-10:30 Constantine Dafermos (Brown University)
Elastodynamics and John Ball
10:30-11:00 Refreshments
11:00-12:00 Stuart Antman (University of Maryland)
Asymptotics of small inertias and large times in nonlinear continuum mechanics
12:00-13:00 Carme Calderer (University of Minnesota)
Nonlinear elasticity and liquid crystals in biological and biomedical application
13:00-14:30 Lunch
Chair: Tony Carbery
14:30-15:30 Maria J. Esteban (CEREMADE)
Critical magnetic field for the Dirac-Coulomb model
15:30-16:00 Refreshments
16:00-17:00 Vladimir Sverak (University of Minnesota)
Liouville theorems for the Navier-Stokes equations
17:00-18:00 Poster Viewing

Tuesday 24 June
Chair: Robin Knops
09:30-10:30 Stefan Müller (Max Planck Institute for Mathematics, Leipzig)
Some thoughts about crystals and the Cauchy-Born rule at positive temperature
10:30-11:00 Refreshments
11:00-12:00 Sergio Conti (Universität Bonn)
Patterns in thin elastic sheets
12:00-13:00 Philippe Ciarlet (City University of Hong Kong)
Intrinsic elasticity
13:00-14:30 Lunch and Poster Viewing
Chair: Dick James
14:30-15:30 Nick Schryvers (University of Antwerp)
Nano- and microstructures charaterized by transmission electron microscopy
15:30-16:00 Refreshments
16:00-17:00 Kaushik Bhattacharya (California Institute of Technology)
Electronic structure and macroscopic properties of solids
17:00-18:00 Poster Viewing

Wednesday 25 June
Chair: Chris Eilbeck
09:30-10:30 Bob Kohn (New York University)
Cloaking by change of variables
10:30-11:00 Refreshments
11:00-12:00 Gero Friesecke (University of Warwick)
A mathematical picture of basic aspects of the periodic table
12:00-13:00 Oliver Penrose (Heriot-Watt University)
Statisical mechanics of nonlinear elasticity in the system of hard disks
13:00-14:30 Lunch
14:30-19:00 Afternoon Free

Thursday 26 June
Chair: Stefan Müller
09:30-10:30 François Murat (Université Pierre et Marie Curie (Paris VI))
A priori estimate and existence for elliptic problems with subquadratic gradient dependent terms
10:30-11:00 Refreshments
11:00-12:00 Laszlo Szekelyhidi (Max Planck Institute, Leipzig)
The h-principle in fluid mechanics
12:00-13:00 Michael Ortiz (California Institute of Technology)
Minimum principles for characterizing the trajectories and microstructural evolution of dissipative systems
13:00-14:30 Lunch
Chair: François Murat
14:30-15:30 Bernard Dacorogna (École Polytechnique Fédérale de Lausanne)
On the pullback equation
15:30-16:00 Refreshments
16:00-17:00 Gilles Francfort (Université Paris-Nord)
Evolution and approximation in brittle fracture
17:00-18:00 Discussions
Host: Robin Knops
19:00-22:30Evening Banquet

Friday 27 June
Chair: Marshall Slemrod
09:30-10:30 Robert Pego (Carnegie Mellon University)
How to compute the pressure in incompressible viscous flow with no-slip boundary
10:30-11:00 Refreshments
11:00-12:00 Antonio DeSimone (SISSA, Trieste)
Excellent swimmers
12:00-13:00 Carsten Carstensen (Humboldt-Universität zu Berlin)
Computational challenges in the calculus of variations
13:00-14:30 Lunch
Chair: Jan Kristensen
14:30-15:30 Yann Brenier (Université de Nice-Sophia-Antipolis)
Convection, magnetic relaxation and optimal transport
15:30-16:00 Refreshments
16:00-17:00 Terry Tao (University of California at Los Angeles)
Global regularity of wave maps via harmonic map heat flow
17:00-18:00 Jack Carr: Close of Meeting

 

 

Presentations

Antman, Stuart
Asymptotics of small inertias and large times in nonlinear continuum mechanics
View Abstract
There are numerous dynamical problems in mechanics in which some inertias
are very small, their smallness signalled by a small parameter. For such
problems it is customary to discard the small terms, in which case some
or all the variables evolve quasistatically, i.e., through a sequence of
equilibrium states parametrized by time. Formal asymptotic expansions
in the small parameter might exhibit the detailed effects of the parameter.
Rigorous asymptotic justifications, which provide error estimates and
are typically far harder to carry out, are used by those compulsive about
mathematical hygiene. They are valuable, however, when the formal theory
exhibits surprising results.

The purpose of this lecture is to illuminate the material behavior supporting
the asymptotic justification of quasistatic response and to examine
some strange and surprising large-time dynamical behavior when such
justification is not available. These results will be described in the context
of a couple of conceptually simple problems from particle and continuum
mechanics.
Bhattacharya, Kaushik
Electronic structure and macroscopic properties of solids
View Abstract
Defects play a critical role in determining the properties of solids. They do so even though they are present at extremely small concentrations because the long-range elastic fields they generate enables them to respond to macroscopic stresses and to interact with other distant defects. At the same time, the strength of the elastic field and other critical aspects of the defect core are determined by quantum mechanical interactions. This direct coupling between electronic and macroscopic scales makes defects both interesting and challenging to understand. We present a computational method for studying the electronic structure at macroscopic scales. We explain the mathematical framework that enables a systematic means of adaptive coarse-graining retaining full resolution where it is necessary and coarsening with no patches, assumptions or structure. We demonstrate its accuracy under modest computational cost and the physical insights it offers using various examples.
Brenier, Yann
Convection, magnetic relaxation and optimal transport.
View Abstract
We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems by a gradient flow approach. We interpret this geometric equation as a generalization of the Darcy-Boussinesq equations for use in Convection Theory. This suggests that the Navier-Stokes-Boussinesq equations, the basic model in Convection Theory, provide a good framework for Optimal Transport related problems, such as Hoskins' Semigeostrophic equations and some fully nonlinear version of some Chemotaxis equations. In a different direction, we introduce a "stringy" generalization of the Angenent Haker Tannenbaum model. This model is closely related to the Magnetic Relaxation model investigated by Arnold and Moffatt. It is a gradient flow for which the equilibrium states are just the (variational) solutions of the Euler equations for incompressible flows. We also discuss a vectorial version of the Burgers equation, the 'cross-Burgers' equation.
Calderer, Carme
Nonlinear elasticity and liquid crystals in biological and biomedical application
View Abstract
Nonlinear elasticity plays a prominent role in the mathematical modeling of gels from the point of view of continuum physics. The coexistence of materials such as synthetic polymers and biological fibers with fluid, naturally yields gel states characterized by macroscopic properties including elasticity, diffusion, transport and dissipation.

The mathematical problems analyzed in this presentation are motivated by
two types of modeling endeavors:
cell motility in fiber matrices and elastic behavior of
body-implantable biomedical devices. Some of these systems are also highly anisotropic and exhibit liquid crystal behavior.

I formulate and analyze boundary value problems arising in the design and implanting of biomedical devices.
I will also discuss time dependent regimes dominated by dissipation. One main difficulty
is the fact that both formulations, Lagrangian and Eulerian are present in the governing systems of partial differential equations. Finally, I will address the early-dynamics regime where chemical potentials dominate over viscosity.
The governing system in such case is hyperbolic with a source term.
Carstensen, Carsten
Computational challenges in the calculus of variations
View Abstract
The difficulties in the numerical simulation of three classes of problems, namely convex, polyconvex, and quasiconvex minimisation problems, guide through the presentation. Simple benchmarks in computational microstructures with finer and finer oscillations allow for a convexified relaxation hull to model the macroscopic behaviour of measure-valued generalised solutions. This motivates a class of degenerate convex minimisation problems with local regularity of stresses and a seemingly optimal a priori error control. Despite some reliability-efficiency gap in the a posteriori error analysis, there exist some convergent adaptive mesh-refining algorithm.
Polyconvex minimisation problems allow for the existence of global minimisers with unclear regularity. For arbitrarily small loads in hyperelasticity, global minimisers coincide with local smooth solutions based on the implicit function theorem. In this situation, finite element methods are convergent as well. For more realistic loads, however, even $Gamma$ convergence of finite element approximations is uncertain.
Finite strain elastoplasticity is based on a multiplicative split of elastic and plastic deformations which results in a non-convex mathematical model. In fact, a typical time-step within some incremental description of the elastoplastic evolution results in a non-quasiconvex minimisation problem. Difficulties with the efficient numerical relaxation are discussed for some single-slip elastoplastic benchmark.
Nonlinear minimisation problems allow exciting phenomena such as repelling singularities. The Lavrentiev phenomena is shown to be equivalent to the failure of finite element methods. This motivates non-conforming discrete spaces and/or modified energies. The presentations reviews some solution strategies and a $L^p$ penalisation scheme with some preliminary affirmative results.
The presentation is based on recent joint work with S. Conti, R. Huth, A. Orlando, C. Ortner.
Ciarlet, Philippe G.
Intrinsic elasticity
View Abstract
In the classical approach to elasticity problems, the deformation field is the primary unknown. In the "intrinsic" approach, new unknowns with more mechanical or geometrical meanings, such as a strain tensor field or a rotation field for instance, are instead taken as the primary unknowns. We survey here recent progress about the mathematical analysis of such methods.
Conti, Sergio
Patterns in thin elastic sheets
View Abstract
Crumpling a sheet of paper leads to the formation of folding patterns over several length scales. This can be understood on the basis of the interplay of a nonconvex elastic energy, which favors locally isometric deformations, and a small singular perturbation, which penalizes high curvature.
Based on three-dimensional nonlinear elasticity and by using a combination of explicit constructions and general results from differential geometry, we prove that, in agreement with previous heuristic results in the physics literature, the total energy per unit thickness of such folding patterns scales at most as the thickness of the sheet to the power $5/3$. We contrast this scaling with the linear scaling obtained in the case of Dirichlet boundary conditions.
Dacorogna, Bernard
On the pullback equation
View Abstract
I discuss the existence of a diffeomorphism for the change of variables formula for differential forms
Dafermos, Constantine
Elastodynamics and John Ball
View Abstract
The seminal ideas introduced by John Ball in elastostatics are also relevant to elastodynamics.
DeSimone, Antonio
Excellent swimmers
View Abstract
We will discuss swimming strategies for microscopic swimmers and recipes to optimize their strokes.
Esteban, Maria J
Critical magnetic field for the Dirac-Coulomb model
View Abstract
In this talk I will present joint work with Jean Dolbeault and Michael Loss in which we investigate the threshold of electron-positron pair creation for the Dirac-Coulomb model, describing the bound states of a relativistic electron submitted to a nuclear electrostatic field and an external constant magnetic field.

Mathematically we are interested in the critical magnetic field at which the lowest energy state gets into the continuum spectrum, describing a destabilizing effect.

A scaling argument allows us to find a simple variational characterization for the critical magnetic field. We use it in order to compare the general critical magnetic field and the one found when taking into account only states which lie in the lowest Landau level. We do this both analytically and numerically. We find an important discrepancy between the two regimes.
Francfort, Gilles A.
Evolution and approximation in brittle fracture
View Abstract
In recent years, quasi-static crack evolution in a brittle solid has been revisited in a variational light. At this time, the only completely predictive model is based on global minimization, together with a surface energy that is proportional to the created crack area (the Griffith model). We will briefly review the main mechanical and mathematical features of this model and discuss its shortcomings. Because the essential ingredient of the model is an unconstrained crack path, the variational model is formulated in a BV type space which does not provide an easy setting for numerical computations. Approximating schemes, which are based on Gamma-convergence, typically fail to follow globally minimizing paths and may only capture paths among critical points. We will describe one dimensional results on the connection between critical points of the approximating functionals and those of the initial one.
Finally, we will examine other approximations that are popular in the mechanics community and discuss their relevance to fracture evolution.
Friesecke, Gero
A mathematical picture of basic aspects of the periodic table
View Abstract
In textbooks, the striking chemical similarities and differences between the elements are discussed on a descriptive, semi-empirical level, based on the "hydrogen orbital configurations'' developed by Bohr, Hund and Slater.

In the talk I will present a mathematical picture (for the first ten elements H, He, Li, Be, B, C, N, O, F, Ne), based on calculating the many-electron Schroedinger ground state explicitly in a natural scaling limit. In this limit, the nontrivial postulates of Bohr, Hund and Slater (electrons filling orbitals, shell and sub-shell formation,
sub-shell ordering rules such as 2s < 2p, Hund's rules) are seen to emerge in a natural way.

In fact, in a small minority of cases (such as ordering of the excited states of the Carbon atom) our mathematical results differ from the semi-empirical theory (and experiment confirms our predictions). Proofs involve hydrogen atom theory, careful use of the symmetry group SU(2) x SO(3) x Z_2 of the many-electron Schroedinger equation and its representation theory, Fourier analysis, and ideas from quantum chemistry.

Joint work with Ben Goddard (Warwick).
Kohn, Robert V.
Cloaking by Change of Variables
View Abstract
We say a region of space is "cloaked" with respect to electromagnetic measurements if its contents -- and even the existence of the cloak -- are inaccessible to such measurements. One recent proposal for achieving cloaking takes advantage of the coordinate-invariance of Maxwell's equations. I shall explain this scheme, including its mathematical basis and its apparent limitations.
Murat, François
A priori estimate and existence for elliptic problems with subquadratic gradient dependent terms
View Abstract
In this lecture I will consider the nonlinear elliptic model problem $$ u in H^1_0 (Omega), qquad - div , A(x) Du + alpha_0 u = gamma |Du|^q + f(x)
; ; hbox{in} ; {cal D}'(Omega),$$ with $A$ a coercive matrix with bounded coefficients, $alpha_0 geq 0$, $0 leq q leq 2$ and $fin L^m (Omega)$ for some suitable~$m$. This is a model problem, and there are many possible variants of it. In particular, I will consider the case where the right-hand side is only bounded by (but not equal to) $gamma |Du|^q + f(x)$.

In the case where $0 leq q < 1$, existence is classical for $f in H^{-1} (Omega)$. When $gamma$ is large, the case where $q = 1$ and $f in H^{-1} (Omega)$ is difficult but has been solved by G. Bottaro and M.E. Marina in 1973. On the other hand, the case $q = 2$ has been treated by many authors, including in particular in a series of papers by L.~Boccardo, J.-P.~Puel and myself. In a more recent paper, V. Ferone and myself proved the existence of a solution $u$ which further satisfies
$ e^{gamma u} -1 in H^1_0 (Omega)$, and an a priori estimate for such solutions, when %$fin L^{{N} over {2}} (Omega)$.

In this lecture I will mainly report about recent joint work with Nathalie Grenon and Alessio Porretta, the announcement of which has been published in C. R. Acad. Sci. Paris, S'erie I 342 (2006), {23-28}. When $1 + (2 / N) leq q <2$ and $fin L^m(Omega)$ with $m = {{N (q-1)} / {q}}$ (we also solved the case where $1leq q < 1 + {{2}over{N}} $, but I will not discuss it since it uses the notion of renormalized solution), and when either $alpha_0 >0$ or $f$ is sufficiently small in $L^m(Omega)$, we prove the existence of a solution $u$ which enjoys the further regularity $|u|^sigma in H^1_0(Omega)$ with, $sigma = {{(N-2) (q - 1)} / {2(2 - q)}}$. Uniqueness results in this class have also been proved by G. Barles and A. Porretta under certain structure conditions. We moreover prove an a priori estimate for every solution in this class. One of the main interests of our work lies in this a priori estimate, the proof of which is non standard.
Ortiz, Michael
Minimum principles for characterizing the trajectories and microstructural evolution of dissipative systems
View Abstract
This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i. e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of energy-dissipation functionals under consideration is derived by first defining a sequence of time-discretized minimum problems, stringing them together with weights that decay exponentially over times of order epsilon, and formally passing to the limit of continuous time. The Gamma-limit of these functionals as epsilon tends to zero is highly degenerate and provides limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. The relaxation of energy-dissipation functionals and optimal scaling can be rigorously derived in a number of examples of application. A first example concerns a one-dimensional bar characterized by a quadratic dissipation function and a bistable energy density. A second example concerns the coarsening kinetics of island growth in thin films exhibiting a preferred slope. In both cases, we present closed-form relaxations in the local limit of the problem and optimal scaling relations for the nonlocal problems. The relaxations rigorously characterize macroscopic properties of complex microstructural evolution by means of well-posed effective problems. The scaling relations rigorously characterize some average properties of the coarsening kinetics of the systems, and lead to predictions on the growth exponents. A third example of application concerns crack propagation. When dissipation is assumed to be concentrated at the crack front, the dissipation must be relaxed in general. We prove a general relaxation formula that gives the surprising result that the effective dissipation is always rate-independent.
Pego, Robert
How to compute the pressure in incompressible viscous flow with no-slip boundary
View Abstract
We describe a formula for the pressure in the Navier-Stokes equations for
incompressible flow with no-slip boundary conditions, that involves the
commutator of the Laplacian and Leray-Helmholtz projection operators.
An estimate for this commutator shows that it is strictly dominated
by the viscous term at leading order. This leads to a number of new developments,
including a well-posedness theorem for an extended Navier-Stokes dynamics
unconstrained by the divergence-free condition, and a number of significant
improvements concerning methods of numerical computation and numerical analysis
for such flows. This is joint work with Jian-Guo Liu (Maryland) and Jie Liu (Irvine).
Penrose, Oliver
Statistical mechanics of nonlinear elasticity in a system of hard disks
View Abstract
The model considered is a two-dimensional system of hard disks, with or
without an attractive short-range interaction. A `constrained free energy'
is defined, using the standard configurational integral of statistical
mechanics subject to two constraints which relate the set of
configurations we integrate over to a given close-packed reference
configuration of the set of disks under consideration. These constraints
are: (i) if two disks are nearest neighbours in the reference
configuration, then their separation is constrained to be at most sqrt{2}
times the disk diameter and (ii) each disk at the boundary of the
reference configuration is constrained to stay within a specified distance
from a specified ``anchor point". It is shown that, if the anchor points
of the boundary particles are related to their reference positions by a
constant deformation matrix A belonging to a certain open set (whose
closure comprises all 2 x 2 matrices which, when applied to the reference
configuration, produce a configuration that is compatible with the
constraint (i) and also with the condition that disks must not overlap),
then the free energy per particle, related to the partition function in
the usual way, has a well-defined thermodynamic limit. It is also shown
that the free energy per particle is a Lipschitz-continuous function of
the matrix A and that it satisfies the integral inequality characterizing
quasi-convex functions.
Schryvers, Dominique (Nick)
Nano- and microstructures characterized by transmission electron microscopy
View Abstract
The present lecture will focus on some typical examples showing the strength and variety of different transmission electron microscopy modes for the characterization of nano- and microstructures that play an important role in the shape memory behaviour of materials undergoing martensitic phase transformations. A chronology of research will be followed, starting with early work on pretransitional modulations in Ni-Al observed by atomic resolution and a series of stress induced martensitic structures surrounding a micro-crack in the thin edge of a TEM sample. The nucleation and growth of the Ni5Al3 phase in the austenite as well as martensite matrix will be discussed after which the emphasis will shift to muliple microtwinned martensite structures as well as bending martensite needles in Ni-Al. In a second part the focus will be on the Ni-Ti system, in which the refinement of the atomic structure of the R-phase and the Ni4Ti3 precipitates will be documented, the determination of strain and concentration fields surrounding these precipitates will be shown as well as the measurement of the elastic constants of these precipitates. To finish some ideas for future work will be proposed.
Sverak, Vladimir
Liouville theorems for the Navier-Stokes equations
View Abstract
It is well-known that every bounded solution of the heat equation defined in in all space and for all negative times is constant. Is a suitable version of this statement true for the Navier-Stokes equations?
We will discuss what is known about this problem and its connections to other questions. The problem is open in the general 3d situation, but results can be obtained in some interesting special cases.
Székelyhidi, László
The h-principle in fluid mechanics
View Abstract
In the talk we discuss parallels between the Nash-Kuiper theory of isometric embeddings and the weak solutions of the Euler equations of ideal incompressible flows, that were constructed by Scheffer and Shnirelman. In particular the latter, when interpreted in the framework of the former, can be largely simplified and extended.
Tao, Terence
Global regularity of wave maps via harmonic map heat flow
View Abstract
We discuss the use of harmonic map heat flow to renormalise the wave maps equation heat flow, and its ramifications for the global regularity problem.

Participants

Name
Institution
Stefan, AdamsUniversity of Warwick
Stuart, AntmanUniversity of Maryland
John, BallHeriot-Watt University
Peter, BatesMichigan State University
Margaret, BeckHeriot-Watt University
Kaushik, BhattacharyaCalifornia Institute of Technology
Natalia, BochkinaUniversity of Edinburgh
David, BourneUniversity of Glasgow
Yann, BrenierEcole Polytechnique
Ken, BrownHeriot-Watt University
Martin, BurnsUniversity of Strathclyde
Carme, CaldererUniversity of Minnesota
Yves, CapdeboscqUniversity of Oxford
Anthony, CarberyUniversity of Edinburgh
Jack, CarrHeriot-Watt University
Carsten, CarstensenHumboldt-Universität zu Berlin
Isaac, ChenchiahUniversity of Bristol
Kirill, CherednichenkoUniversity of Bath
Miroslav, ChlebikUniversity of Sussex
Adam, ChmajWarsaw University of Technology
Philippe G., CiarletCity University of Hong Kong
Sergio, ContiUniversität Bonn
Daniel, CoutandHeriot-Watt University
Elaine, CrooksUniversity of Swansea
Bernard, DacorognaÉcole Polytechnique Fédérale de Lausanne
Constantine, DafermosBrown University
Fordyce, DavidsonUniversity of Dundee
Penny, DaviesUniversity of Strathclyde
Antonio, DeSimoneScuola Internazionale Superiore di Studi Avanzati
Wlodzimierz, DomanskiPolish Academy of Sciences
Patrick, DondlDurham University
Robert, DouglasAberystwyth University
Dugald, DuncanHeriot-Watt University
Angel, DuranUniversity of Valladolid
J. Chris, EilbeckHeriot-Watt University
Maria J, EstebanUniversité Paris-Dauphine
Jozsef, FarkasUniversity of Stirling
Gilles A., FrancfortUniversité Paris-Nord
Gero, FrieseckeTU Munich
Michael, GrinfeldUniversity of Strathclyde
Mark, HaskinsImperial College London
Duvan, HenaoUniversity of Oxford
Viet Ha, HoangNanyang Technological University
Gunnar, HornigUniversity of Dundee
Peter, HornungUniversity of Bath
Cheng-Hsiung, HsuNational Central University,Taiwan
Wen-Guei, HuNational Chiao Tung University
Richard, JamesUniversity of Minnesota
Abdelouahab, KademUniversity of Setif
Agnieszka, KalamajskaUniversity of Warsaw
Biswajit, KarmakarMax Planck Institute
Abdul Rahim, KhanKing Fahd University of Petroleum and Minerals
Robin, KnopsHeriot-Watt University
Gabriel, KochUniversity of Chicago
Robert V., KohnNew York University
Gerasim, KokarevUniversity of Leeds
Michalis, KontovourkisMax Planck Institute
Konstantinos, KoumatosUniversity of Oxford
Jan, KristensenUniversity of Oxford
Andrew, LaceyHeriot-Watt University
Ben, LeimkuhlerUniversity of Edinburgh
Gabriel, LordHeriot-Watt University
Alex, MahalovArizona State University
Emmanuel, MaitreUniversité de Grenoble
Maroje, MarohnicUniversity of Zagreb
Karsten, MatthiesUniversity of Bath
Christof, MelcherUniversity of Oxford
Roger, MoserUniversity of Bath
Benson, MuiteUniversity of Oxford
Stefan, MüllerMax Planck Institute
François, MuratUniversité Pierre et Marie Curie (Paris VI)
Camil, MuscaluCornell University
Sergiy, NesenenkoDarmstadt University of Technology
Stefan, NeukammTU München
Igor, NeygebauerNational University of Rwanda
Barbara, NiethammerUniversity of Bonn
Michael, OrtizCalifornia Institute of Technology
Christoph, OrtnerUniversity of Warwick
Felix, OttoMax Planck Institute for Mathematics in the Sciences, Leipzig
Iryna, PankratovaNarvik University College
Robert, PegoCarnegie Mellon University
Oliver, PenroseHeriot-Watt University
Mircea, PetracheScuola Normale Superiore
Andrey, PiatnitskiNarvik University College
Andrey, RekaloHeriot-Watt University
David, RuleUniversity of Edinburgh
Bryan, RynneHeriot-Watt University
Witold, SadowskiUniversity of Warwick
Dominique (Nick), SchryversUniversity of Antwerp
Yasemin, SengulUniversity of Oxford
Gregory, SereginUniversity of Oxford
Hélia, SerranoUniversidad de Castilla-La Mancha
Eugene, ShargorodskyKing's College London
Michael, ShearerNorth Carolina State University
Michael, SingerUniversity College London
Jeyabal, SivaloganathanUniversity of Bath
Valeriy, SlastikovUniversity of Bristol
Marshall, SlemrodUniversity of Wisconsin
Wolfgang, StaubachHeriot-Watt University
Philipp Emanuel, StelzigTU München/Università di Trento
Pawel, StrzeleckiUniversity of Warsaw
Endre, SuliUniversity of Oxford
Vladimir, SverakUniversity of Minnesota
László, SzékelyhidiLeipzig University
Ali, TaheriUniversity of Sussex
Josip, TambacaUniversity of Zagreb
Antoine, TambueHeriot-Watt University
Terence, TaoUniversity of California at Los Angeles
John, TolandIsaac Newton Institute for Mathematical Sciences
Basang, TseringUniversity of Oxford
Igor, VelcicUniversity of Zagreb
David, WagnerUniversity of Houston
Stephen, WatsonUniversity of Glasgow
Tzi-Sheng, YangTunghai University
Ting-Hui, YangTamkang University
Arghir, ZarnescuUniversity of Sussex
Kewei, ZhangUniversity of Swansea