Speaker:
Antonio Giorgilli, antonio@matapp.unimib.it, (Italy)
Location & Time:
Cameron Auditorium
(1:45 - 2:45,
Friday, May 24, 2002)
Title:
Localization of
energy in FPU chains
Abstract:
The celebrated model of
Fermi, Pasta and Ulam is revisited with the aim of investigating the relaxation of the
system towards equipartition. We perform an extensive numerical exploration starting with
all the energy $E$ initially concentrated on the lowest frequency mode, as in the original
FPU report. We produce evidence of the existence of two well separated different time
scales. In a short time we observe the formation of a packet of low frequency modes which
share most of the total energy,while the high frequency modes appear to be frozen in a
exponentially decreasing distribution. After this fast relaxation a second phase begins
during which the energy flows at a definitely slower rate towards the higher frequency
modes. The numerical exploration shows that the fraction of the total number $N$ of modes
that take active part in the formation of the initial packet is a function of the specific
energy $E/N$. Furthermore, the speed of the energy flow during the second phase
tends quite rapidly to zero with $E/N$. Thus, this phenomenon appears to be relevant for
the thermodynamic limit.
Speaker:
Thomas Y. Hou, hou@ama.caltech.edu, (Applied and Comp. Math, Caltech, USA)
Location & Time:
Cameron Auditorium
(9:00
- 10:00, Saturday, May 25, 2002)
Title:
Singularity
Formation in 3-D Vortex Sheets
Abstract:
One of the
classical examples of hydrodynamic instability occurs when two fluids are separated by a
free surface across which the tangential velocity has a jump discontinuity. This is called
Kelvin-Helmholtz instability. Kelvin-Helmholtz instability is a fundamental instability of
incompressible fluid flow at high Reynolds number. The idealization of a shear
layered flow as a vortex sheet separating two regions of potential flow has often been
used as a model to study mixing properties, boundary layers and coherent structures of
fluids. It is well known that small initial perturbations on a vortex sheet may grow
rapidly due to Kelvin-Helmholtz instability. The problem is ill-posed in the Hadamard
sense. Most analytical studies of vortex sheet singularity to date rely heavily on
complexifying the interface variables. It is not clear how to generalize this
technique to 3-D vortex sheets in a natural way. In a joint work with G. Hu and P. Zhang,
we study the singularity of 3-D vortex sheets using a new approach. First, we derive a
leading order approximation to the boundary integral equation governing the 3-D vortex
sheet. This leading order equation captures the most singular contribution of theintegral
equation. Moreover, after applying a transformation to the physical variables, we found
that this leading order 3-D vortex sheet equation de-generates into a two-dimensional
vortex sheet equation in the direction of the tangential velocity jump. This rather
surprising result confirms that the tangential velocity jump is the physical driving force
of the vortex sheet singularities. We show that the singularity type of the
three-dimensional problem is similar to that of the two-dimensional problem. Moreover, we
introduce a generalized Moore's approximation to 3-D vortex sheets. This model equation
captures the same singularity structure of the full 3-D vortex sheet equation, and it can
be computed efficiently using Fast Fourier Transform. This enables us to perform
well-resolved calculations to study the generic type of 3-D vortex sheet singularities. We
will provide detailed numerical results to support the analytic prediction, and to reveal
the generic form of the vortex sheet singularity.
Speaker:
Hiroshi Matano, matano@ms.u-tokyo.ac.jp, (University of Tokyo, Japan)
Location & Time:
Cameron Auditorium
(9:00
- 10:00, Sunday, May 26, 2002)
Title:
A Variational Characterization of
Travelling Waves in Quasi-periodic Media
Abstract:
Travelling
waves in heterogeneous media have gained much attention in the past decade in various
fields of science such as ecology, physiology and combustion theory. Previously most of
the mathematical studies were focused on spatially periodic cases, and little was known
about the nature of traveling waves in spatially aperiodic media. This is in cotrast with
the case of temporally varying media, for which there is a comprehensive study by Shen
(1999). Recently I have introduced the notion of travelling waves in spatially
almost-periodic media, including quasi-periodic ones as special cases. The concept is a
natural extension of the classical notion of travelling waves, and I have discussed
existence, uniqueness and stability of those travelling waves. To be more precise, a
travelling wave is defined to be a solution whose current profile depends continuously on
its current landscape. Here, roughly speaking, the current profile means the shape of the
solution (at each time moment) viewed from the postion of the ``front", and the
current landscape refers to the spatial environment viewed from that positon. In this
lecture I will discuss two variational problems associate with travelling waves for
nonlinear diffusion equations. The first is the mini-max characterization of propagation
speed, which is introduced by Volpert et al for homogeneous problems and later extended to
periodic problems by Heinze, Papanicolaou and Stevens in the case of bistable
nonlinearity, and by Berestycki and Hamel in the case of KPP nonlinearity. This method
enables one to obtain fine rigorous estimates of propagation speed. One can extend this
method to quasi-priodic problems with bistable nonlinearity, but it raises a very
intriguing question, which I will discuss in my lecture. The second is concerned with the
minimal speed of travelling waves for KPP type equations. As conjectured by
Kawasaki-Shigesada (1986),and later proved by Hudson-Zinner (1995 for 1-dim) and
Berestycki-Hamel (2002 for higher dim), the minimal speed is characterized by a positive
eigenfunction of a certain elliptic eigenvalue problem. In the case of
quasi-periodicinhomogeneity, I can formulate a similar eigenvalue problem on a torus, but
this eigenvalue problem may not have a positive eigenfunction because of the degeneracy of
the differential operator. Despite this difficulty, this eigenvalue problems can still
play a principal role for proving the existence of pseudo-travelling waves and to obtain
estimates of their minimal speed.
Speaker:
Wei-Ming Ni, ni@math.umn.edu, (University of Minnesota, USA)
Location & Time:
Cameron Auditorium
(8:00
- 9:00, Saturday, May 25, 2002)
Title:
Diffusion and Cross-Diffusion in Pattern Formation: from Single Equations to Systems
Abstract:
In this lecture I plan to explain how diffusions and cross-diffusions are used
in modeling pattern formation, both from a practical point of view and from a mathematical
point of view in studying the potential of these equations and systems. Examples will be
used to illustrate various approaches.
Speaker:
Hans Othmer, othmer@math.umn.edu, (University of Minnesota, USA)
Location & Time:
Cameron Auditorium
(11:00 -
12:00, Friday, May 24, 2002)
Title:
Macroscopic Equations from Microscopic Models: The Bacterial Example
Abstract:
In
recent years it has become clear that a reductionist approach to biological systems is
generally inadequate; complex systems cannot be understood simply by dissecting them into
their components. In a number of systems detailed information on transduction of
extracellular signals into a behavioral change is now available, and a current problem
is to incorporate this information into macroscopic equations that describe
population-level behavior. We use bacterial chemotaxis as an example to describe several
different levels at which this can be done, and show how a general theory can be
developed. We also describe a new computational approach aimed at bridging the gap between
microscopic models and macroscopic evolution.
Speaker:
C.V. Pao, cvpao@math.ncsu.edu, (North Carolina State University, USA)
Location & Time:
Cameron Auditorium
(8:00
- 9:00, Monday, May 27, 2002)
Title:
Method of Upper and Lower Solutions for Reaction-Diffusion
Abstract:
The method of upper and lower solutions and its associated monotone iteration are powerful
tools for the treatment of reaction diffusion systems both analytically and numerically.
The analytical treatment includes the existence and uniqueness of the time dependent
solution, existence, multiplicity, and bifurcation of steady-state solutions,
stability or instability of a steady state solution, and blow-up and quenching of
time-dependent solutions. Numerically, this method can be used to develop stable and
reliable computational algorithms for numerical solutions of both time-dependent and
steady state solutions, convergence of discrete solutions to their corresponding
continuous solutions, and bounds and error estimates of the discrete solutions. The
aim of this talk is to give an overview of the method, and to describe some of the
basic ideas and main elements for various types of reaction diffusion systems, including
systems with nonlinear or nonlocal boundary conditions, systems with finite or infinite
time delays, and periodic solutions of time dependent problems. Applications and numerical
results are given to some specific reaction diffusion equations to illustrate the
practical aspect of the method.
Speaker:
Nikolas Papageorgiou, npapg@math.ntua.gr, (National Tech. University, Greece)
Location & Time:
Cameron Auditorium
(9:00
- 10:00, Monday, May 27, 2002)
Title:
Nonlinear Boundary Value
Problems with Multivalued Terms
Abstract:
We examine
vector nonlinear differential inclusions driven by the p-Laplacian or by more general
differential operators which are not necessarily homogeneous.We impose nonlinear
set-valued boundary conditions which incorporate the Dirichlet, Neumann and periodic
conditions.We prove existence and relaxation results. We also consider evolution equations
with multivalued terms defined in the framework of evolution triples.We obtain
existence theorems for nonmonotone problems, while for the monotone problem we prove the
topological triviality of the solution set.We also prove sensitivity results using the
G-convergence of operators. Finally we examine optimal control problems (sensitivity
analysis and minimax control).
Speaker:
Paul Rabinowitz, rabinowi@math.wisc.edu, (University of Wisconsin, USA)
Location & Time:
Cameron Auditorium
(10:00 -
11:00, Friday, May 24, 2002)
Title:
Spatial Heteroclinics for a
Semilinear Elliptic PDE
Abstract:
The
existence of spatially heteroclinic solutions for a semilinear elliptic PDE in an infinite
cylinder in R^n will be discussed. Minimization arguments will be used to obtain the
solutions.
Speaker:
Marcelo Viana, viana@impa.br, (IMPA, Brazil)
Location & Time:
Cameron Auditorium
(8:00
- 9:00, Sunday, May 26, 2002)
Title:
Dynamics - a Panorama of Recent
Results
Abstract:
There has been
substantial recent progress in the field of Dynamics, from which an understanding of the
typical behaviour of very general systems is emerging, extending the scope of the
classical hyperbolic theory. I'll discuss some of these developments. In particular, I'll
report some very recent results in the theory of Lyapunov exponents of smooth systems and
linear cocycles.