Stuart S. Antman, University of Maryland,
USA. Nonlinear
evolution equations from solid mechanics. The nonlinear
partial differential equations from solid mechanics furnish a rich and important
source of open problems awaiting analysis. Many of their difficulties are
inherently geometrical, reflecting that the equations must (i) describe
onetoone deformations of regions of Euclidean space, and (ii) meet certain
invariance requirements, which complicate the geometrical descriptions. These
lectures treat a handful of geometrically exact problems in which there is but
one independent spatial variable. These problems are typically governed by
quasilinear parabolichyperbolic systems and generalizations thereof. The
governing equations are briefly by carefully derived. The main emphasis is on
how standard methods of nonlinear analysis, like monotoneoperator theory in the
context of the FaedoGalerkin method, must be significantly modified to
accommodate the intrinsic difficulties of solid mechanics.
Luis
Caffarelli, University of Texas at Austin, USA. Recent
developments in nonlinear PDE.
Equations with non divergence structure rely heavily on perrons
method for the theory of existence and regularity. Therefore, the functional space theory related to conservation laws is not
available to study homogenization properties of equations
in periodic or random media.
We will discuss several examples where issues of non uniquenes, instability and other phenomena arise, as well as some common approaches to treat these problems.
Walter Craig, McMaster University, Canada.
The fluid dynamics of ocean waves.
Lecture 1: surface water waves and tsunamis
Lecture 2: interactions between nonlinear solitary water waves
Because of the enormous earthquake in Sumatra, and the devastating tsunami which followed, I am changing the topic of my minicourse lectures at this year's PASI meeting. I will speak on the following two topics which involve the dynamics of surface water waves. This topic is also of interest to Chilean scientists, I believe,
because of Chile's presence on the tectonically active Pacific Rim.
My first lecture will describe the equations of fluid dynamics for the
free surface above a body of fluid (the ocean surface), and the linearized equations of motion. From this we can predict the travel time of the recent tsunami from its epicenter off of the north Sumatra
coast to the coast of nearby Thailand, the easy coasts of Sri Lanka and south India, and to Africa. We will further formulate the full nonlinear fluid dynamical equations as a Hamiltonian system (Zakharov 1968), we will describe the Greens function and the
DirichletNeumann operator for the fluid domain, along with the harmonic analysis of the theory of their regularity. From an asymptotic theory of scaling transformations we will derive the known Boussinesqlike systems and the KdV and KP equations which govern the asymptotic behavior of tsunami waves over an idealized flat bottom.
When the bottom is no longer assumed to be perfectly flat, a related
theory (Rosales & Papanicolaou 1983)(Craig, Guyenne, Nicholls & Sulem 2004) gives a family of model equations taking this into account.
My second lecture will describe a series of recent results in PDE,
numerical results, and experimental results on the nonlinear interactions of solitary surface water waves. In contrast with the case of the KdV equations (and certain other integrable PDE),
the Euler equations for a free surface do not admit clean ('elastic')
interactions between solitary wave solutions. This has been a classical concern of oceanographers for several decades, but only recently have there been sufficiently accurate and thorough numerical
simulations which quantify the degree to which solitary waves lose
energy during interactions (Cooker, Wiedman & Bale 1997) (Craig, Guyenne, Hammack, Henderson & Sulem 2004). It is remarkable that this degree of 'inelasticity' is remarkably small. I'll describe this work, as well as recent results on the initial value problem which are very relevant to this phenomenon (Schneider & Wayne 2000) (Wright 2004).
Rafael De La Llave, University of Texas at Austin, USA. An
introduction to variational methods in mechanics. There are many
formulations in mechanics in which orbits are selected as minimizers of
functionals. The minicourse will discuss not only problems of existence but
also how to obtain qualitative properties. Some of the methods apply both to
discrete systems as well as to partial differential equations. Part 1: The
classical theory. 1. Tonellis theorem. Lower bounds on entropy of geodesic
flows. 2. Morses theory of broken geodesics. Chaperons proof of Conley 
Zehnder theorem. 3. WeinsteinMoser theorem. Part 2: Instroduction to
Aubry Mather theory. 4. Variational principles for well ordered orbits. 5.
Minimax orbits. 6. Existence of quasiperiodic orbits. 7. Formulation in
terms of invariant measures.
Ivar Ekeland, PIMS,
Canada. Hedonic models in economic theory. Lecture 1: An optimal
matching problem. Lecture 2: Reservation utilities. For certain economic
goods, such as homes or cars, quality is more important than quantity. If the
price of homes or cars decreases, people do not buy more homes or cars, they buy
a better one. Hedonic models take into account this quality component, and the
economic equilibrium for such models raise very interesting mathematical
questions, related to mass transportation.
Jack Hale, Georgia
Tech, USA. Dissipation and Compact Attractors. Compact global
attractors are playing an important role in the discussion of flows in infinite
dimensional spaces. We discuss how dissipation and the analysis of specific
types of systems have led to the present characterization of those dynamical
systems for which there exists the compact global attractor. Specific
illustrations will be taken from delay differential equations, quasilinear
parabolic and hyperbolic partial differential equations.
John MalletParet, Brown University, USA. Dynamics of
lattice dynamical systems. The course will discuss recent
results in the theory of lattice differential equations. Such equations are
continuoustime infinitedimensional dynamical systems (that is, infinite
systems of ODE's) which possess a discrete spatial structure modeled on a
lattice, for example on Z^{d}. As it is seen, even for rather simply
constructed systems a rich variety of dynamical phenomena are present. Of
particular interest are spontaneously generated patterns (for example stripes or
checks), spatial chaos, and travelling front solutions between equilibria which
may either be spatially homogeneous or which exhibit regular patterns. Also of
interest are the effects of anisotropy of the lattice, in particular propagation
failure of fronts, and the effect of random imperfections in the lattice.
Konstantin Mischaikow, Georgia Tech, USA. Dispersal and
SpatialTemporal Heterogeneity. The minicourse will describe a
series of results that are meant to elucidate the role that spatial and temporal
heterogeneity plays in the evolution of dispersal. A starting point is the
assumption that the organisms are in competition, but identical except for the
rate or form of dispersal. It is shown that in the case of spacial heretogeneity
the slower dispersal rate in preferred, though this can be reversed in the case
of appropriate temporal periodicity. It will also discuss the case when the
reaction terms differ to understand relative importance of the above mentioned
results. Using integrodifferential models it will discuss the relative advantage
and/or disadvantage of 'spread'.
Gustavo Ponce, University of
California at Santa Barbara, USA. Periodic and quasiperiodic solutions of
Hamiltonian PDE. The motion of Npoint vortices in the flow plane
is described by a (Hamiltonian) system of ODEs. The same system models the
evolution of Nlines filaments perpendicular to the plane flow. The equation
representing the motion of a selfinteraction vortex filament was derived by in
works of Da Rois, Hama, and ArmHama. (This is related with the Hasimoto
transformation which reduces this equation to the famous cubic 1D Schrodinger
equation). Klein, Majda and Damodaran proposed a system to describe the
interaction of nearly parallel vortex filaments. This model reflects both
equations commented above. In a recent work (jointly with C. E. Kenig and L.
Vega), some results concerning this system are deduced. In particular, (1) local
existence for the associated initial value problem with data which are small
perturbations of perfect parallel filaments is established, (2) assumptions on
the unperturbed configurations and on the initial parameters which guarantee the
existence of global regular solutions (no finite time collapse) are obtained,
and (3) explicit stationary solutions are found.
Takis
Souganidis, University of Texas at Austin, USA. Recent advances to the
homogenization theory of fully nonlinear first and second order
PDE. There has been recently a great interest in the homogenization
(averaging) of fully nonlinear first and second order PDE in random ergodic
media. This is not just a theoretical problem. On the contrary, it has
applications to areas like large deviations of Markov processes, front
propagation and phase transitions, material science, etc. Although there is a
well developed theory for periodic and almost periodic settings, passing to
stationary ergodic environments poses several challenges due to the inherited
lack of compactness. New methods and techniques need to be developed. The
first lecture will be devoted to a review of the classical homogenization theory
for (nonlinear) first and second order PDE. Some of the recent advances as
well as remaining open problems will be presented in the second lecture.
Yingfei Yi, Georgia Tech, USA. On almost automorphic
phenomena of differential equations. Almost automorphy is a notion
first introduced by S. Bochner in 1955 to generalize the almost periodic one. It
is proved to be a fundamental notion in characterizing multifrequency phenomena
and their generating dynamical complexity in dynamical systems especially those
generated from differential equations. The lectures will give an introduction of
the topic along with some discussions on related problems arising in both
monotone and nonmonotone differential systems.
James Yorke,
University of Maryland, USA. Prevalence, a mathematical concept of "almost
every" in Banach spaces. It is often valuable to speak of typical
behavior of functions, and the lectures will review the ideas behing this
theory, and will give applications. Several applications will be discussed
including a current interest of ours in the theory of observation. When a
laboratory experiment (like a moving fluid) is oscillating chaotically, the
state of the experiment is revealed to the scientist only by simultaneously
measuring a limited number m of physical variables in the experiment, such as
fluid flow rates at different points, or temperatures. Socalled "embedding"
techniques have been developed where in the chaotic attractor can apparently
sometimes be reconstructed. Is the number of variables m large enough to
reconstruct the dynamics? The goal is to justify these embedding methods, or
rather to what is necessary for them to work. Ruelle and Takens introduced the
notion of measuring the dimension of a chaotic attractor using such ideas. Is
the dimension that we compute representative of the actual dimension of the
attractor? Are Lyapunov exponents that are computed from data real; if several
are computed, which are real and which are numerical artifacts? In The Republic,
Plato has Socrates discussing the very limited nature of observation. He says we
do not see reality but only limited images or shadows of reality. One must use
these shadows to understand reality.
