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Mini-courses   PDF  Print 

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 one-to-one 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 parabolic-hyperbolic systems and generalizations thereof. The governing equations are briefly by carefully derived. The main emphasis is on how standard methods of nonlinear analysis, like monotone-operator theory in the context of the Faedo-Galerkin 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 mini-course 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 Dirichlet-Neumann 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 Boussinesq-like 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 mini-course 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. Weinstein-Moser theorem.
Part 2: Instroduction to Aubry Mather theory.
4. Variational principles for well ordered orbits.
5. Minimax orbits.
6. Existence of quasi-periodic 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, quasi-linear parabolic and hyperbolic partial differential equations.

John Mallet-Paret, Brown University, USA.
Dynamics of lattice dynamical systems.
The course will discuss recent results in the theory of lattice differential equations. Such equations are continuous-time infinite-dimensional dynamical systems (that is, infinite systems of ODE's) which possess a discrete spatial structure modeled on a lattice, for example on Zd. 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 Spatial-Temporal Heterogeneity.

The mini-course 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 quasi-periodic solutions of Hamiltonian PDE.

The motion of N-point vortices in the flow plane is described by a (Hamiltonian) system of ODEs. The same system models the evolution of N-lines filaments perpendicular to the plane flow. The equation representing the motion of a self-interaction vortex filament was derived by in works of Da Rois, Hama, and Arm-Hama. (This is related with the Hasimoto transformation which reduces this equation to the famous cubic 1-D 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 multi-frequency 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 non-monotone 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. So-called "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.

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