Plenary Speakers, Titles and Abstracts
In the
Fourth International Conference on Dynamical Systems and Differential Equations

Speaker:   Michael Ghil,,   (University of California Los Angeles, USA)
  Successive Bifurcations in the Oceans' Wind-Driven Circulation
  The large-scale flow of the mid-latitude atmosphere and oceans is governed by systems of PDEs that approximate the Euler and Navier-Stokes equations in the presence of rotation and stratification. These PDEs include the quasi-geostrophic and shallow-water equations in two and three dimensions (2-D and 3-D), as well as the so-called primitive equations in 3-D; the latter are used, in discretized form, to formulate the general circulation models used in numerical weather prediction and climate studies.

We illustrate the mathematical issues that arise in solving these PDEs by the specific problem of the oceans' wind-driven circulation. This circulation is dominated by a large, anticyclonic and a smaller, cyclonic gyre in each mid-latitude basin on Earth. These two gyres are induced by the shear in the winds that cross the respective ocean basins. They share the eastward extension of western boundary currents, such as the Gulf Stream or Kuroshio. The boundary currents and eastward jets carry substantial amounts of heat and momentum. The jets also contribute to mixing in the oceans by their "whiplashing" oscillations and the detachment of eddies from them.

We study the low-frequency variability of this double-gyre circulation, for time-constant and purely periodic wind stress. Both analytical and numerical methods of nonlinear dynamics are applied in our study. Symmetry-breaking bifurcations occur, from steady to periodic and aperiodic flows, as wind stress increases or dissipation decreases. The first bifurcation is of pitchfork or perturbed-pitchfork type, depending on the model's degree of realism. Two types of oscillatory instabilities arise by supercritical Hopf bifurcation, with periods of a few months and a few years, respectively. Numerical evidence points to homoclinic orbits that connect high- and low-energy branches of steady-state solutions. The results are compared with decade-long in situ and more recent, satellite observations of three ocean basins, the North and South Atlantic, and the North Pacific, and their significance for climate variability is discussed. 

Speaker:  Antonio,  (Italy)
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,,  (Applied and Comp. Math, Caltech, USA)
  Singularity Formation in 3-D Vortex Sheets
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 the integral 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,  (University of Tokyo, Japan)
Title:   A variational characterization of travelling waves in quasi-periodic media
    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-periodic inhomogeneity, 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,  (University of Minnesota, USA)
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,   (University of Minnesota, USA)
   Macroscopic Equations from Microscopic Models:  The Bacterial Example
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.,   (North Carolina State University, USA)
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,   (National Tech. University, Greece) 
  Nonlinear boundary value problems with multivalued terms
   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, ,   (University of Wisconsin, USA)
Spatial heteroclinics for a semilinear elliptic PDE
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, ,  (IMPA, Brazil)
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.