(Sept.7) R. Grauer Singular structures in fluids, plasmas and nonlinear optics---
The formation of singular structures in incompressible hydrodynamic and related systems is still a controversial issue. Up to now, numerical simulations do not show a coherent picture, the main reason being the insufficient resolution due to limited computing resources. Here we investigate such problems using adaptive mesh refinement where each grid is integrated using an upwind-scheme combined with a second order projection method. Our findings are that the magnetohydrodynamic equations in two and three dimensions show only exponential growth of vorticity and current density (an effective resolution of $4096^3$ mesh points could be reached). This depletion of nonlinearity results from the formation of current sheets. The growth of vorticity and current density is consistent with a scaling ansatz for the stream functions. The simulations for the three dimensional incompressible Euler equations indicate the formation of a finite time singularity (here, an effective resolution of $2048^3$ mesh points was achieved). In addition, we present results concerning the question of multisplitting in media with anisotropic dispersion. Simulations corresponding to $8192^2 \times 12288$ will be presented. From a numerical point of view we discuss the details of the AMR implementation and our impressions on different integration methods (including recent central schemes) with emphasis on future projects concerning magnetic reconnection.

(Sept.14) J. Colin Surface Instability of Stressed Solids ---
The development of surface roughness has been observed in a number of stressed materials: thin films on substrates, superalloys, multifilamentary conductors for high pulsed magnetic fields, etc. In this work, the morphological stability of a solid under stress has been first studied considering the energy variation induced by the free surface fluctuations of the solid. The time evolution of these fluctuations has been then investigated when surface diffusion is actived.

(Sept.20) Fred Weissler Finite time blow up of complex valued, spatially periodic solutions of the KdV equation ---
We show that certain complex-valued solutions of the Korteweg-de Vries equation blow up in finite time. For example, this is the case if the initial value is a large multiple of exp(ix). This behavior is in marked contrast to real valued solutions, all of which are global in time. Our results extend to a large class of nonlinear dispersive equations. (joint work with Jerry Bona)

(Sept.28) K. Trivisa On the Navier Stokes' equations for compressible reacting flow with large discontinuous initial data: ---
The existence, regularity and large time behavior of global discontinous solutions to the Navier Stokes' equations for compressible, reacting mixture with large discontinuous initial data are discussed. The model under consideration describes dynamic combustion. We show that the velocity and the internal decay asymptotically, while the discontinuities of the density, the pressure and the reactant mass fraction persist even asymptotically. Finally, we identify the necessary and sufficient conditions on the initial data for the complete burning as t-->infinity. (This joint work with G.-Q Chen and D. Hoff.)

(Oct.5) Mark Williams Singular pseudodifferential operators, symmetrizers, and shocks: ---
We introduce a class of singular pseudodifferential operators depending on wavelength that can be used to study several of the singular quasilinear hyperbolic systems arising in nonlinear geometric optics, the study of incompressible limits, and perhaps elsewhere.  The operators permit simultaneous microlocalization in both slow and fast variables.  We sketch how the operators can be used to prove the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength as the wavelength tends to zero.

(Oct.12) J. Stalker Estimates for the Wave Equation with a Potential: ---
I will talk about recent work on the equation $$ {partial^2 u\over\partial t^2} - {partial^2 u\over\partial x^2_1} - \cdots {partial^2 u\over\partial x^2_n} = V(x_1,\ldots,x_n) u. $$ This equation typically arises as the linearization of a non-linear problem, but I will discuss only the linear problem in this talk. The estimates I will discuss are pointwise estimates, dispersive inequalities (decay estimates), and Strichartz estimates. The scale invariant potentials are those which are homogeneous of order~$-2$. This is in many respects the hardest case to analyze. The results I will present are joint work with Sergiu Klainerman and Matei Machedon and with A. Shadi Tahvildar-Zadeh and Fabrice Planchon.

(Oct.19) Ed Seidel Solving Einstein's Equations on the Grid: Using Supercomputers to Collapse Gravitational Waves, Collide Black Holes: ---
Einstein's equations of general relativity govern such exotic phenomena as black holes, neutron stars, and gravitational waves. Unfortunately they are among the most complex in physics, and require very large scale computational power --- which we are just on the verge of achieving --- to solve. I will motivate and describe these equations, and the worldwide effort to develop advanced computational tools to solve them in their full generality for the first time since they were written down nearly a century ago. I will focus on applications of these tools to the study of black hole collisions, considered to be promising sources of observable gravitational waves that may soon be seen for the first time by the worldwide network of gravitational wave detectors (LIGO, VIRGO, GEO, and others) currently under construction. I will show movies of large scale simulations of black hole formation and black hole collisions. I will discuss the new scientific research possibilities opened up by the emergence of high performance computing as a tool for basic physics research, and the impact of the study of Einstein's equations on high performance computing technology. Finally, I will describe the emerging "Grid" of networked computational resources, and the techniques being developed to exploit it for scientific computing.

(Oct.26) N. Masmoudi Existence Results for Some Polymeric Flows: ---
Polymeric liquids can be modeled by some kinetic equations (Elastic Dumbbell model, Rigid Dumbbell model) or by some macroscopic models (such as the Oldroyd model for non-Newtonian fluids). We present here some global existence result of weak solutions for these models.

(Nov.2) Luc Tartar Microlocal measures and their use in partial differential equations: ---
Microlocal methods have been introduced in the early 70s, independently by SATO and by Lars H\"ORMANDER, I believe. These initial ideas are not adapted to solving questions related to the partial differential equations from Continuum Mechanics or Physics, despite the use of catch words like ``propagation of singularities''. What is really being proved in H\"ORMANDER's approach are results of propagation of microlocal regularity, and that is hardly a relevant question in Continuum Mechanics or Physics; another defect, shared by the theory of pseudo-differential operators, is that one assumes that coefficients of the partial differential equations are infinitely differentiable; another defect is that systems of partial differential equations are not considered. I was led to introducing microlocal measures for a different question of Homogenization (and I give to this term a much more general meaning than most other people do), and therefore I called these new objects H-measures. Then, I tried to use them for problems of propagations of oscillations and concentration effects, which are important questions for Continuum Mechanics or Physics. It worked, and it gave a new point of view on old things like the Geometrical Optics approximation for the wave equation; my results are not like the classical approach using an amplitude and a phase, as H-measures do not see any phase and describe limits when frequency tends to infinity, but the classical approach can at best show that there are solutions of the wave equations for which the energy almost propagates along light rays (away from caustics), while my results says that all oscillating solutions have this property (and no problem with caustics). The method can be used for systems, and it also gives a qualitative answer to a few puzzling facts (for mathematicians) about some physicists' computations; one of them is that some computations done in a periodic framework can be used with success in a nonperiodic setting; another one is the classical puzzle of Quantum Mechanics about particles which sometimes are waves, and it suggests a different point of view, where there are only waves satisfying some partial differential equations (of hyperbolic type) but the limit for infinite frequency leads to an ordinary differential equation for a microlocal measure, which can be interpreted as describing some kind of ``particles''. The same objects have been introduced independently by Patrick GERARD, for a different reason. For variants using one characteristic length, I will describe my approach and the independent idea of Patrick GERARD (semi-classical measures), together with the later work, and mistake, of Pierre-Louis LIONS and Thierry PAUL. A lot remains to be done, and possible developments will be mentioned.

(Nov. 9) S-M Sun Non-existence of truly solitary waves in water with small surface tension: ---
The talk considers permanent capillary-gravity waves on the free surface of a two-dimensional incompressible, inviscid fluid of finite depth. It has been known for more than a century that there are solitary waves on the surface of the flow with or without surface tension, whose approximations are the solutions of the famous Korteweg-De Vries (KdV) equation. The solitary waves are single hump waves that decay to zero exponentially at infinity. However, in this talk we will show rigorously that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity, if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth-order nonlinear ordinary differential equation, when the solution is extended to the complex plane. Then the non-existence of truly solitary waves is obtained using a contradiction on this constant.

(Nov. 15) E. Tadmor High Resolution Methods with Unresolved Small Scales: ---
Spontaneous evolution of different scales leads to challenging difficulties of stable computations with unresolved small scales. We discuss how modern algorithms address these difficulties: detection of edges, high-resolution reconstruction of piecewise-smooth data between edges, and the interplay between the theory and computational aspects of high-resolution in low regularity spaces. We focus our attention on two particular examples. We discuss non-oscillatory central schemes for computing piecewise smooth solutions of hyperbolic conservation laws, Hamilton-Jacobi equations and related nonlinear problems. The high-resolution of these locally based algorithms is gained by coupling nonlinear edge detectors, and turning the piecewise polynomials reconstructions into the direction of smoothness. The second part is an example for global methods. Here we discuss the reconstruction of piecewise smooth data from (pseudo-) spectral global information. To avoid spurious Gibbs oscillations and to regain the superior exponential accuracy we proceed in two separate steps: a detection procedure which identifies (the location and amplitude of finitely many) edges, followed by a family of spectrally accurate mollifiers which recover the data between those edges. We demonstrate applications from CFD (-- formation of shocks), geometrical optics, MHD problems, image processing and more.

(Nov. 16) K. Hoffman Stability Results for Elastic Rods: ---
The variational structure of the elastic rod model will be exploited to predict which equilibrium configurations are stable, that is, correspond to local minima. Three techniques to determine the stability of the equilibrium configurations will be discussed and illustrated using two examples: the twisted elastic loop and an untwisted elastic loop with inherent curvature. The stability properties of equilibria depend upon the material parameters of the rod.

(Nov. 30) G. Kostakis Electromagnetic scattering by a homogeneous chiral obstacle in a chiral environment: ---
Chiral media are examples of media responding with both electric and magnetic polarization to either electric or magnetic excitation.They can be characterized by a set of constitutive relations in which the electric and magnetic fields are coupled.We consider the scattering of time-harmonic electromagnetic waves by a bounded three-dimensional chiral obstacle surrounded by a chiral environment.We assume that the obstacle and its host medium are homogeneous.This allows us to reduce the arising transmission problem to an integral equation over the interface between the obstacle and its surroundings.We study the solvability of this integral equation by using Fredholm theory.This is a join work with C.Athanasiadis and I.Stratis from the univ. of Athens in Greece.

(Dec. 6) John Weeks Current Induced Step Dynamics and Pattern Formation on Crystal Surfaces: ---
We review experimental and theoretical work on two dimensional step patterns that form on crystal surfaces in response to nonequilibrium driving forces. Steps are viewed as a set of non crossing interacting lines. The derivation and solution of effective differential equations describing their coupled motion will be discussed.
 

(Jan. 11) Lance Leslie Theory and Applications of a New Coupled Atmospheric Model: ---
The first part of this seminar will focus on recent work in the development of the speaker's coupled atmospheric-landsurface-ocean-hydrology-pollutant transport model system. This system has a range of unique features and has been applied to problems ranging from severe local weather, through routine numerical weather prediction on the short to medium range, as a tool for agricultural prediction on a seasonal scale, and most recently as a coupled AGCM-OGCM as part of a major new initiative.
The second part of the presentation will describe recent work in four key research areas: 4D assimilation of data from the rapidly expanding new observational platforms (satellites, radar etc.); new accurate and stable modeling algorithms that allow very long term integrations; two-way coupled systems; and applications of the system to problems including a supercell simulation, soil ersoion, soil moisture and temperature prediction and some very recent work on understanding and prediction of ENSO related problems in the Indian Ocean. The last problem is of particular interest because of the overwhelming emphasis hitherto on the Pacific Ocean.
 

(Jan. 18) Steve White The Density Matrix Renormalization Group: ---
A crucial ingredient in the modern theoretical study of strongly correlated electron systems, such as the high temperature superconductors, is the numerical simulation of quantum lattice models. One of the most recent and useful techniques for these simulations is the density matrix renormalization group (DMRG). In the first half of the talk the basic DMRG method will be introduced using a simple "particle in a box" system as an example. In the second half, results will be presented for models representing high temperature superconductors. In particular, the competing tendencies for the formation of striped patterns of holes versus Cooper pairs and superconductivity will be explored.