(Jan 28) J. Maddocks, Lausanne: Global curvature and ideal shapes ---  The notion of global curvature of a space curve will be introduced, and then used to discuss certain ideal shapes of knots, as have been investigated within the context of DNA.

(Feb 8) T.-P. Liu, Stanford: Well-Posedness Theory for Hyperbolic Conservation Laws ---  The existence theory for the conservation laws was established by James Glimm in 1965. This and the recent L1 stability analysis finally settle the well-posedness theory. Besides presenting the major ideas of interaction functionals and the generalized entropy functionals, possible future research directions will also be speculated.

(Feb 15) Andrea Bertozzi, Duke: Undercompressive Shocks in Driven Film Flow --- Recent experiments in thermal/gravity driven thin film flow show a transition, as the film thickness is varied, from unstable fronts that finger to stable fronts. This phenomenon can be explained by the shock dynamics of a scalar conservation law lubrication model of the experiment. For thicker films, the advancing front evolves into an `undercompressive' capillary shock structure which is stable to contact line perturbations. A phase space study of the dynamical system describing traveling waves shows that the undercompressive front is an accumulation point for a family of compressive waves with the same speed.

(Feb 18) R. Krasny, Ann Arbor: The Onset of Chaos in Vortex Sheet Roll-Up --- Computations are presented for the roll-up of a planar and an axisymmetric vortex sheet into a vortex pair and a vortex ring, respectively. The results indicate the onset of chaos due to resonance phenomena in the vortex core and a heteroclinic tangle at the rear of the vortex ring. We conclude that the vortex sheet flow resembles a perturbed integrable Hamiltonian system although the chaos is induced here by self-sustained oscillations in the vortex core rather than external forcing. This is joint work with Monika Nitsche (University of New Mexico).

(Mar 4) D. Hoff, Bloomington: The global attractor for the Navier-Stokes equations of one-dimensional, compressible flow --- First I will describe a well-posedness theory for the system of the title: unique, global solutions with large, discontinuous initial data and with large external forces are shown to exist, with densities bounded above and below, uniformly in time. These pointwise bounds lead to a kind of hyperbolic/parabolic dissipation, sufficient for the asymptotic compactness required in the study of the global dynamics. The existence of a compact global attractor then follows, and the regularity properties of its elements can be related to the aforementioned "hyperbolic" dissipation. I will conclude with a discussion of the dimension of the attractor and the number of determining modes.

(Apr 8) C. Elmer, NIST: Analysis and computation of traveling wave solutions for bistable nonlinear differential-difference equations --- In this talk we consider traveling wave solutions to nonlinear bistable differential-difference equations. We study propagation failure, lattice anisotropy, bifurcation points in the wave speed and detuning parameter relation, and step-like solution profiles.    Traveling wave solutions satisfy a two-point boundary value problem but standard BVP codes do not handle the discrete difference operator. We present two relaxation techniques to deal with the delay terms.  The first is fixed point iteration which allows us to solve equations where we represent the bistable nonlinearity with a piecewise linear function. The second is a variation of Newton's method which allows us to solve equations containing smooth bistable nonlinearities.  We also present a series of numerical examples along with current and future research directions.

(Apr 15) T. Grava, UMCP & SISSA: The theory of the Whitham equations and dispersive shock waves--- We study the Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion considering initial data rapidly decreasing at infinity. The solution of the Cauchy problem is characterized by the appearance of an expanding region filled with rapid modulated oscillations which are called dispersive shock waves. It turns out that for given x and t these oscillations are approximately described by the so called g-phase solution of KdV. The modulation in time and space of the oscillation parameters is described by the Whitham equations. We make an overview of recent results in the theory of the Whitham equations. We explain their variational formulation. Using this approach we solve the Cauchy problem for the Whitham equations and we determine the conditions for the existence of a global solution.

(Apr 22) I. Gamba, Austin:

(Apr 29) T. Gill, Howard Univ: Relativistic Theory of Particles and Fields --- We provide an overview of a new generalization of classical mechanics and electrodynamics, which is simple, technically correct, and requires no additional work for the quantum case. We first formulate a new implementation of Einstein's two postulates which fixes the proper-time of the source for all observers. This leads to a new invariance group and an equivalent formulation of Maxwell's equations left covariant under the action of this group. This approach is distinct from the Minkowski implementation in that it does not require the use of time as the fourth component of a four vector. It also allows us to construct a general theory of directly interacting relativistic particles in which the Lorentz force contains the expected dissipative term corresponding to Newton's third law; and need not be put in as an adjunct to the theory. The wave equation derived from Maxwell's equations contains an additional term, first order in the proper-time which arises instantaneously with acceleration. This shows explicitly that the long-sought origin of radiation reaction is inertial resistance to changes in particle motion. The field equations carry intrinsic information about the velocity and acceleration of the particles in the system, so that our theory introduces an arrow for the proper-time of the global system.

(May 6) J.-G. Liu, UMCP: Convergence of Point Vortex Method for 2-D Vortex Sheet --- I will present a convergence result of the point vortex method to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity is a finite Radon measure of distinguished sign and the kinetic energy is locally bounded. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the PVM don't have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.