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PDE/Applied Math Seminar - Abstracts Fall 2002

(Sept. 12) Steven A. Gabriel Analyzing Forecasting Strategies in Retail Electrical Power Markets

In this presentation, we describe recent work analyzing load forecasting strategies for electrical power retailers. We analyze both the risk and reward of these strategies using a Monte Carlo simulation approach as well as provide results from an optimization perspective based on a mixed integer programming formulation. This is joint work with several co-authors.

(Sept. 26) Sylvia Serfaty Optimal energy estimates and patterns for some models for micromagnetics

We study the asymptotic limit of an energy-functional related to the theory of micromagnetics. It involves S^1-valued vector fields whose limits are divergence-free vector fields which jump along line singularities. We give some explicit formulas for the cost of the jumps, and show how they are optimal through 2D patterns called cross-tie walls in physics. These are joint works with Tristan Riviere, and Francois Alouges and Tristan Riviere.

(Oct. 3) Daniel M. Anderson Modeling the solidification of ternary alloys in mushy layers

We describe a model for the solidification of a ternary (three component) alloy cooled from below at a planar boundary. The modeling extends previous theory for binary alloy solidification by including a conservation equation for the additional solute component and coupling the conservation equations for heat and species to equilibrium relations from the ternary phase diagram. We focus on growth conditions under which the solidification path (liquid line of descent) through the ternary phase diagram gives rise to two distinct mushy layers. A primary mushy layer, which corresponds to solidification along a liquidus surface in the ternary phase diagram, forms above a secondary (or cotectic) mushy layer, which corresponds to solidification along a cotectic line in the ternary phase diagram. These two mushy layers are bounded above by a liquid layer and below by a eutectic solid layer. The mathematical model is comprised of a system of partial differential equations in each layer, coupled through interfacial boundary conditions between each layer. We obtain a one-dimensional similarity solution and investigate numerically the role of the control parameters on the growth characteristics. In the special case of zero solute diffusion and zero latent heat an analytical solution can be obtained. We compare our predictions with previous experimental results. Finally, we discuss the potentially rich convective behavior anticipated for other growth conditions.

(Oct. 17) Steve Shkoller The Navier-Stokes equations with a free-surface and surface tension

The incompressible Navier-Stokes equations (NSE) on time-dependent domains arise in the study of either a single fluid with a free-surface, or in the context of multi-phase flows wherein the motion of two or more immiscible fluids is considered. In the presence of surface tension, the mathematical analysis (as well as the numerical computation) is a challenging task, because the mean curvature vector, given in the surface tension term, appears to induce too much derivative loss on the boundary; in fact, Newton iteration will fail to converge, and other iteration schemes must be constructed. For short-time well-posedness of the NSE, I will present a technique, based on new types of energy laws of the linearized system.

(Oct. 31) Hans Engler Limiting self-similar solutions for integro-differential equations

Consider a solution, defined on R^n, of a linear partial integrodifferential equation u_t = a_0 Au + a*Au, where A is the Laplacian, a_0 is nonnegative, and a is a scalar kernel. The topic of the talk is the question whether solutions converge to a limiting profile, u(x,t) ~ k(t)U(x/m(t)), as t goes to infinity: what are the possible limits, what properties of the original equation determine the limit, in what sense does the convergence occur? It turns out that there are two extreme cases, corresponding to the wave equation and the heat equation, and a one parameter continuum of intermediate cases. The growth behavior of the primitive of the kernel a determines which case applies.

(Nov. 18) Graeme Milton The exact photonic band structure for a class of periodic media: a spectral problem

An exact formula for the photonic band structure (and hence the dispersion relation) is found for a certain class of periodic media. These media have the special property that the spatially fluctuating part of their complex electrical permittivity and magnetic permeability, when Fourier transformed, vanishes in half of Fourier space. As a result the band structure is proved to be exactly the same as for a homogeneous medium. Or put another way, the operator associated with the equations has a trivial solution for its eigenvalues, but not its eigenfunctions.

(Dec. 5) John Ball Varying volume fractions

In elasticity models of materials which can undergo phase transformations involving a change of shape, different phases or variants of the material are associated with energy wells comprising corresponding sets of deformation gradients. Zero-energy microstructures can then be identified with gradient Young measures supported on these sets and the corresponding phase fractions are the masses of these measures restricted to each well.

The talk will describe a general result for gradient Young measures that allows one to vary the volume fractions of such microstructures, together with applications, open questions and related results.

(Dec. 6) John Ball The regularity of minimizers in elasticity

It is a major open problem of nonlinear elasticity theory to decide whether or not energy minimizers are smooth or can have singularities. Although some singular minimizers related to phase transformations or fracture are known, there remains the possibility that there is a large class of realistic stored-energy functions for which minimizers are smooth. On the other hand there is apparently not a single example known of a stored-energy function for which smoothness can be proved for arbitrary large boundary data.

The talk will survey what is known about this problem, and about related questions such as satisfaction of the Euler-Lagrange equation and uniform positivity of the Jacobian.

(Dec. 12) Silvère Akamatsu Dynamics of 2D eutectic solidification fronts

Directional solidification of a eutectic alloy in thin sample gives rise to 2D lamellar patterns in the solid due to diffusion controlled growth dynamics. The stability range of the well-known stationary periodic symmetrical state is bounded by instabilities corresponding to thresholds of symmetry breaking bifurcations. I will present an experimental study of these bifurcations.

Last modified: Tue Dec 10 14:24:44 EST 2002