Friday, January 18, 2008 at 3:00 PM in HA 330
Adrian Tudorascu, Georgia Institute of Technology
Pressureless Euler/Euler-Poisson systems in the Wasserstein space
We shall present recent results on the existence, uniqueness
and regularity of action-minimizing solutions to the pressureless Euler-Poisson system with uniform background charge in spatial dimension one. Also, we shall discuss the existence of "sticky particles" solutions to the initial-value problem for 1-d pressureless Euler/Euler-Poisson systems. As conjectured by various authors (E, Rykov, and Sinai; Brenier and Grenier), the solution to pressureless Euler satisfies the Oleinik entropy condition. Central to our approach is a flow-map formula relating the velocity of an absolutely continuous curve in the Wasserstein space (of probability measures) to the family of optimal maps that rearrange a given reference probability into the measures on the curve. Joint work with: W. Gangbo and T. Nguyen.
Faculty, students and post-docs are encouraged to attend.
Dr. Tudorascu is a candidate for an assistant professorship in the mathematics department.Friday, February 1, 2008 at 3:00 PM in HA 330
Mark Hoefer, National Institute of Standards and Technology , Boulder, Colorado
Dispersive Shock Waves and Their Interactions
Thunder, the crack of a whip, and the boom heard from a jet plane surpassing the speed of sound are familiar occurrences in human experience and all result from the generation of "classical" shock waves. This talk will focus on a very different type of shock wave that propagates through dispersive media such as the superfluidic Bose-Einstein condensate (BEC) and defocusing, nonlinear photonic crystals. These shock waves cannot be heard but have been observed in recent experiments and represent a relatively new and exciting area of physical and applied mathematical research. In contrast to the well known theory of classical shock waves in a compressible fluid where dissipation plays the dominant role in regularizing the shock solution and a strong limit exists, the propagation of a dispersive shock wave (DSW) through a dispersive fluid requires a dispersive regularization where only a weak limit exists. Whitham averaging theory will be used to study dispersive shock waves in the context of the Nonlinear Schrodinger equation. Applications to physically relevant problems including recent BEC blast wave experiments, DSW interactions, and the piston shock wave problem will be discussed.
Faculty, students and post-docs are encouraged to attend.
Dr. Hoefer is a candidate for an assistant professorship in the mathematics department.
Tea will be served at 2:30pm before the seminar, in HA 243.Monday, February 11, 2008 at 3:00 PM in HA 335
Jim Kelliher, Brown University
Vanishing viscosity in the presence of a boundary
The behavior of an incompressible fluid as we let its viscosity approach zero is poorly understood when a boundary is present. Understanding this behavior is of both theoretical and practical interest. The most natural question to ask is whether in this vanishing viscosity limit solutions to the Navier-Stokes equations, which describe an incompressible viscous fluid, converge in the energy norm to a solution to the Euler equations, which describe an incompressible inviscid fluid. Under the assumption that the viscous fluid remains stationary on the boundary (no-slip boundary conditions) this is one of the oldest and most important questions in mathematical fluid mechanics, and one whose full solution does not appear likely to arrive anytime soon. I will describe some of the existing partial results concerning the vanishing viscosity limit and related problems, and will mention some potentially tractable approaches to extend some of these partial results.
Faculty, students and post-docs are encouraged to attend.
Dr. Kelliher is a candidate for an assistant professorship in the mathematics department.
Tea will be served at 2:30pm before the seminar, in HA 243.Tuesday, February 12, 2008 at 3:00 PM in HA 335
William Ott, Courant Institute of Mathematical Sciences
Dissipative homoclinic loops and rank one chaos
I will discuss my recent work with Qiudong Wang on the existence of strange attractors with SRB measures for periodic perturbations of differential equations that admit homoclinic loops. Our work is based on the recent theory of rank one maps developed by Wang and Young.
Faculty, students and post-docs are encouraged to attend.
Dr. Ott is a candidate for an assistant professorship in the mathematics department.
Tea will be served at 2:30pm in HA 243.Wednesday, February 13, 2008 at 3:00 PM in HA 330
Thomas Ivey, College of Charleston
Cable knot solutions of the vortex filament flow
A naive model of vortex filament motion in an ideal fluid leads to an integrable nonlinear evolution equation, known as the localized induction approximation or vortex filament flow (VFF), closely related to the cubic focussing nonlinear Schroedinger (NLS) equation. In particular, spatially closed filaments may be constructed from solutions the AKNS scattering system for certain periodic NLS potentials, characterized in terms of their Floquet spectra. In this talk, I will discuss joint work with Annalisa Calini, describing how to generate a family of closed VFF solutions of increasing topological complexity via a sequence of deformations of the Floquet spectrum of the multiply-covered circle. We prove that every step in this sequence corresponds to constructing a cable on previous filament; moreover, the cable\'s knot type (which is invariant under the evolution) can be read off from data generating the deformation sequence.Wednesday, March 26, 2008 at 3:00 PM in HA 330
Anne Catlla, Duke University
Mathematical modeling of interactions between cells in the brain
Our brains are composed of networks of cells, including neurons and glial cells. While the significance of neurons has been established by biologists, the role of glial cells is less understood. One hypothesis is that glial cells facilitate neural communication in nearby neurons, while suppressing communication among more distant neurons via a reaction-diffusion process. I consider this proposed mechanism using
partial and ordinary differential equation models. By analyzing the ordinary differential equation model, I can determine a condition for this hypothesis to hold. I then compare the results of this analysis with simulations of the partial differential equation model and discuss the biological implications.Wednesday, April 9, 2008 at 3:00 PM in HA 330
Jason Metcalfe, UNC Chapel Hill
Strichartz estimates for wave equations on Schwarzschild black hole
This is a joint work with J. Marzuola, D. Tataru, and M. Tohaneanu. We prove Strichartz estimates for wave equations on Schwarzschild black hole backgrounds. This is done by combining some local energy estimates with a global-in-time outgoing parametrix for
small perturbations of the d'Alembertian. Particular care needs to be taken near the regions of trapping, namely the event horizon and the photon sphere.Wednesday, April 16, 2008 at 3:30 PM in HA 330
Karen Daniels, NC State
Not-so-Continuum Behaviors in Granular Material
From bowls of nuts to eroding soil, granular materials are all around us. While for many years they have been modelled using continuum equations, some of their properties and dynamics challenge this approach. In this talk, I will both introduce some characteristic features of granular materials (such as force chains) and describe recent results from several experiments in progress in my lab in the Physics Department: granular acoustics, size-segregation, and fault-like failure.
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Seminar Organizer: Dmitry Zenkov