Georgia Institute of Technology

The emergence of travelling waves for reaction-diffusion equations under a co-moving change of coordinates

Mon, 11/30/2009 - 11:00am, Skiles 269

Maria Lopez, Consejo Superior de Investigaciones Cientificas Madrid, Spain        Organizer: Yingfei Yi

We introduce a change of coordinates allowing to capture in a fixed reference frame the profile of travelling wave solutions for nonlinear parabolic equations. For nonlinearities of bistable type the asymptotic travelling wave profile becomes an equilibrium state for the augmented reaction-diffusion equation. In the new equation, the profile of the asymptotic travelling front and its propagation speed emerge simultaneously as time evolves. Several numerical experiments illustrate the effciency of the method.

Spectral methods in Hamiltonian PDE

Mon, 11/16/2009 - 11:00am, Skiles 269

Wei-Min Wang, Universite Paris-Sud, France        Organizer: Yingfei Yi

We present a new theory on Hamiltonian PDE. The linear theory solves an old spectral problem on boundedness of L infinity norm of the eigenfunctions of the Schroedinger operator on the 2-torus. The nonlinear theory develops Fourier geometry, eliminates the convexity condition on the (infinite dimension) Hamiltonian and is natural for PDE.

Quasi Non-Integrable Hamiltonian System and its Applications

Mon, 11/09/2009 - 11:00am, Skiles 269

Dongwei Huang, Tianjin Polytechnic University, China and School of Mathematics, Georgia Tech        Organizer: Yingfei Yi

Many dynamical systems may be subject to stochastic excitations, so to find an efficient method to analyze the stochastic system is very important. As for the complexity of the stochastic systems, there are not any omnipotent methods. What I would like to present here is a brief introduction to quasi-non-integrable Hamiltonian systems and stochastic averaging method for analyzing certain stochastic dynamical systems. At the end, I will give some examples of the method.

Stable sets and unstable sets in positive entropy systems

Mon, 11/02/2009 - 11:00am, Skiles 269

Wen Huang, USTC, China and SoM, Georgia Tech        Organizer: Yingfei Yi

Stable sets and unstable sets of a dynamical system with positive entropy are investigated. It is shown that in any invertible system with positive entropy, there is a measure-theoretically ?rather big? set such that for any point from the set, the intersection of the closure of the stable set and the closure of the unstable set of the point has positive entropy. Moreover, for several kinds of specific systems, the lower bound of Hausdorff dimension of these sets is estimated. Particularly the lower bound of the Hausdorff dimension of such sets appearing in a positive entropy diffeomorphism on a smooth Riemannian manifold is given in terms of the metric entropy and of Lyapunov exponent.

A Hepatitis B virus model with age since infection that exhibits backward bifurcation

Mon, 10/19/2009 - 11:00am, Skiles 269

Redouane Qesmi, York University, Canada and SoM, Georgia Tech        Organizer: Yingfei Yi

Despite advances in treatment of chronic hepatitis B virus (HBV) infection, liver transplantation remains the only hope for many patients with end-stage liver disease due to HBV. A complication with liver transplantation, however, is that the new liver is eventually reinfected in chronic HBV patients by infection in other compartments of the body. We have formulated a model to describe the dynamics of HBV after liver transplant, considering the liver and the blood of areas of infection. Analyzing the model, we observe that the system shows either a transcritical or a backward bifurcation. Explicit conditions on the model parameters are given for the backward bifurcation to be present, to be reduced, or disappear. Consequently, we investigate possible factors that are responsible for HBV/HCV infection and assess control strategies to reduce HBV/HCV reinfection and improve graft survival after liver transplantation.

Fourier's Law, a brief mathematical review - Continued

Mon, 09/28/2009 - 11:00am, Skiles 269

Federico Bonetto, School of Mathematics, Georgia Tech        Organizer: Yingfei Yi

Fourier's Law assert that the heat flow through a point in a solid is proportional to the temperature gradient at that point. Fourier himself thought that this law could not be derived from the mechanical properties of the elementary constituents (atoms and electrons, in modern language) of the solid. On the contrary, we now believe that such a derivation is possible and necessary. At the core of this change of opinion is the introduction of probability in the description. We now see the microscopic state of a system as a probability measure on phase space so that evolution becomes a stochastic process. Macroscopic properties are then obtained through averages. I will introduce some of the models used in this research and discuss their relevance for the physical problem and the mathematical results one can obtain.

This talk continues from last week's colloquium.

Fourier's Law, a brief mathematical review

Mon, 09/21/2009 - 11:00am, Skiles 269

Federico Bonetto, School of Mathematics, Georgia Tech        Organizer: Yingfei Yi

Fourier's Law assert that the heat flow through a point in a solid is proportional to the temperature gradient at that point. Fourier himself thought that this law could not be derived from the mechanical properties of the elementary constituents (atoms and electrons, in modern language) of the solid. On the contrary, we now believe that such a derivation is possible and necessary. At the core of this change of opinion is the introduction of probability in the description. We now see the microscopic state of a system as a probability measure on phase space so that evolution becomes a stochastic process. Macroscopic properties are then obtained through averages. I will introduce some of the models used in this research and discuss their relevance for the physical problem and the mathematical results one can obtain.

Asymptotic dynamics of reaction-diffusion equations in dumbbell domains

Mon, 09/14/2009 - 11:00am, Skiles 269

Jose M. Arrieta, Universidad Complutense de Madrid        Organizer: Yingfei Yi

We study the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain, which, roughly speaking, consists of two fixed domains joined by a thin channel. We analyze the behavior of the stationary solutions (solutions of the elliptic problem), their local unstable manifold and the attractor of the equation as the width of the connecting channel goes to zero.

Bendixson conditions for differential equations in Banach spaces

Mon, 08/24/2009 - 4:30pm, Skiles 255

Qian Wang, School of Mathematics, Georgia Tech        Organizer: Yingfei Yi

The Bendixson conditions for general nonlinear differential equations in Banach spaces are developed in terms of stability of associated compound differential equations. The generalized Bendixson criterion states that, if some measure of 2-dimensional surface area tends to zero with time, then there are no closed curves that are left invariant by the dynamics. In particular, there are no nontrivial periodic orbits, homoclinic loops or heteroclinic loops. Concrete conditions that preclude the existence of periodic solutions for a parabolic PDE will be given. This is joint work with Professor James S. Muldowney at University of Alberta.

The Minimal Period Problem for the Classical Forced Pendulum Equation

Mon, 04/20/2009 - 4:30pm, Skiles 255

Jianshe Yu, Guangzhou University        Organizer: Hao Min Zhou

In the talk I will discuss the periodicity of solutions to the classical forced pendulum equation y" + A sin y = f(t) where A= g/l is the ratio of the gravity constant and the pendulum length, and f(t) is an external periodic force with a minimal period T. The major concern is to characterize conditions on A and f under which the equation admits periodic solutions with a prescribed minimal period pT, where p>1 is an integer. I will show how the new approach, based on the critical point theory and an original decomposition technique, leads to the existence of such solutions without requiring p to be a prime as imposed in most previous approaches. In addition, I will present the first non-existence result of such solutions which indicates that long pendulum has a natural resistance to oscillate periodically.

Building Databases of the Global Dynamics of Multiparameter Systems

Mon, 04/13/2009 - 4:30pm, Skiles 255

Konstantin Mischaikow, Rutgers University        Organizer: Hao Min Zhou

I will discuss new computational tools based on topological methods that extracts coarse, but rigorous, combinatorial descriptions of global dynamics of multiparameter nonlinear systems. These techniques are motivated by the fact that these systems can produce an wide variety of complicated dynamics that vary dramatically as a function of changes in the nonlinearities and the following associated challenges which we claim can, at least in part, be addressed. 1. In many applications there are models for the dynamics, but specific parameters are unknown or not directly computable. To identify the parameters one needs to be able to match dynamics produced by the model against that which is observed experimentally. 2. Experimental measurements are often too crude to identify classical dynamical structures such as fixed points or periodic orbits, let alone more the complicated structures associated with chaotic dynamics. 3. Often the models themselves are based on nonlinearities that a chosen because of heuristic arguments or because they are easy to fit to data, as opposed to being derived from first principles. Thus, one wants to be able to separate the scientific conclusions from the particular nonlinearities of the equations. To make the above mentioned comments concrete I will describe the techniques in the context of a simple model arising in population biology.

Dispersive properties of surface water waves

Mon, 04/06/2009 - 4:30pm, Skiles 255

Vera Mikyoung Hur, MIT        Organizer: Yingfei Yi

I will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. I will begin by giving a precise account of the formulation of the surface water-wave problem and discussion of its distinct features. They include the dispersion relation, its severe nonlinearity, traveling waves and the Hamiltonian structure. I will describe the recent work of Hans Christianson, Gigliola Staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. If time permits, I will explore some open questions regarding long-time behavior and stability.

On capacity allocation in queueing networks

Mon, 03/30/2009 - 4:30pm, Skiles 255

Ton Dieker, ISyE, Georgia Tech        Organizer: Yingfei Yi

Allocation of service capacity ('staffing') at stations in queueing networks is both of fundamental and practical interest. Unfortunately, the problem is mathematically intractable in general and one therefore typically resorts to approximations or computer simulation. This talk describes work in progress with M. Squillante and S. Ghosh (IBM Research) on an algorithm that serves as an approximation for the 'best' capacity allocation rule. The algorithm can be interpreted as a discrete-time dynamical system, and we are interested in uniqueness of a fixed point and in convergence properties. No prior knowledge on queueing networks will be assumed.

Local entropy theory

Mon, 03/23/2009 - 4:30pm, Skiles 255

Xiangdong Ye, University of Science and Technology of China        Organizer: Yingfei Yi

In this talk we will review results on local entropy theory for the past 15 years, introduce the current development and post some open questions for the further study.

On the theory and applications of the longtime dynamics of 3-dimensional fluid flows on thin domains

Fri, 03/13/2009 - 2:00pm, Skiles 255

George Sell, University of Minnesota        Organizer: Yingfei Yi

The current theory of global attractors for the Navier-Stokes equations on thin 3D domains is motivated by the desire to better understand the theory of heat transfer in the oceans of the Earth. (In this context, the thinness refers to the aspect ratio - depth divided by expanse - of the oceans.) The issue of heat transfer is, of course, closely connected with many of the major questions concerning the climate. In order to exploit the tools of modern dynamical systems in this study, one needs to know that the global attractors are "good" in the sense that the nonlinearities are Frechet differentiable on these attractors. About 20 years ago, it was discovered that on certain thin 3D domains, the Navier-Stokes equations did possess good global attractors. This discovery, which was itself a major milestone in the study of the 3D Navier-Stokes equations, left open the matter of extending the theory to cover oceanic-like regions with the appropriate physical boundary behavior. In this lecture, we will review this theory, and the connections with climate modeling, while placing special emphasis on the recent developments for fluid flows with the Navier (or slip) boundary conditions