LCDS Seminars 2005

Monday, February 14, 2005

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Tremendous advances have been made in cataloguing the structures and motifs of genetic regulatory networks. However, our understanding of the implications of these structures on the dynamic response of the network is more limited. I will discuss our efforts to build a simple scalable model based on the Nitrogen Catabolite Repression (NCR) circuit in Saccharomyces cerevisiae and provide a mathematical analysis of its dynamics. In particular, I will touch on five topics:
1) An introduction to the biology
2) Comments on mathematical results that allow one to make statements about the dynamics of a system without detailed knowledge of the nonlinear interactions of the system.
3) Our model and its comparison with experimental data.
4) Mathematical theorems about the asymptotic behavior of a sub-circuit of the NCR circuit.
5) Some tantalizing numerical results.

Monday, February 28, 2005

Lefschetz Center for Dynamical Systems Seminar

Abstract :   The pressure term has always created difficulties in treating the Navier-Stokes equations of incompressible flow, reflected in the lack of a useful evolution equation or boundary conditions to determine it. In joint work with Jian-Guo Liu and Jie Liu, we show that in bounded domains with no-slip boundary conditions, the pressure can be determined in such a way that it is strictly dominated by viscosity. As a consequence, in a general domain with no-slip boundary conditions, we can treat the Navier-Stokes equations as a perturbed vector diffusion equation instead of as a perturbed Stokes system. We illustrate the advantages of this view by providing simple proofs of (i) the stability of a difference scheme that is implicit only in viscosity and explicit in both pressure and convection terms, requiring no solutions of stationary Stokes systems or inf-sup conditions, and (ii) existence and uniqueness of strong solutions based on the difference scheme.

Monday, March 7, 2005

Lefschetz Center for Dynamical Systems Seminar

Abstract :   I will discuss the problem of speed selection for traveling wave solutions to a class of scalar second order reaction diffusion equations. One of the main problems is whether the minimal wave speed is determined by linear or nonlinear considerations. Another problem is the actual determination the minimal speed. I will present the geometric characterization of the minimal speed, as well as some recent work on its variational characterization.

Wednesday, March 9, 2005

Special PDE & Lefschetz Center for Dynamical Systems Seminar

Monday, March 14, 2005

Special PDE & Lefschetz Center for Dynamical Systems Seminar

Lefschetz Center for Dynamical Systems Seminar

•  Abstract :   As an experimental law, Darcy's law plays an important role in the investigation on compressible flows through porous medium. It was conjectured that Darcy's law can be verified by basic balance laws of mechanics time asymptotically. Previous attempts are able to justify the conjecture for small smooth flows away from vacuum. I will show a proof to the conjecture valid for all physical isentripic flows. The approach is then generalized to uniformly bounded adiabatic flows.

Monday, March 21, 2005

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We show that for Smale solenoids the fractal (Hausdorff or box) dimensions of all stable slices are the same. Our approach should lead to a proof that the fractal dimension of hyperbolic sets can be computed by adding those of their stable and unstable slices.

Monday, April 11, 2005

Lefschetz Center for Dynamical Systems Seminar

Monday, April 18, 2005

Brown University Graduate School Dissertation Defense Information Lefschetz Center for Dynamical Systems Seminar

April 21 - April 23, 2005

Workshop "Infinite-dimensional dynamical systems: structures and patterns"

(Bernold Fiedler, John Mallet-Paret)

Friday, June 17, 2005

Special LCDS/PDE Seminar

Abstract :   In my introduction I will describe lattice differential equations (LDE) and their similarities and differences with partial differential equations (PDE). In particular, I will focus on reaction-diffusion equations with bistable nonlinearities. I will then describe the phenomena of pinning and crystallographic pinning (CP), which do not occur in the PDE setting but which do occur in the LDE setting. The remainder of the talk will be devoted to outlining a proof that crystallographic pinning is generic in the horizontal and vertical directions on a two dimensional lattice.

Monday, October 31, 2005

Special Division/Lefschetz Center for Dynamical Systems Colloquium

Abstract :   Flow separation is a major cause of performance loss in engineering devices such as diffusers, airfoils and jet engines. In a landmark 1904 paper on boundary layers, L. Prandtl derived a criterion for flow separation in steady two-dimensional incompressible flows. Despite widespread effort, however, no unsteady or three-dimensional extension of Prandtl's criterion has emerged in the fluid dynamics literature.

In this talk, I discuss a recent extension of Prandtl's criterion using invariant manifold concepts from dynamical systems. I show numerical and experimental results confirming the extended separation criterion for unsteady and three-dimensional flows. I also discuss applications of the new criterion to flow control

Monday, November 7, 2005

Special Joint LCDS and PDE Seminar

Abstract :   In this talk, I will discuss several micro-macro coupling models for viscoelastic fluids. The focus will be on the transport and the induced elastic stress.