Seminars and Colloquia by Series

On the Hamilton-Jacobi variational formulation of the Vlasov equation

Series
CDSNS Colloquium
Time
Monday, November 7, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Phil Morrison – Univ. of Texas at Austin
The Vlasov-Poisson and Vlasov-Maxwell equations possess various variational formulations1 or action principles, as they are generally termed by physicists. I will discuss a particular variational principle that is based on a Hamiltonian-Jacobi formulation of Vlasov theory, a formulation that is not widely known. I will show how this formu- lation can be reduced for describing the Vlasov-Poisson system. The resulting system is of Hamilton-Jacobi form, but with nonlinear global coupling to the Poisson equation. A description of phase (function) space geometry will be given and comments about Hamilton-Jacobi pde methods and weak KAM will be made.Supported by the US Department of Energy Contract No. DE-FG03- 96ER-54346.H. Ye and P. J. Morrison Phys. Fluids 4B 771 (1992).D. Pfirsch, Z. Naturforsch. 39a, 1 (1984); D. Pfirsch and P. J. Morrison, Phys. Rev. 32A, 1714 (1985).

Generic properties of scalar parabolic equations

Series
CDSNS Colloquium
Time
Friday, April 1, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Genevieve Raugel – Universite Paris-Sud
In this talk, we generalize the classical Kupka-Smale theorem for ordinary differential equations on R^n to the case of scalar parabolic equations. More precisely, we show that, generically with respect to the non-linearity, the semi-flow of a reaction-diffusion equation defined on a bounded domain in R^n or on the torus T^n has the "Kupka-Smale" property, that is, all the critical elements (i.e. the equilibrium points and periodic orbits) are hyperbolic and the stable and unstable manifolds of the critical elements intersect transversally. In the particular case of T1, the semi-flow is generically Morse-Smale, that is, it has the Kupka-Smale property and, moreover, the non-wandering set is finite and is only composed of critical elements. This is an important property, since Morse-Smale semi-flows are structurally stable. (Joint work with P. Brunovsky and R. Joly).

Liquid-crystals are intermediate phases between solid and liquid states

Series
CDSNS Colloquium
Time
Monday, March 14, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Weishi Liu – University of Kansas –
They may flow like fluids but under constraints of mechanical energies from their crystal aspects. As a result, they exhibit very rich phenomena that grant them tremendous applications in modern technology. Based on works of Oseen, Z\"ocher, Frank and others, a continuum theory (not most general but satisfactory to a great extent) for liquid-crystals was formulated by Ericksen and Leslie in 1960s. We will first give a brief introduction to this classical theory and then focus on various important special settings in both static and dynamic cases. These special flows are rather simple for classical fluids but are quite nonlinear for liquid-crystals. We are able to apply abstract theory of nonlinear dynamical systems upon revealing specific structures of the problems at hands.

Lorenz flow and random effect

Series
CDSNS Colloquium
Time
Friday, March 11, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Weiping Li – Oklahoma State University
In this talk, I will explain the correspondence between the Lorenz periodic solution and the topological knot in 3-space.The effect of small random perturbation on the Lorenz flow will lead to a certain nature order developed previously by Chow-Li-Liu-Zhou. This work provides an answer to an puzzle why the Lorenz periodics are only geometrically simple knots.

Ramified optimal transportation in geodesic metric spaces

Series
CDSNS Colloquium
Time
Monday, March 7, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Qinglan Xia – University of California Davis
An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensional distance", on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.

Recent Progress in Delay-Differential Equations

Series
CDSNS Colloquium
Time
Monday, December 6, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
John Mallet-Paret – Brown University
We examine a variety of problems in delay-differential equations. Among the new results we discuss are existence and asymptotics for multiple-delay problems, global bifurcation of periodic solutions, and analyticity (or lack thereof) in variable-delay problems. We also plan to discuss some interesting open questions in the field.

Markov Perfect Nash Equilibria: Some Considerations on Economic Models, Dynamical Systems and Statistical Mechanic

Series
CDSNS Colloquium
Time
Monday, November 29, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Federico Bonetto – Georgia Tech
Modern Economic Theory is largely based on the concept of Nash Equilibrium. In its simplest form this is an essentially statics notion. I'll introduce a simple model for the use of money (Kiotaki and Wright, JPE 1989) and use it to introduce a more general (dynamic) concept of Nash Equilibrium and my understanding of its relation to Dynamical Systems Theory and Statistical Mechanics.

Normally Elliptic Singular Perturbations and persistence of homoclinic orbits

Series
CDSNS Colloquium
Time
Monday, November 22, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Nan Lu – Georgia Tech
We consider a dynamical system, possibly infinite dimensional or non-autonomous, with fast and slow time scales which is oscillatory with high frequencies in the fast directions. We first derive and justify the limit system of the slow variables. Assuming a steady state persists, we construct the stable, unstable, center-stable, center-unstable, and center manifolds of the steady state of a size of order $O(1)$ and give their leading order approximations. Finally, using these tools, we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems.

Localized planar patterns

Series
CDSNS Colloquium
Time
Monday, November 15, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Björn Sandstede – Brown University
In this talk, I will discuss localized stationary 1D and 2D structures such as hexagon patches, localized radial target patterns, and localized 1D rolls in the Swift-Hohenberg equation and other models. Some of these solutions exhibit snaking: in parameter space, the localized states lie on a vertical sine-shaped bifurcation curve so that the width of the underlying periodic pattern, such as hexagons or rolls, increases as we move up along the bifurcation curve. In particular, snaking implies the coexistence of infinitely many different localized structures. I will give an overview of recent analytical and numerical work in which localized structures and their snaking or non-snaking behavior is investigated.

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