Seminars and Colloquia by Series

Mixed Models for Traffic Flow and Crowd Dynamics

Series
PDE Seminar
Time
Tuesday, October 12, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Prof. Benedetto Piccoli – Rutergs University –
Motivated by applications to vehicular traffic, supply chains and others, various continuous models for traffic flow on networks were recently proposed. We first present some results for theory of conservation laws on graphs. Then we focus on recent mixed models, involving continuous-discrete spaces and ode-pde systems. Then a time evolving measures approach is showed, with applications to crowd dynamics.

Nonlinear Schroedinger equation with a Magnetic Potential

Series
PDE Seminar
Time
Tuesday, October 5, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Prof. Shijun Zheng – Georgia Southern University –
The dissipative mechanism of Schroedinger equation is mathematically described by the decay estimate of solutions. In this talk I mainly focused on the use of harmonic analysis techniques to obtain suitable time decay estimates and then prove the local wellposedness for semilinear Schroedinger equation in certain external magnetic field. It turns out that the scattering with a potential may lead to understanding of the wellposedness of NLS in the presence of nonsmooth or large initial data. Part of this talk is a joint work with Zhenqiu Zhang.

Turbulence: a walk on the wild side

Series
PDE Seminar
Time
Tuesday, September 28, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Prof. Predrag Cvitanović – Physics, Georgia Institute of Technology –
In the world of moderate Reynolds number, everyday turbulence of fluids flowing across planes and down pipes a velvet revolution is taking place. Experiments are almost as detailed as the numerical simulations, DNS is yielding exact numerical solutions that one dared not dream about a decade ago, and dynamical systems visualization of turbulent fluid's state space geometry is unexpectedly elegant. We shall take you on a tour of this newly breached, hitherto inaccessible territory. Mastery of fluid mechanics is no prerequisite, and perhaps a hindrance: the talk is aimed at anyone who had ever wondered why - if no cloud is ever seen twice - we know a cloud when we see one? And how do we turn that into mathematics? (Joint work with J. F. Gibson)

An unusual duality principle for fully nonlinear equations

Series
PDE Seminar
Time
Tuesday, September 21, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Professor Scott Armstrong – University of Chicago
We discuss how to solve a Hamilton-Jacobi-Bellman equation ``at resonance." Our characterization is in terms of invariant measures and is analogous to the Fredholm alternative in the linear case.   

Small solutions of nonlinear Schrodinger equations near first excited states

Series
PDE Seminar
Time
Tuesday, August 31, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Professor Tai-Peng Tsai – Department of Mathematics, University of British Columbia –
Consider a nonlinear Schrodinger equation in $R^3$ whose linear part has three or more eigenvalues satisfying some resonance conditions. Solutions which are initially small in $H^1 \cap L^1(R^3)$ and inside a neighborhood of the first excited state family are shown to converge to either a first excited state or a ground state at time infinity. An essential part of our analysis is on the linear and nonlinear estimates near nonlinear excited states, around which the linearized operators have eigenvalues with nonzero real parts and their corresponding eigenfunctions are not uniformly localized in space. This is a joint work with Kenji Nakanishi and Tuoc Van Phan.The preprint of the talk is available at http://arxiv.org/abs/1008.3581

A uniqueness result for the continuity equation in dimension two

Series
PDE Seminar
Time
Tuesday, April 20, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Gianluca Crippa – University of Parma (Italy)
In the simplest form, our result gives a characterization of bounded,divergence-free vector fields on the plane such that the Cauchyproblem for the associated continuity equation has a unique boundedsolution (in the sense of distribution).Unlike previous results in this directions (Di Perna-Lions, Ambrosio,etc.), the proof does not rely on regularization, but rather on adimension-reduction argument which allows us to prove uniqueness usingwell-known one-dimensional results (it is indeed a variant of theclassical method of characteristics).Note that our characterization is not given in terms of functionspaces, but using a qualitative property which is completelynon-linear in character, namely a suitable weak formulation of theSard property.This is a joint work with Giovanni Alberti (University of Pisa) andStefano Bianchini (SISSA, Trieste).

Fokker-Planck equation on graphs with finite number of vertices

Series
PDE Seminar
Time
Tuesday, April 13, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yao Li – Georgia Tech
Fokker-Planck equation is a linear parabolic equation which describes the time evolution of of probability distribution of a stochastic process defined on a Euclidean space. Moreover, it is the gradient flow of free energy functional. We will present a Fokker-Planck equation which is a system of ordinary differential equations and describes the time evolution of probability distribution of a stochastic process on a graph with a finite number of vertices. It is shown that there is a strong connection but also substantial differences between the ordinary differential equations and the usual Fokker-Planck equation on Euclidean spaces. Furthermore, the ordinary differential equation is in fact a gradient flow of free energy on a Riemannian manifold whose metric is closely related to certain Wasserstein metrics. Some examples will also be discussed.

On Landau Damping

Series
PDE Seminar
Time
Tuesday, April 6, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Clement Mouhot – Ecole Normale Superieure
Landau damping is a collisionless stability result of considerable importance in plasma physics, as well as in galactic dynamics. Roughly speaking, it says that spatial waves are damped in time (very rapidly) by purely conservative mechanisms, on a time scale much lower than the effect of collisions. We shall present in this talk a recent work (joint with C. Villani) which provides the first positive mathematical result for this effect in the nonlinear regime, and qualitatively explains its robustness over extremely long time scales. Physical introduction and implications will also be discussed.

The eigenvalue problem of singular ergodic control

Series
PDE Seminar
Time
Tuesday, March 30, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ryan Hynd – University of California
We discuss a non-linear eigenvalue problem where the eigenvalue has a natural control-theoretic interpretation as an optimal "long-time averaged cost." We also show how such problems arise in financial market models with small transaction costs.

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