Effective dynamics of the nonlinear Schroedinger equation on large domains

Speaker: 

Zaher Hani

Institution: 

Georgia institute of Technology

Time: 

Tuesday, October 10, 2017 - 3:00pm

Location: 

RH 306

We consider the nonlinear Schroedinger equation posed on a large box of characteristic size $L$, and ask about its effective dynamics for very long time scales. After pointing out some “more or less” trivial time scales along which the effective dynamics can be easily described, we start inspecting some much longer time scales where we notice some non-trivial dynamical behaviors. Particularly, the end goal of such an analysis is to reach the so-called “kinetic time scale”, at which it is conjectured that the effective dynamics is governed by a kinetic equation called the “wave kinetic equation”. This is the subject of wave turbulence theory. We will discuss some recent advances towards this end goal. This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah. 
 

Anomalous diffusion in passive scalar transport

Speaker: 

Gautam Iyer

Institution: 

Carnegie Mellon University

Time: 

Thursday, January 11, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

Consider a diffusive passive scalar advected by a two
dimensional incompressible flow. If the flow is cellular (i.e.\ has a
periodic Hamiltonian with no unbounded trajectories), then classical
homogenization results show that the long time behaviour is an effective
Brownian motion. We show that on intermediate time scales, the effective
behaviour is instead a fractional kinetic process. At the PDE level this
means that while the long time scaling limit is the heat equation, the
intermediate time scaling limit is a time fractional heat equation. We
will also describe the expected intermediate behaviour in the presence
of open channels.
 

Derivation of multi-layered interface system and its application

Speaker: 

Hiroyoshi Mitake

Institution: 

Hiroshima University

Time: 

Thursday, September 21, 2017 - 3:00pm

Host: 

Location: 

410M

 In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation.  By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. 

This is a joint work with H. Ninomiya, K. Todoroki. 

Properties of minimizers of the average-distance problem

Speaker: 

Dejan Slepcev

Institution: 

Carnegie Mellon University

Time: 

Tuesday, October 17, 2017 - 3:00pm

Host: 

Location: 

RH 306

The general average distance problem, introduced by Buttazzo, Oudet, and Stepanov,  asks to find a good way to approximate a high-dimensional object, represented as a measure, by a one-dimensional object. We will discuss two variants of the problem: one where the one-dimensional object is a measure with connected one-dimensional support and one where it is an embedded curve. We will present examples that show that even if the data measure is smooth the nonlocality of the functional can cause the minimizers to have corners. Nevertheless the curvature of the minimizer can be considered as a measure. We will discuss a priori estimates on the total curvature and ways to obtain information on topological complexity of the minimizers. We will furthermore discuss functionals that take the transport along the network into account and model best ways to design transportation networks. (Based on joint works with Xin Yang Lu and Slav Kirov.)

 

Topology of the set of singularities of viscosity solutions of the Hamilton-Jacobi equation

Speaker: 

Albert Fathi

Institution: 

Georgia Institute of Technology

Time: 

Tuesday, December 5, 2017 - 3:00pm

Host: 

Location: 

RH 306

This is a joint work with Piermarco Cannarsa and Wei Cheng. 

We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian. 

The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property is far from generic in the class of semi-concave functions.

We also “identify” the connected components of this set S. 

This work relies on the idea of Cannarsa and Cheng to use the positive Lax-Oleinik operator to construct a global propagation of singularities (without necessarily obtaining uniqueness of the propagation).

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