Applied/GPS: Difference between revisions
mNo edit summary |
No edit summary |
||
| Line 63: | Line 63: | ||
|- | |- | ||
|Apr 30 | |Apr 30 | ||
| | |Antonio Ache | ||
| | |''Asymptotics for solutions of parabolic systems'' | ||
|- | |- | ||
|May 7 | |May 7 | ||
| Line 129: | Line 129: | ||
The system of equations describing nonlinear and weakly dispersive internal waves between two layers of inviscid fluids bounded by top and bottom walls is known to be mathematically ill-posed despite the fact that physically stable internal waves have been observed. We obtain a stable nonhydrostatic model for this system and illustrate our results with solitary waves. | The system of equations describing nonlinear and weakly dispersive internal waves between two layers of inviscid fluids bounded by top and bottom walls is known to be mathematically ill-posed despite the fact that physically stable internal waves have been observed. We obtain a stable nonhydrostatic model for this system and illustrate our results with solitary waves. | ||
===Monday, April 30: Antonio Ache=== | |||
''Asymptotics for solutions of parabolic systems'' | |||
We will start by discussing the statement and some key of the key ideas involved in a | |||
result proved by Leon Simon in 1983 about the solution of certain elliptic variational | |||
problems by following the curves of "steepest descent". More importantly we will discuss | |||
applications to 3 problems | |||
(1) The unique asymptotic limit problem for gradient flows, | |||
(2) The uniqueness of the cone structure at infinity for Ricci-flat manifolds, | |||
(3) The unique tangent problem for harmonic maps. | |||
Some of the central aspects of the theory include the Lojasiewicz-Simon inequality, | |||
monotonicity formulas and the Liapunov-Schmidt reduction. | |||
Revision as of 20:53, 26 April 2012
GPS Applied Mathematics Seminar
The GPS (Graduate Participation Seminar) is a weekly seminar by and for graduate students. If you're interested in presenting a topic or your own research, contact the organizers: Sarah Tumasz, Li Wang, and Zhennan Zhou.
All seminars are on Mondays from 2:25 to 3:15 in B235 Van Vleck. Speakers should aim for their talk to last no longer than 45 minutes.
Spring 2012
| date | speaker | title |
|---|---|---|
| Feb 6 | Zhan Wang | Hydroelastic solitary wave and its application in ice problem |
| Feb 13 | Sarah Tumasz | What is Topological Mixing? |
| Feb 20 | Zhennan Zhou | Semi-classical analysis of Schrodinger equation with periodic potential |
| Feb 27 | Li Wang | A Jin-Xin-Glimm scheme for hyperbolic conservation laws |
| Mar 5 | ||
| Mar 12 | ||
| Mar 19 | Xiaoqian Xu | Interpolation of Linear Operators |
| Mar 26 | Yu Sun | Matching-pursuit/split-operator-Fourier-transform simulations of quantum dynamics |
| Apr 2 | Spring Break | |
| Apr 9 | Lei Li | Landau–Zener formula |
| Apr 16 | Leland Jefferis | Wigner Decomposition Methods for High Frequency Symmetric Hyperbolic Systems |
| Apr 23 | Tete Boonkasame | A stable nonhydrostatic shallow-water model for two-layer flows |
| Apr 30 | Antonio Ache | Asymptotics for solutions of parabolic systems |
| May 7 |
Abstracts
Monday, February 6: Zhan Wang
Hydroelastic solitary wave and its application in ice problem
The study of deformation of a floating ice sheet has applications in polar regions where ice cover is used for roads or runways and there is an interesting on the safe use of these transport links. We use the full potential model to study the forced and unforced dynamics of hydroelastic waves near the minimum phase speed in deep water. This is a joint work with Paul Milewski and J.-M. Vanden-Broeck.
Monday, February 13: Sarah Tumasz
What is Topological Mixing?
In this talk, I hope to provide an answer to the question 'What is Topological Mixing?' This will be a very introductory level talk, and I'll attempt to give intuitive, rather than technical, definitions. I'll discuss the basics of mixing, topology, and braids (as they apply) and then give some examples of applications.
Monday, February 20: Zhennan Zhou
Semi-classical analysis of Schrodinger equation with periodic potential
abstract In this talk, I plan to (at least try to) convince you that for Schrodinger equations, we need more analytic insight to build up reliable numerical schemes. I will introduce basic asymptotic methods for semiclassical limits, and explain why the situation changes dramatically for highly oscillatory potentials.
Monday, February 27: Li Wang
A Jin-Xin-Glimm scheme for hyperbolic conservation laws
We present a class of numerical scheme(called Jin-Xin-Glimm scheme) for scalar conservation law, which will be extended to hyperbolic system later. This scheme contains the advantages of Jin-Xin relaxation scheme, which is free of Riemann solver, and Glimm scheme, which is a sharp shock capturing method. This is a joint work with Frederic Coquel, Shi Jin and Jian-guo Liu.
Monday, March 19: Xiaoqian Xu
Interpolation of Linear Operators
I'll try to introduce the definition of linear operators of strong type(p,q), weak type(p,q) and restricted weak type (p,q). I'll also give the definition of weak Lp spaces and general Lorentz spaces. In my view, there is no relationship between this and applied math.
Monday, April 9: Lei Li
Landau-Zerner formula
Landau-Zerner formula describes the transition rate for a 2-level quantum mechanical system. It has many applications in various models. In BO approximation, it's the basis for the surface-hopping algorithm. I'll briefly talk about how to derive this formula following the original idea of Zener proposed in 1930s.
Monday, April 16: Leland Jefferis
Wigner Decomposition Methods for High Frequency Symmetric Hyperbolic Systems
We develop a method for computing physical observable for general symmetric hyperbolic systems using the high frequency limit of the Wigner Transform. Symmetric hyperbolic systems can represent many physically relevant systems of partial differential equations (PDE) such as Maxwell's equations, the elastic equations and the acoustic equations. Using the Wigner transform to analyze high frequency behavior has been done in special cases, but here we extend its application to the fully genearl case.
Monday, April 23: Tete Boonkasame
A stable nonhydrostatic shallow-water model for two-layer flows
The system of equations describing nonlinear and weakly dispersive internal waves between two layers of inviscid fluids bounded by top and bottom walls is known to be mathematically ill-posed despite the fact that physically stable internal waves have been observed. We obtain a stable nonhydrostatic model for this system and illustrate our results with solitary waves.
Monday, April 30: Antonio Ache
Asymptotics for solutions of parabolic systems
We will start by discussing the statement and some key of the key ideas involved in a
result proved by Leon Simon in 1983 about the solution of certain elliptic variational
problems by following the curves of "steepest descent". More importantly we will discuss
applications to 3 problems
(1) The unique asymptotic limit problem for gradient flows, (2) The uniqueness of the cone structure at infinity for Ricci-flat manifolds, (3) The unique tangent problem for harmonic maps.
Some of the central aspects of the theory include the Lojasiewicz-Simon inequality, monotonicity formulas and the Liapunov-Schmidt reduction.