Overview
The focus of this week will be on general nonlinear wave equations that
are not directly related to the theory of relativity. Techniques developed
for linear and non-linear wave equations arising from Lagrangian theories
have been central tools in the dynamical study of Einstein's equations.
While the mathematical and numerical community working in this eld is very
large, we hope to host a small number of people to build interest in Einstein's
equations.
Participants
as of June 8, 2015
* Indicates
not yet confirmed
|
Full Name
|
University/Affiliation
|
| Thomas Alazard |
Ecole Normale Superieure, Paris |
| Xinliang An |
Rutgers University |
| Nicolas Burq |
Université Paris Sud-Orsay |
| Hans Christianson |
University of North Carolina - Chapel Hill |
| Grigorios Fournodavlos |
University of Toronto |
| Cécile Huneau |
Ecole Normale Superieure |
| Alexander Kiselev |
Rice University |
| Joachim Krieger |
Ecole Polytechniques Federale de Lausanne |
| Andrew Lawrie |
University of California, Berkeley |
| Luis Lehner |
Perimeter Institute |
| Adam Lewis |
University of Toronto |
| Jason Metcalfe |
University of North Carolina - Chapel Hill |
| Andrea Nahmod |
University of Massachusetts - Amherst |
| Sung-Jin Oh |
University of California, Berkeley |
| Benoît Pausader |
Princeton University |
| Natasha Pavlovic |
University of Texas at Austin |
| Nataša Pavlovi? |
University of Texas, Austin |
| Oana Pocovnicu |
Princeton University |
| Fabio Pusateri |
Princeton University |
| Volker Schlue |
University of Toronto |
| Sohrab Shahshahani |
University of Michigan |
| Arick Shao |
Imperial College London |
| Gigliola Staffilani |
Massachusetts Institute of Technology |
| Mihai Tohaneanu |
Georgia Southern University |
| Klaus Widmayer |
New York University |
| Monday,
June 22 |
|
8:30-9:15
|
On site registration and morning coffee |
|
9:15-9:30
|
Welcoming remarks |
|
9:30-10:30
|
Jason Metcalfe, University of North
Carolina - Chapel Hill
Local energy decay for wave equations on nontrapping
asymptotically flat space-times |
|
10:30-11:00
|
Coffee break |
|
11:00-12:00
|
Walter Craig, The Fields Institute
and McMaster University
Birkhoff normal forms for null forms |
|
12:00-2:00
|
Lunch break |
|
2:00-3:00
|
Arick Shao, Imperial College London
Uniqueness Theorems for Waves from Infinity
|
|
3:00-3:30
|
Coffee break |
|
3:30-4:30
|
Thomas Alazard, Ecole Normale Superieure,
Paris
Controllability of water waves |
| Tuesday, June 23 |
|
9:30-10:30
|
Sung-Jin Oh, University
of California, Berkeley
Global well-posedness of the energy critical Maxwell-Klein-Gordon equation
|
|
10:30-11:00
|
Coffee break |
|
11:00-12:00
|
Andrew Lawrie, University
of California, Berkeley
Wave maps on hyperbolic space |
|
12:00-2:00
|
Lunch break |
|
2:00-3:00
|
Hans Christianson,
University of North Carolina - Chapel Hill
Trapping and high frequency resolvent estimates
on warped products |
|
3:00-3:30
|
Coffee break |
|
3:30-4:30
|
Alexandru Ionescu,
Princeton University |
| Wednesday,
June 24 |
|
9:30-10:30
|
Natasha Pavlovic, University
of Texas at Austin
Many Body Dynamics, Nonlinear Dispersive PDE
and Quantum De Finetti Theorems |
|
10:30-11:00
|
Coffee break |
|
11:00-12:00
|
Joachim Krieger, Ecole
Polytechniques Federale de Lausanne
On the critical MKG equation |
|
12:00-2:00
|
Lunch break |
|
2:00-3:00
|
Cécile Huneau,
Ecole Normale Superieure
Stability in exponential time of Minkowski space-time
with a space-like translation symmetry |
|
3:00-3:30
|
Coffee break |
|
3:30-4:30
|
Yannis Angelopoulos,
University of Toronto
Nonlinear waves on extremal black hole spacetimes
and other applications |
| Thursday,
June 25 |
|
09:30-10:30
|
Mihai Tohaneanu,
Georgia Southern University
Global existence for quasilinear wave equations
close to Schwarzschild |
|
10:30-11:00
|
Coffee break |
|
11:00-12:00
|
Gigliola Staffilani,
Massachusetts Institute of Technology |
|
12:00-2:00
|
Lunch break |
|
2:00-3:00
|
Any sessions after 2:00
p.m will be held in the Stewart Library of the Fields Institute |
|
3:00-3:30
|
Coffee break |
|
3:30-4:30
|
Any sessions after 2:00 p.m
will be held in the Stewart Library of the Fields Institute |
Abstracts
Yannis Angelopoulos, University of Toronto
Nonlinear waves on extremal black hole spacetimes and other applications
I will discuss some recent results for nonlinear wave equations on extremal
black hole spacetimes, and for nonlinear wave equations on more general
asymptotically flat spacetimes with growing weights at infinity.
Thomas Alazard, Ecole Normale Superieure,
Paris
Controllability of water waves
The question we discuss in this talk is the following: which water waves
can be generated by blowing on a localized portion of the free surface of
a liquid. Our main result asserts that one can generate any small amplitude,
periodic in x, two-dimensional, gravity-capillary water waves. This is a
result from control theory. More precisely, we study the exact local controllability
of the incompressible Euler equation with free surface. This is a joint
work with Pietro Baldi and Daniel Han-Kwan.
Nicholas Burq, Université Paris Sud-Orsay
Long time dynamics for damped Klein-Gordon equations
For general semilinear Klein-Gordon equations with dissipation we show
that any finite energy radial solution either blows up in finite time or
asymptotically approaches a stationary solution in $H^1\times L^2$. In particular,
any global solution is bounded. The result applies to standard energy subcritical
focusing nonlinearities $|u|^{p-1} u$, $1<p<(d+2)/(d-2)$ as well as
any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz
type. The argument involves both techniques from nonlinear dispersive PDEs
and dynamical systems. This is a joint work with G. Raugel (CNRS and University
Paris-Sud-Orsay) and W. Schlag (University of Chicago)
Hans Christianson, University of North
Carolina - Chapel Hill
Trapping and high frequency resolvent estimates on warped products
Understanding trapping is an important aspect of the study of evolution
equations. For example, black hole geometries in general relativity can
exhibit very complicated trapping. In this talk, I will survey recent resolvent
estimates in the presence of trapping in a very simple geometric situation:
warped product spaces. As with any high-frequency resolvent estimates, there
are many applications, and I will explain a few of these applications.
Walter Craig, The Fields Institute and McMaster
University
Birkhoff normal forms for null forms
Theorems on global existence of small data solutions of nonlinear wave
equations in ${\mathbb R}^n$ depend upon a competition between the time
decay of solutions and the degree of the nonlinearity. Decay estimates are
more effective when inessential nonlinear terms are able to be removed through
a well-chosen transformation. In this talk, we construct canonical transformations
into Birkhoff normal form for the class of wave equations which are Hamiltonian
PDEs and null forms, giving a new proof via canonical transformations of
the global existence theorems for null form wave equations of S. Klainerman,
J. Shatah and F. Pusateri. These results are work-in-progress with A. French
and C.-R. Yang.
Cécile Huneau, Ecole Normale Superieure
Stability in exponential time of Minkowski space-time with a space-like
translation symmetry
In the presence of a translation symmetry, the 3+1 vacuum Einstein equations
reduce to the 2+1 Einstein equations with a scalar field. We work in generalized
wave coordinates. In this gauge Einstein equations can be written as a system
of quasilinear quadratic wave equations. The main difficulty is due to the
weak decay of free solutions to the wave equation in 2+1 dimensions, compared
to
3+1 dimensions. To prove the stability in exponential time, we have to
rely on the particular structure of Einstein equations in wave coordinates.
We also have to carefully chose an approximate solution with a non trivial
behaviour at space-like infinity to enforce convergence to Minkowski space-time
at time-like infinity.
Joachim Krieger, Ecole Polytechniques Federale
de Lausanne
On the critical MKG equation
I'll discuss recent work with J. Luehrmann on global existence and a priori
bounds for arbitrary large solutions of the energy critical Maxwell-Klein-Gordon
equation.
Andrew Lawrie, University of California, Berkeley
Wave maps on hyperbolic space
We begin with a survey of some recent results on energy critical wave maps
from Minkowski space with an emphasis on the effect of the geometry of the
target manifold on dynamics and the special role played by finite energy
harmonic maps. Then we will consider equivariant wave maps from the hyperbolic
plane into rotationally symmetric targets and discuss new phenomena that
arise in this model problem due to the hyperbolic geometry of the domain.
Jason Metcalfe, University of North Carolina
- Chapel Hill
Local energy decay for wave equations on nontrapping asymptotically flat
space-times
This is a joint work with J. Sterbenz and D. Tataru on local energy decay
for the wave equation. In the stationary case, we show that trapping and
eigenvalues / resonances are the only obstructions to local energy decay.
Moreover, we show that these results are stable for time-dependent perturbations.
Sung-Jin Oh, University of California, Berkeley
Global well-posedness of the energy critical Maxwell-Klein-Gordon equation
The massless Maxwell-Klein-Gordon system describes the interaction
between an electromagnetic field (Maxwell) and a charged massless scalar
field (massless Klein-Gordon, or wave). In this talk, I will present
a recent proof, joint with D. Tataru, of global well-posedness and scattering
of this system for arbitrary finite energy data in the (4+1)-dimensional
Minkowski space, in which the PDE is energy critical.
Natasha Pavlovic, University of Texas at
Austin
Many Body Dynamics, Nonlinear Dispersive PDE and Quantum De Finetti Theorems
The derivation of nonlinear dispersive PDE, such as the nonlinear Schr\"{o}dinger
(NLS) from many body quantum dynamics is a central topic in mathematical
physics, which has been approached by many authors in a variety of ways.
In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy,
which is an infinite system of coupled linear non-homogeneous PDE.
Motivated by the idea that techniques from nonlinear PDE might be useful
at the level of the GP, we obtained a new proof of the unconditional uniqueness
of solutions to the cubic Gross-Pitaevskii hierarchy in ${\mathbb{R}}^3$.
In our work, we employ the quantum de Finetti's theorem (which a quantum
analogue of the Hewitt-Savage theorem in probability theory) as a direct
link between the NLS and the GP hierarchy. In another application of the
quantum de Finetti's theorem we prove the existence of scattering states
for the defocusing cubic GP hierarchy in ${\mathbb{R}}^3$. In the talk,
we will focus on these two applications of the quantum de Finetti's theorem,
both obtained in joint works with T. Chen, C.Hainzl and R. Seiringer.
Oana Pocovnicu, Princeton University
A modulated two-soliton with transient turbulent regime for a focusing
cubic nonlinear half-wave equation on the real line
In this talk we discuss work in progress regarding a nonlocal focusing
cubic half-wave equation on the real line. Evolution problems with nonlocal
dispersion naturally arise in physical settings which include models for
weak turbulence, continuum limits of lattice systems, and gravitational
collapse. The goal of the present work is to construct an asymptotic global-in-time
modulated two-soliton solution of small mass, which exhibits the following
two regimes: (i) a turbulent regime characterized by an explicit growth
of high Sobolev norms on a finite time interval, followed by (ii) a stabilized
regime in which the high Sobolev norms remain stationary large forever in
time. This talk is based on joint work with P. Gerard (Orsay, France), E.
Lenzmann (Basel, Switzerland), and P. Raphael (Nice, France).
