# SCIENTIFIC PROGRAMS AND ACTIVITIES

November 13, 2018

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

## Nonlinear Wave equations and their numerical study June 22 - 26, 2015

 Organizing Committee Focus Week Organizers Spyros Alexakis, University of Toronto Mihalis Dafermos, Princeton University Luis Lehner, Perimeter Institute for Theoretical Physics and University of Guelph Harald Pfeiffer, Canadian Institute for Theoretical Astrophysics (CITA) Eric Poisson, University of Guelph Alexandru Ionescu, Princeton University

 On-line Registration open to June 14 also on-site during Focus Weeks $100 registration fees, students and PDF$50 *Please note that a nominal registration fee is required of all participants for this Focus Program. Your contributions allow us to provide the Program with refreshments and social events for each of the Focus Weeks.nts for each of the Focus Weeks. Application for Participant support Deadline to apply April 30, 2015 Accommodation in Toronto Information for speakers Reimbursement information for funded participants Map to Fields

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).

Arick Shao, Imperial College London
Uniqueness Theorems for Waves from Infinity

Unique continuation has been a general problem of interest in the study of partial differential equations for several decades. We survey some recent uniqueness results regarding linear and nonlinear wave equations with data given at infinity. We focus on methods, in particular Carleman estimates, which are geometrically robust and apply to a wide range of asymptotically flat and asymptotically Anti-de Sitter spacetimes. We also discuss some applications of these results, both inside and outside of general relativity.

Mihai Tohaneanu, Georgia Southern University
Global existence for quasilinear wave equations close to Schwarzschild

We study the quasilinear wave equation $\Box_{g} u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of extra assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.