The APDE seminar on Monday, 04/29 will be given by Dejan Gajic in Evans 740 from 4:10 to 5pm.
Resonances on asymptotically flat black holes
A fundamental problem in the context of Einstein’s equations of general relativity is to understand the dynamical evolution of small perturbations of stationary black hole solutions. It is expected that there is a discrete set of characteristic frequencies that play a dominant role at late times and carry information about the nature of the black hole, much like how the normal frequencies of a vibrating guitar string play an important role in the resulting sound wave. These frequencies are called quasinormal frequencies or resonant frequencies and they are closely related to scattering resonances in the study of Schrödinger-type equations. I will consider the linear wave equation on black hole backgrounds as a toy model for Einstein’s equations and give an introduction to resonances in this setting. Then I will discuss a new method of defining and studying resonances on asymptotically flat spacetimes, developed from joint work with Claude Warnick, which puts resonances on the same footing as normal modes by showing that they are eigenfunctions of a natural operator acting on a Hilbert space.
The APDE seminar on Monday, 04/22 will be given by Martin Taylor in Evans 740 from 4:10 to 5pm.
The nonlinear stability of the Schwarzschild family of black holes .
I will present a theorem on the full finite codimension asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.
The APDE seminar on Monday, 04/15 will be given by Annalaura Stingo in Evans 740 from 4:10 to 5pm.
Global existence of small amplitude solutions for a model quadratic quasilinear wave-Klein-Gordon system in 2D.
In this talk we discuss the problem of global existence of solutions to a coupled wave-Klein-Gordon system in space dimension two when initial data are small, smooth and mildly decaying at infinity. Several physical models related to general relativity have shown the importance of studying such systems but very few results are known at present in low space dimension, where linear solutions slowly decay in time.
We study here a model quadratic quasilinear two-dimensional system, in which the nonlinearity writes in terms of “null forms”, and prove global existence by propagating a-priori energy estimates and optimal uniform estimates on the solution. In proving such estimates one has to deal with several issues such as the quasilinear nature of the problem, the very low decay in time of quadratic nonlinearities, the fact that initial data are not compactly supported…
We will show how to obtain energy estimates by using systematically quasilinear normal forms, in their para-differential version. Uniform estimates will instead be recovered by deducing a new coupled system of a transport equation and an ordinary differential equation from the starting PDE system by means of a semiclassical microlocal analysis of the problem.
The next APDE seminar will be given on Monday, 03/04 by Mihaela Ifrim in Evans 740 from 4:10 to 5pm.
Title: Dispersive decay of small data solutions for the KdV equation
Authors: Mihaela Ifrim, Herbert Koch, Daniel Tataru
We consider the Korteweg-de Vries (KdV) equation, and prove that small
localized data yields solutions which have dispersive decay on a quartic
time-scale. This result is optimal, in
view of the emergence of solitons at quartic time, as predicted by
inverse scattering theory.
The APDE seminar on Monday, 04/08 will be given by Maciej Zworski in Evans 740 from 4:10 to 5pm.
Rough control for Schr\”odinger operators on 2-tori.
Abstract: I will explain how the results of Bourgain, Burq and the speaker ’13 can be used to obtain control and observability by rough functions and sets on 2-tori. We show that for the time dependent Schr\”odinger equation, any set of positive measure can be used for observability and controllability. For non-empty open sets this follows from the results of Haraux ’89 and Jaffard ’90, while for sufficiently long times and rational tori this can be deduced from the results of Jakobson ’97. Other than tori (of any dimension; cf. Komornik ’91, Anantharaman–Macia ’14) the only compact manifolds for which observability holds for any non-empty open sets are hyperbolic surfaces. That follows from results of Bourgain–Dyatlov ’16 and Dyatlov–Jin ’17 and I will discuss the difficulty of passing to rougher rougher sets in that case. Joint work with N Burq.
The next APDE seminar will be given on Monday, 02/11 by Jean-Michel Coron in Evans 740 from 4:10 to 5pm.
Some methods to use the nonlinearities in order to control a system
A control system is a dynamical system on which one can act thanks to what is called the control. For example, in a car, one can turn the steering wheel, press the accelerator pedal etc. These are the control(s). One of the main problems in control theory is the controllability problem. One starts from a given situation and there is a given target. The controllability problem is to see if, by using some suitable controls depending on time, one can move from the given situation to the desired target. We study this problem with a special emphasis on the case where the nonlinearities play a crucial role. We first recall some classical results on this problem for finite dimensional control systems. We explain why the main tool used for this problem in finite dimension, namely the iterated Lie brackets, is difficult to use for many important control systems modeled by partial differential equations. We present methods to avoid the use of these iterated Lie brackets. We give applications of these methods to various physical control systems (Euler and Navier-Stokes equations of incompressible fluids, shallow water equations, Korteweg-de Vries equations).
The next APDE seminar will be given on Monday, 12/11 by Benjamin Harrop-Griffiths in Evans 740 from 4:10 to 5pm.
Title: Vortex filament solutions of the Navier-Stokes equations
Abstract: From Helmholtz to vaping hipsters, the dynamics of vortex filaments, i.e. fluids with vorticity concentrated along a smooth curve, has been a topic of significant interest in fluid dynamics. The global well-posedness of vortex filaments with small circulation follows from the theory of mild solutions of the 3d Navier-Stokes equations at critical regularity. However, for filaments with large circulation these results no longer apply. In this talk we discuss a proof of well-posedness (in a suitable sense) for vortex filaments of arbitrary circulation. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of the 3d Navier-Stokes without additional symmetry assumptions. This is joint work with Jacob Bedrossian and Pierre Germain.
The next APDE seminar will be given by Suncica Canic in Evans 740 from 4:10 to 5pm.
Title: A mathematical framework for proving existence of weak solutions to a class of nonlinear parabolic-hyperbolic moving boundary problems
Abstract: The focus of this talk will be on nonlinear moving-boundary problems involving incompressible, viscous fluids and elastic structures. The fluid and structure are coupled via two sets of coupling conditions, which are imposed on a deformed fluid-structure interface. The main difficulty in studying this class of problems stems from the strong geometric nonlinearity due to the nonlinear fluid-structure coupling. We have recently developed a robust framework for proving existence of weak solutions to this class of problems, allowing the treatment of various structures (Koiter shell, multi-layered composite structures, mesh-supported structures), and various coupling conditions (no-slip and Navier slip). The existence proofs are constructive: they are based on Rothe’s method (semi- discretization in time), and on our generalization of the Lions-Aubin-Simon’s compactness lemma to moving boundary problems. Applications of this strategy to the simulations of real-life problems will be shown. A new problem involving a design of bioartificial pancreas (together with Dr. Roy of UCSD Bioengineering) will be discussed.
The next APDE seminar will take place Wednesday, Nov 28, in 740 Evans from 3-4pm.
Title: Illusions: curves of zeros of Selberg zeta functions
Abstract: It is well known (since 1956) that the Selberg Zeta function
for compact surfaces satisfies the “Riemann Hypothesis”: any zero in the
critical strip 0<R(s)<1 is either real or Im(s)=1/2. The question of
location and distribution of the zeros of the Selberg Zeta function
associated to a noncompact hyperbolic surface attracted attention of the
mathematical community in 2014 when numerical experiments by
D. Borthwick showed that for certain surfaces zeros seem to lie on
smooth curves. Moreover, the individual zeros are so close to each other
that they give a visual impression that the entire curve is a zero set.
We will give an overview of the computational methods used, present
recent results, justifying these observations as well as state open
The next APDE seminar will take place Monday, Nov 19, in 740 Evans from 4-5pm.
Title: Rough control for Schr\”odinger operators on 2-tori.
Abstract: I will explain how the results of Bourgain, Burq and the
speaker ’13 can be used to obtain control and observability by rough
functions and sets on 2-tori. We show that for the time dependent
Schrödinger equation, any set of positive measure can be used for
observability and controllability.
For non-empty open sets this follows from the results of Haraux ’89
and Jaffard ’90, while for sufficiently long times and rational tori
this can be deduced from the results of Jakobson ’97.