Title: Dispersive decay of small data solutions for the KdV equation

Authors: Mihaela Ifrim, Herbert Koch, Daniel Tataru

Abstract:
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.

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 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.

]]>Title:

Some methods to use the nonlinearities in order to control a system

Abstract:

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).

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.

]]>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

conjectures.

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.

title: The Marked Length Spectrum of Anosov manifolds

Abstract: We discuss new results on the geometric problem of determining a Riemannian metric with negative curvature on a closed manifold from the lengths of its periodic geodesics. We obtain local rigidity results in all dimensions using combination of dynamical system results with microlocal analysis. Joint work with Thibault Lefeuvre.

]]>Title: Wave maps on (1+2)-dimensional curved spacetimes

Abstract: I will discuss recent joint work, with Cristian Gavrus and Daniel Tataru, in which we consider wave maps on a (1+2)-dimensional nonsmooth background. Our main result asserts that in this variable-coefficient context, the wave maps system is wellposed at almost-critical regularity.

]]>Title: The effect of threshold energy obstructions on the $L^1 \to L^\infty$

dispersive estimates for some Schrödinger type equations

Abstract: In this talk, I will discuss the differential equation $iu_t

= Hu, H := H_0 + V$ , where $V$ is a decaying potential and $H_0$ is a

Laplacian related operator. In particular, I will focus on when $H_0$

is Laplacian, Bilaplacian and Dirac operators. I will discuss how the

threshold energy obstructions, eigenvalues and resonances, effect the

$L^1 \to L^\infty$ behavior of $e^{itH} P_{ac} (H)$. The threshold

obstructions are known as the distributional solutions of $H\psi = 0$

in certain dimension dependent spaces. Due to its unwanted effects on

the dispersive estimates, its absence have been assumed in many

work. I will mention our previous results on Dirac operator and recent

results on Bilaplacian operator under different assumptions on

threshold energy obstructions.