Title: On the Cauchy problem for the Hall-magnetohydrodynamics equations

Absract:

In this talk, I will describe a recent series of work with I.-J. Jeong on the Cauchy problem for the Hall-MHD equation without resistivity. This PDE, first investigated by the applied mathematician M. J. Lighthill, is a one-fluid description of magnetized plasmas with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. Curiously, we demonstrate ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field). Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character.

Title: Diabatic Surface Hopping, Marcus Rate and Ehrenfest dynamics

Abstract: Surface hopping algorithms are popular tools to study dynamics of the quantum-classical mixed systems. In this talk, we will present a surface hopping algorithm in diabatic representations, in the view point of time dependent perturbation theory and semiclassical analysis. The algorithm is validated numerically in both weak coupling and avoided crossing regimes. We then discuss some recent progress on the asymptotics of the algorithm in weak and large coupling regimes.

Title:

Skyrmions and stability of degree ±1 harmonic maps from the plane to the two-dimensional sphere.

Abstract: Skyrmions are topologically nontrivial patterns in the magnetization of extremely thin ferromagnets. Typically thought of as stabilized by the so-called Dzyaloshinskii-Moriya interaction (DMI), or antisymmetric exchange interaction, arising in such materials, they are of great interest in the physics community due to possible applications in memory devices.

In this talk, I will characterize skyrmions as local minimizers of a

two-dimensional limit of the full micromagnetic energy, augmented by DMI and retaining the nonlocal character of the stray field energy. In the regime of dominating Dirichlet energy, I will provide rigorous predictions for their size and “wall angles”. The main tool is a quantitative stability result for harmonic maps of degree ± 1 from the plane to the two-dimensional sphere, relating the energy excess of any competitor to the homogeneous H¹-distance to the closest harmonic map. This is joint work with Anne Bernand-Mantel and Cyrill B. Muratov.

Title:

Concentration and Growth of Laplace Eigenfunctions.

Abstract: In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration including Weyl laws; in each case obtaining quantitative improvements over the known bounds.

]]>Title:

Resonances on asymptotically flat black holes

Abstract:

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.

Title:

The nonlinear stability of the Schwarzschild family of black holes .

Abstract:

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.

Title:

Global existence of small amplitude solutions for a model quadratic quasilinear wave-Klein-Gordon system in 2D.

Abstract:

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.

]]>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).