Hélène Barucq : Current trends in solving full-wave equations for imaging complex media
Waves have the outstanding faculty of recording characteristics of the medium they go through. By studying their propagation, we can thus obtain all the information that is necessary for mapping hidden media. The world provides plenty of applications for wave imaging and this talk will focus on numerical methods that are efficient for exploring complex media such as the subsurface. Time-dependent and time-harmonic equations can be used, each with their respective advantages and drawbacks. Discontinuous Galerkin (DG) methods have recently demonstrated their efficiency for solving such problems and they are now finding their way in industrial applications. This talk will be the opportunity of showing numerical experiments illustrating the performances of DG approximations for simulating waves and their impact on improving full-waveform inversion.
Jean-François Bony : Résolvante des opérateurs faiblement non-autoadjoints
Cet exposé sera consacré à l’étude spectrale d’opérateurs faiblement non-autoadjoints, c’est à dire d’opérateurs dont la partie imaginaire est petite par rapport à la partie réelle. On montrera d’abord des estimations à poids de la résolvante. Leur preuve repose sur une nouvelle approche du principe d’absorption limite. Enfin, on donnera des applications qui sont spécifiques au cas non-autoadjoint. Il s’agit d’une collaboration avec Guy Métivier.
Piotr Chruściel : Space-time singularities with controlled asymptotics and without symmetries
In this talk I will make a review of known results about the existence and nature of singularities in general relativity, and present a construction of singular solutions of Einstein vacuum equations without any symmetries. The talk is based on joint work with Paul Klinger.
Eric Darrigrand : OSRC preconditioner and Fast Multipole Method for 3-D Helmholtz equation: a spectral analysis
In acoustic scattering, integral equation methods are widely used to study wave propagation outside a bounded 3-D domain. Thanks to these techniques, the governing boundary-value problem is reduced to an integral equation on the surface of the scatterer ([Colton-Kress, 83]). We consider here impenetrable bodies with smooth boundaries and incident time-harmonic plane waves. The Combined Field Integral Equation (CFIE) is uniquely solvable for all wavenumber, and implies the first and second traces of the double-layer potential respectively denoted M and D.
In terms of numerical iterative resolution, this equation does not provide a good spectral behavior due to the strongly singular and non-compact operator D. A strategy consists in preconditioning the operator D by introducing an efficient approximation of the exterior Neumann-to-Dirichlet map, using On-Surface Radiation Condition (OSRC) methods ([Antoine-Darbas, 07]). The preconditioned equation is uniquely solvable for all wavenumbers and exhibits very interesting spectral properties. The OSRC technique only involves local operators, so that the numerical implementation of the preconditioning operator requires only the use of a sparse direct solver and does not really affect the cost of the iterative resolution of the CFIE equation. The most expensive part of the resolution is still consequent on the integral operators. For this reason, the Fast Multipole Method (FMM) ([Coifman-Rokhlin-Wandzura, 93]) is used to accelerate the computation involving the operators M and D.
A thorough study of the eigenvalues behavior is realized in order to illustrate the impact of the OSRC-preconditioning technique on the spectrum of the CFIE operator. The resolution scheme is applied to several numerical test-cases (sphere, cube, trapping domains). The convergence of the GMRES corroborates the spectral analysis. Only a few GMRES iterations are required for both high frequencies and refined meshes. The computation cost follows the FMM behavior. Combining the OSRC preconditioner and the FMM proves to be a very efficient approach to solve the CFIE at high frequencies ([Darbas-Darrigrand-Lafranche, 13]).
Stephan De Bièvre : Aspects of approach to equilibrium
That macroscopic isolated systems tend to thermal equilibrium is a basic tenet of thermodynamics. Explaining why and how this happens in terms of the underlying dynamics of the constituents of such systems remains a difficult and largely unsolved problem to which several different approaches are actively pursued. I will review some recent results on this question, obtained for simple model systems.
Bruno Després : Electromagnetic waves in fusion plasmas
Time harmonic waves in plasmas receive increasing interests due to their scientific importance for the heating of magnetic fusion plasma. Recent progresses are reported on the development of a convenient mathematical theory for time harmonic waves in plasmas near the hybrid resonance. After presenting the cold-plasma dielectric tensor, a basic analytic solution is constructed that captures the essential singularity of the problem. This information is used to construct manufactured solutions in the context of the limit absorption principle. Manufactured solutions have the ability to capture the singular limit in a non singular way. In dimension two, manufactured solutions exhibit an additional highly oscillating behavior.
Jérémy Faupin : Asymptotic completeness in dissipative scattering theory
In this talk, we will consider a dissipative operator of the form H=HV-iC*C, where HV=H0+V is self-adjoint and C is a bounded operator. We will recall conditions implying existence of the wave operators associated to H and H0, and we will see that they are asymptotically complete if and only if H does not have spectral singularities on the real axis. For Schrödinger operators, the spectral singularities correspond to real resonances. This is joint work with Jürg Fröhlich.
Christian Gérard : Quelques remarques sur l’état de Hartle-Hawking pour des trous noirs éternels
Un trou noir éternel est décrit par un espace-temps (M,g) possédant à la fois un champ de Killing global et un horizon de Killing qui délimite l’intérieur du trou noir. Hartle et Hawking ont conjecturé il y a longtemps l’existence d’un état stationnaire pour un champ de Klein-Gordon quantique sur M, qui dans l’extérieur du trou noir est un état thermal à température à la température de Hawking TH, reliée à la gravité de surface du trou noir. L’existence et les propriétés de l’état de Hartle-Hawking n’ont été démontrées que récemment par Sanders en 2013. Nous montrerons une manière simple de construire l’état de Hartle-Hawking et de montrer qu’il vérifie la propriété de Hadamard, en utilisant la rotation de Wick et la notion du projecteur de Calderon associé à un problème aux limites elliptique.
Benoît Grébert : Reducibility of the quantum harmonic oscillator in d-dimensions with time dependent perturbation
We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimensions perturbed by a linear operator which is a polynomial of degree two in xj, -i∇j with coefficients depending quasiperiodically on time. As a consequence any solution of such a non homogeneous linear Schrödinger equation is almost periodic in time and remains bounded in all Sobolev norms.
Dietrich Häfner : Some aspects of the scientific work of Alain Bachelot: black hole radiation and time machines
Sergiu Klainerman : On the reality of Black holes
I will be discussing recent results concerning the rigidity, stability and formation of Black Holes.
Philippe LeFloch : Nonlinear dynamical stability of matter. A potpourri of recent mathematical advances
I will present recent mathematical advances on the nonlinear dynamical stability of matter fields, especially on the following problems:
- Ergodicity of spherically symmetric fluid flows outside of a Schwarzschild black hole with random boundary forcing
- Formulation and convergence of the finite volume method for scalar conservation laws on spacetimes with boundary
- Global nonlinear stability of Minkowski spacetime for the Einstein equations in presence of massive fields
- Global evolution of spherically symmetric self-gravitating matter in f(R)–gravity
Jean-Philippe Nicolas : Superradiance of charged black holes, a numerical study
Superradiance in black hole spacetimes is a phenomenon by which a field of spin 0 or 1 can extract energy from the background. Typically, one can imagine sending a wave packet with a given energy towards a black hole and receiving in return a superposition of wave packets carrying a total amount of energy that is larger than the energy sent in. It can be caused by rotation or by interaction between the charges of the black hole and the field. In the first case, the region where superradiance takes place (the ergoregion) has a clear geometrical localization depending only on the physical parameters of the black hole. For charge induced superradiance, this is not the case and we have a generalized ergoregion depending also on the physical properties of the field (mass, charge, angular momentum). In the most severe cases, the generalized ergoregion may cover the whole exterior of the black hole. We focus on charge-induced superradiance for spin 0 fields in spherically symmetric situations. Alain wrote a thorough theoretical study of this question in 2004, which, to my knowledge, is the only work of its kind. When I was in Bordeaux, he and I discussed the possibility of investigating superradiance numerically. Over the years it became an actual research project (involving Laurent Di Menza and more recently Mathieu Pellen) of which this talk is an account. We shall discuss the Penrose process, which is a simpler phenomenon for particles, then show its exact analogue with the superradiance of wave packets and a slightly different behaviour for fields "emerging" inside the ergoregion. Finally we shall explore the related question of black hole bombs and present some recent observations.
Vesselin Petkov : Cauchy problem for effectively hyperbolic operators with triple characteristics
We consider the Cauchy problem for hyperbolic operators with characteristics of variable multiplicities r ≤ 3 assuming that the fundamental matrix of the principal symbol has two non-vanishing real eigenvalues. The last condition is necessary for the Cauchy problem to be well posed for every choice of lower order terms. The operators with this property are called strongly hyperbolic and it was conjectured that every effectively hyperbolic operator is strongly hyperbolic. This conjecture has been proved in the case r = 2. In this talk we present a survey of the results in the case r = 3. The proofs are based on the energy estimates with a big loss of derivatives.
Didier Robert : Some recent results on growth of Sobolev norms for time dependent Schrödinger evolutions
When a time independent Schrödinger Hamiltonian H0 is perturbed by a time dependent potential V(t) the modes of H0 are not preserved by the evolution generated by H0+V(t) and exchange of energies may occur. We shall present a general approach connecting spectral properties of H0, the size and the oscillations in time of the perturbation V(t).
Jérémie Szeftel : On the stability of black holes
Are black holes stable? This fascinating question has attracted a lot of attention in the physics community from the 60’s on. Over the last 10 years, it has also generated an intense activity in mathematical relativity. I will present the main stability conjecture, the history of the problem as well as some of the main challenges. I will also discuss a work in progress with Sergiu Klainerman concerning a particular case of this conjecture.
Michał Wrochna : Quantum Klein-Gordon fields on asymptotically AdS spacetimes
On asymptotically AdS spacetimes, if suitable boundary conditions are imposed, it is possible to assign to each smooth solution of the Klein-Gordon equation a "trace" on the conformal horizon. In order to get a quantum version of that result one rather has to work with bi-solutions or inverses, the Schwartz kernel of which is singular. The main difficulty is that the description of singularities of parametrices provided by Duistermaat and Hörmander is not available in this setting. I will demonstrate how this can be overcome using Vasy's propagation of singularities theorems in the framework of Melrose's b-calculus. I will then sketch the construction of quantum fields in the bulk and explain how this induces a conformal theory on the boundary.
Maciej Zworski : Resonances for obstacles in hyperbolic space
We consider scattering by star-shaped obstacles in hyperbolic space and show that resonance width satisfy a universal bound 1/2 which is optimal in dimension 2. In odd dimensions we also show that the resonance width is also bouded by m/R for a universal constant m, where R is the (hyperbolic) diameter of the obstacle; this gives an improvement for small obstacles. In dimensions 3 and higher the proofs follow the classical vector field approach of Morawetz, while in dimension 2 we obtain our bound by working with spaces coming from general relativity. We also show that in odd dimensions resonances of small obstacles are close, in a suitable sense, to Euclidean resonances. The talk is based on joint work with Peter Hintz.