Résumés

Agis Athanassoulis (University of Dundee)

Landau damping for the Alber equation and observability of unidirectional wave spectra

The Alber equation is a moment equation for the nonlinear Schrodinger equation, widely used in ocean engineering to investigate the stability of stationary and homogeneous sea states through their power spectrums. More specifically, a sufficient condition for instability is known. In this work we present the first ever well-posedness theory for the Alber equation with the help of an appropriate equivalent reformulation. Moreover, we show linear Landau damping, i.e. we prove that under a stability condition any perturbations of the homogeneous sea state disperse and decay in time. The idea of the proof is related to recent work in Landau damping for unconfined Vlasov equations [Bedrossian, Masmoudi, Mouhot 2018], although there are substantial differences as well. It is worth mentioning that we manage to avoid the mean-zero assumption of [Bedrossian, Masmoudi, Mouhot 2018], thus including the physically interesting case of incoming wavetrains interacting a homogeneous background.  Finally, the sufficient condition for stability is resolved, and compared to the known sufficient condition for instability. An algorithm for the practical checking of stability for measured spectra is also provided, and the physical implications for ocean waves are discussed.

Corentin Audiard (LJLL)

On the time of existence of solutions of the Euler-Korteweg system with vorticity
The Euler-Korteweg system is a dispersive perturbation of the usual (compressible) Euler equations that accounts for capillary forces. For small irrotational initial data global well-posedness is known to hold in dimension at least 3. In this talk we discuss the case of small initial data with nonzero vorticity, where the dispersive system becomes a coupled dispersive-transport system. The main result is that the time of existence only depends on the size of the initial vorticity.

Valeria Banica (LJLL)

Construction and description of a class of singular binormal flow solutions

The binormal flow is a standard model for the evolution in time of a 3-D curve representing the location of a vortex filament. In this talk I will show how to construct a class of singular solutions and display some of their special properties, to be linked with turbulence features. These results are obtained in collaboration with Luis Vega. 

Federico Cacciafesta (Università di Padova)

Linear and nonlinear Dirac equation on spherically symmetric manifolds

In this talk we will present the Dirac equation on spherically symmetric manifolds, showing how it is possible to exploit the symmetric structure to obtain dispersive (in particular local and global weighted Strichartz) estimates. The main idea consists in using the partial wave decomposition to reduce the problem to a radial Dirac equation with a potential term in order to exploit the existing theory on potential perturbations for dispersive flows. This is a joint work with Anne Sophie de Suzzoni.

Roland Donninger (University of Vienna)

Strichartz estimates for the one-dimensional wave equation
I will report on work in progress with Irfan Glogic on one-dimensional wave evolution in hyperboloidal coordinates. We prove a set of Strichartz estimates for equations perturbed by a general potential. I will also outline possible applications, e.g. Yang-Mills fields on wormhole spacetimes.

Luigi Forcella (EPFL)

Scattering for Dipolar Quantum Gases below the ground state energy threshold

In this talk we will give a result about the dynamics for large times of solutions to the Gross-Pitaevskii equation describing a dipolar Bose-Einstein Condensate. The equation is a nonlinear Schrödinger equation with local and nonlocal interaction terms, the first given by a cubic nonlinearity, the second given by a nonlocal long-range potential expressed in term of convolution with a singular kernel. In this context defocusing/focusing regimes are meaningless, while a notion of stable/unstable regime plays the crucial role. In the unstable regime, where standing states do exist, we can prove that below the energy threshold given by the ground state, all global in time solutions behave as free waves asymptotically in time. The main ingredients of the proof are topological methods and a profile decomposition theorem enabling to implement a Kenig & Merle scheme.

Julien Sabin (Université Paris 11)

The semi-classical limit of the Hartree equation at positive density

We consider the Hartree equation around a translation-invariant background, and show that the limit of the Wigner transforms of its solutions as the Planck constant goes to zero converge to solutions to the nonlinear Vlasov equation around the classical version of the translation-invariant background. We also discuss the well-posedness of this Vlasov equation at positive density (joint work with M. Lewin).

Andrea Sacchetti (Università di Modena e Reggio Emilia)

Bifurcation tree of stationary solutions of nonlinear Schrodinger equations

In this talk we discuss some recent results for a class of nonlinear models in Quantum Mechanics. In particular we focus our attention to the nonlinear one-dimensional Schrodinger equation with a periodic potential and a Stark-type perturbation. In the limit of large periodic potential the stationary solutions of the linear equation bifurcate when the nonlinear term is introduced; in particular, a cascade of bifurcations occurs when the ratio between the effective nonlinearity strength and the tilt of the external field increases. This model has many interesting features; e.g.: the measurement of the value of the gravity acceleration g, using ultracold Strontium atoms confined in a vertical optical lattice.

References:

- A. Sacchetti, Nonlinear Stark-Wannier equation, SIAM JOURNAL ON MATHEMATICAL ANALYSIS (2018).

- A. Sacchetti, Bifurcation trees of Stark-Wannier ladders for accelerated Bose-Einstein condensates in an optical lattice, PHYSICAL REVIEW E (2017).

Jérémie Szeftel (LJLL)

The nonlinear stability of Schwarzschild

I will discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data  subject to a certain symmetry class.

Leonardo Tolomeo (University of Edinburgh)

Ergodicity for stochastic wave equations

In this talk, we study the long time behaviour of stochastic damped wave equation with cubic nonlinearity and additive space-time white noise forcing, posed on the 1 and 2dimensional torus. 

After introducing the notions of ergodicity, unique ergodicity and convergence to equilibrium, we will discuss how these have been proven for a very large class of parabolic SPDEs.

We will then shift our attention to the wave equation(s), where the general strategy for the parabolic case fails, due to the dispersive nature of the equations. 
We will describe how this failure happens, and then briefly discuss how to circumvent these problems and show (unique) ergodicity even in this case.

Nikolay Tzvetkov (Université de Cergy-Pontoise)

Random data wave equations 

We will start by presenting two basic probabilistic effects for questions concerning the regularity of functions and nonlinear operations on functions. We will then overview well-posedenss results for the nonlinear wave equation, the nonlinear Schr\"odinger equation and the nonlinear heat equation in the presence of singular randomness. In the remaining part of the lectures, we will focus on the nonlinear wave equation with initial data distributed according to non degenerate gaussian measures on Sobolev spaces of varying regularity. We will introduce the notation of super-ctitical regularity by presenting the relevant ill-posedness results. We will then prove probabilistic well-posedness for data of super-critical regularity, by constructing the dynamics almost surely with respect to the corresponding gaussian measure. We will then turn to the transport properties of our gaussian measures under the flow of the non linear wave equation. We will finally discuss the proof of a very resent result constructing the relevant dynamics when the measures are supported by Sobolev spaces of negative indexes.