Abstracts
Monday 28
14h00 Isabelle Gallagher
Title: An excursion to Patrick's works
15h00 Thomas Alazard
Title: Entropies of free surface flows in fluid dynamics
Abstract: I will discuss recent works with Didier Bresch, Nicolas Meunier and Didier Smets on the dynamics of a free surface carried by an incompressible flow obeying Darcy's law.
This talk focuses on monotonicity properties of different kinds: maximum principles, Lyapunov functions and entropies. The analysis is based on exact identities which in turn allow us to study the Cauchy problem.
16h30 Pierre Germain
Title: Boundedness of spectral projectors on Riemannian manifolds
Abstract: Given the Laplace(-Beltrami) operator on a Riemannianmanifold, consider the spectral projector on (generalized)eigenfunctions whose eigenvalue lies in [L^2,(L+delta)^2] - where L >> 1and delta << 1. We ask the question of optimal L^2 to L^p bounds forthis operator. Some cases are classical: for the Euclidean space, thisis equivalent to the Stein-Tomas theorem; and for general manifolds,bounds due to Sogge are optimal for delta = 1. The case delta < 1 isparticularly interesting since it is connected with the global geometryof the manifold. I will present new results in this direction for thehyperbolic space (joint with Tristan Leger), and the Euclidean torus(joint with Simon Myerson).
Tuesday 29
9h00 Sandrine Grellier
Title: Turbulent cascades for a family of damped Szegö equations
Abstract: Patrick and I introduced the cubic Szegö equation around ten years ago as a toy model of a totally non dispersive degenerate Hamiltonian equation. Despite of the fact that it is a complete integrable system, we proved that this equation develops some cascades phenomena. Namely, for a dense set of smooth initial data, the Szegö solutions have unbounded high Sobolev trajectories, detecting transfer of energy from low to high frequencies. However, this dense set has empty interior and a lot of questions remain opened to understand turbulence phenomena. Among others, we would like to understand how interactions of Fourier coefficients interfere on it. In a recent work, Biasi and Evnin explore the phenomenon of turbulence on a one parameter family of equations which goes from the cubic Szegö equation to what they call the "truncated Szegö equation". In this latter, most of the Fourier mode couplings are eliminated. However, they prove the existence of unbounded trajectories for simple
rational initial data. In this talk, I will explain how, paradoxically, the turbulence phenomena may be promoted by adding a damping term.
10h00 Alexander Pushnitski
Title: The inverse spectral theory for non-compact Hankel operators
Abstract: The inverse spectral theory for COMPACT Hankel operators was developed by Patrick Gerard and Sandrine Grellier during 2010-2017. A beautiful feature of this theory is that the spectral map (the map from the set of compact Hankel operators to the set of spectral data) is bijective. Recently, Patrick Gerard, Sergei Treil and myself were able to make further progress in this direction by extending most aspects of this theory to (a subclass of) bounded NON-COMPACT Hankel operators. It turns out that in this case the natural extension of the spectral map is injective, but not surjective. The loss of surjectivity is quite subtle and is related to the appearance of the absolutely continuous spectrum in this problem. I will describe the main features of our work and mention some consequences for the dynamics of the cubic Szego equation with the initial data in BMO.
11h30 Thomas Kapeller
Title: On the stability of periodic multi-solitons of the KdV equation
Abstract: I will report on recent results on the stability of periodic (in space) multi-solitons of the KdV equation, also referred to as finite gap solutions. We prove that multi-solitons, whose frequencies satisfy certain diophantine conditions, are stable under perturbation of the equation and the initial data, both of size $\varepsilon$, over an interval of time of length $O(\varepsilon^{-2})$.
This is joint work with Riccardo Montalto.
Title: About the Benjamin-Ono hierarchy
Abstract: We introduce the Birkhoff coordinates for the Benjamin-Ono constructed by Gérard, Kappeler and Topalov in 2020, then explain how this construction extends to the equations from the Benjamin-Ono hierarchy. For the third order Benjamin-Ono hierarchy, we deduce the well-posedness threshold, determine the traveling wave solutions and study their orbital stability properties.
15h00 Nader Masmoudi
16h30 Cécile Huneau
Title: High frequency limit for Einstein equations with a U(1) symmetry.
Abstract: Due to the nonlinear character of the equations, a sequence of metrics, solutions of vacuum Einstein equations, which oscillate with higher and higher frequency, may converge to a solution to Einstein equations coupled to some effective energy momentum tensor. This effect is called backreaction, and has been studied by physicists (Isaacson, Burnett, Green and Wald). It has been conjectured by Burnett, under some definition of the high frequency limit, that the only effective energy momentum tensor that could appear corresponds to a massless Vlasov field. I will present a proof of this conjecture, in the context of U(1) symmetry, which uses dramatically Patrick Gérard's microlocal deffect measures.This is a work in collaboration with Jonathan Luk (Stanford).
Wednesday 30
9h00 Jean-Yves Chemin
Title: Fourier Analysis on the Heisenberg groupAbstract: In this talk, we are going to construct the frequency space of theHeisenberg group starting from the classical definition of the Fouriertransform. This consists in finding a relevant parametrization of thefamily of groups of unitary transformation associated with the Schrödinger representations of the Heisenberg group. This provides in particular a description of the range of the Schwarz class by the Fourier transform which allows to extend the Fourier transform to temperd distribution of the Heisenberg group. Some applications will be explained.
10h00 Frédéric Rousset
Title: Well-posedness of singular Vlasov equations
Abstract: We shall study the well-posedness of singular kinetic equations of Vlasov type arising in plasma physics (quasineutral limit) or fluid mechanics (variational solutons of the incompressible Euler equation). Singular means that the force field is given by the gradient of a macroscopic quantity (the density for example) without smoothing. A stability condition on the initial data is necessary for well-posedness in Sobolev spaces and microlocal tools are used in order to obtain a priori estimates. (joint work with D. Han-Kwan (Ecole Polytechnique).
11h30 Laure Saint-Raymond
Title: Internal waves and pseudo differential operators of order 0
Abstract: TBA
Thursday 1
10h00 Nalini Anantharaman
Title: The bottom of the spectrum of a random hyperbolic surface
Abstract: In an ongoing project with Laura Monk, we are trying to prove that with high probability (and in the limit when the volume goes to infinity) there are no small eigenvalues. Most of the talk will be dedicated to giving motivations, constructing the probabilistic model, and describing the strategy to solve a similar question for random regular graphs (due to Friedman and Bordenave). We will then state our current results.
11h00 Belhassen Dehman
Title: Observation, Control and Measures
Abstract: In the beginning of the 90’, there has been a major development in the study of control and observation of the wave equation. Indeed, the introduction of microlocal analysis tools made it possible to obtain optimal geometrical results. In this lecture, I will run through some ( published or still in progress ) results of control/stabilization whose proofs are essentially based on microlocal defect measures. We will try to note ( and appreciate !) the flexibility and efficiency of this tool.
13h30 Thomas Duyckaerts
Title: Profile decomposition and soliton resolution for energy-critical wave equations
Abstract: TBA
14h30 Nikolay Tzvetkov
Title: Concerning the pathological set in the context of probabilistic well-posedness
Abstract: We will discuss a complementary result to the probabilistic well-posedness for the nonlinear wave equation. More precisely, we show that there is a dense set S of the Sobolev space of super-critical regularity such that in sharp contrast with the probabilistic well-posedness results the family of global smooth solutions, generated by the convolution with an approximate identity of the elements of S, does not converge in the space of super-critical Sobolev regularity. This is a joint work with Chenmin Sun.
16h00 Wilhelm Schlag
Title: On the long-term dynamics of nonlinear wave equations and the uniqueness of solitons
Abstract: We will discuss the problem of existence and uniqueness of nonzero solutions of finite energy to semilinear elliptic PDEs. The uniqueness question, which is often delicate, has consequences for the spectral properties of the linearized operators. This in turn is of essence for the long-term dynamics of solutions. In particular, I will describe recent work with Alex Cohen and Kevin Li at Yale on the long-standing problem of uniqueness of the first few excited states for the cubic problem in three dimensions.
Friday 2
9h00 Pierre Raphaël
Title: On blow up for the defocusing NLS
Abstract: In a desperate attempt to try and have Patrick forget about our first meeting, I will be present a series joint works with Merle, Rodnianski and Szeftel on singularity formation for energy super critical NLS models. The talk will focus on connecting various works related in particular to compressible and incompressible fluids and the non linear heat equation.
10h00 Oana Pocovnicu
Title: On the probabilistic well-posedness of the 3D cubic nonlinear wave equation in negative Sobolev spaces
Abstract: We consider the cubic nonlinear wave equation (NLW) on the three-dimensional torus with random initial data. For random initial data in L^2, Burq and Tzvetkov (2014) proved almost sure global well-posedness of NLW. In this talk, we extend their result, locally in time, to random initial data of negative regularity. More precisely, by introducing a renormalization of the cubic nonlinearity, we prove an almost sure local well-posedness result for the renormalized NLW. We then justify the necessity of this renormalization by showing a new instability result for NLW without renormalization, which is in the spirit of the so-called triviality from singular stochastic PDEs. This talk is based on a joint work with Tadahiro Oh (The University of Edinburgh) and Nikolay Tzvetkov (CY Cergy Paris University).
11h30 Galina Perelman
Title: Global well-posedness for the derivative nonlinear Schrödinger equation
Abstract: I will review some recents results regarding global well-posedness of the derivative nonlinear Schrödinger equation. The talk will be based on joint works with Hajer Bahouri and Trevor Leslie.
14h00 Enno Lenzmann
Title: Solitons for Calogero-Moser NLS and the prescribed $Q$-curvature problem
Abstract: In this talk, we exploit a new link between solitons for Calogero-Moser type NLS and the prescribed $Q$-curvature problem in one space dimension. The later problem can be seen as the canonical one-dimensional analogue of Liouville’s equation for prescribing Gaussian curvarture of a surface. Inspired by features of complete integrability for Calogero-Moser NLS, we develop a general method to prove uniqueness results for the prescribed $Q$-curvature problem in one space dimension. This talk is based on joint work with Maria Ahrend (Basel).
15h00 Jean-Marc Delort
Title: Microlocal partition of energy for linear wave or Schrödinger equations
Abstract: In the study of problems of asymptotic behavior of large solutions to certain semi-linear wave equations, a key step is to show that, for solutions of the linear wave equation (in odd dimensions), half of the energy of the initial data remains outside the wave cone for all positive times or all negative times. We revisit this last property, proved by Duyckaerts and Merle, in a more microlocal way. This allows us to state and prove similar partition of energy results for linear half-wave or Schrödinger equations. Moreover, for the wave equation, we extend the radial equipartition theorem in even dimension due to Côte et al. to nonradial data.
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