Patrick Gérard (Université Paris-Saclay)
Title: Integrable equations of Benjamin-Ono type.
Abstract: In this mini-course, I will study two integrable nonlinear dispersive equations on the line : the Benjamin-Ono equation and the more recently introduced Calogero-Moser derivative nonlinear Schrödinger equation.
The following topics will be discussed.
1. Local wellposedness in Sobolev spaces, Lax pair, explicit formulae.
2. Conservation laws and global wellposedness.
3. Traveling waves and multi-solitons.
4. Zero-dispersion limit.
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Valeria Banica (Sorbonne Université)
Title: Multifractality in the limit evolution of polygonal vortex filaments.
Abstract: With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality of a family of generalized Riemann’s non-differentiable functions.
These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow, the classical model for vortex filaments dynamics. The result that I will present in this talk is that we determine for some of them the spectrum of singularities. The proofs rely on a careful design of Diophantine sets, which we study by using the Duffin-Schaeffer theorem and the Mass Transference Principle.
This is a joint work with Daniel Eceizabarrena, Andrea Nahmod and Luis Vega.
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Yannick Bonthonneau (CNRS, Université Sorbonne Paris Nord)
Title: Inverse analytic scattering.
Abstract: Let M be Riemannian manifold with boundary. For z=(x,v) an incoming vector on the boundary, we denote by S(x,v) the tangent vector to the geodesic starting from z, at its next point of intersection with the boundary.
If M is supposed analytic, under geometric and dynamical assumptions, the geodesic scattering map S determines M, even without prior knowledge of its topology.
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Nicolas Camps (Nantes Université)
Title: Do Energy Cascade Mechanisms for NLS Persist on Generic Tori?
Abstract: When the nonlinear Schr¨odinger equation (NLS) is posed on the square torus, exact resonances can lead to energy transfers and growth of higher-order Sobolev norms.
In this presentation, we investigate whether these mechanisms persist under perturbations of the domain, when the exact resonances are replaced by quasi-resonances.
Under generic Diophantine conditions on the torus, we prove that energy cascades are absent over longer time scales compared to those expected on the square torus. These results emphasize the importance of domain properties in the study of nonlinear waves.
The proofs are based on normal form methods, and the analysis of the quasi-resonant interactions involves a frequency separation property.
These are joint works with Joackim Bernier and Gigliola Staffilani.
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Alberto Enciso (Instituto de Ciencias Matemáticas, Madrid)
Title: Fluid equilibria with compact support and the fearful symmetry of Laplace eigenfunctions.
Abstract: Schiffer’s conjecture in spectral geometry asserts that the only planar domains with a Neumann eigenfunction that is constant on the boundary are disks. If the hypothesis is relaxed so that the Neumann eigenfunction is only assumed to be locally constant on the boundary, we show that there are nontrivial doubly connected domains with this property, in spite of the fact that that the relaxed problem shares many rigidity properties with Schiffer. Furthermore, this has a nice application in fluid mechanics, as one can use these domains to construct nonradial Lipschitz continuous stationary solutions to the 2d incompressible Euler equations with compactly supported velocity.
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Gregory Faye (CNRS, Université Paul Sabatier, Toulouse)
Title: The local limit theorem for complex valued sequences: the parabolic case.
Abstract: We give a complete expansion, at any accuracy order, for the iterated convolution of a complex valued integrable sequence in one space dimension. The remainders are estimated sharply with generalized Gaussian bounds. The result applies in probability theory for random walks as well as in numerical analysis for studying the large time behavior of numerical schemes. This is joint work with Jean-François Coulombel.
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Pierre Germain (Imperial College, Londres)
Title: Asymptotic stability of solitons in 1D dispersive problems.
Abstract: A soliton is asymptotically stable if, for small perturbations, the solution decomposes into soliton + (decaying) radiation as time goes to infinity. I will present results on the asymptotic stability of solitons of NLS and the Phi4 model (in which case the soliton is the ‘kink’). A key idea is to take advantage of nonlinear resonances. This is based on articles with Charles Collot and Fabio Pusateri.
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Bernard Helffer (Nantes Université)
Title: On Courant and Pleijel theorems for sub-Riemannian Laplacians.
Abstract: We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel's theorem, which states that, as soon as the dimension is strictly larger than 1, the number of nodal domains of an eigenfunction corresponding to the k-th eigenvalue is strictly (and uniformly, in a certain sense) smaller than k for large k.
In the first part of the talk we reduce this question from the case of general sub-Riemannian manifolds to that of nilpotent groups.
In the second part, we analyze in detail the case where the nilpotent group is a Heisenberg group times a Euclidean space. Along the way we improve known bounds on the optimal constants in the Faber--Krahn and isoperimetric inequalities on these groups.
This is a joint work with Rupert Frank (LMU München)
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Kevin Le Balc’h (INRIA, Sorbonne Université)
Title: Quantitative unique continuation for real-valued solutions to second order elliptic equations in the plane.
Abstract: In this talk, I will first present the Landis conjecture on exponential decay for solutions to second order elliptic equation in the Euclidean setting. While for complex-valued functions, the Landis conjecture was disproved by Meshkov in 1992, the question is still open for real-valued functions. I will present one way to tackle the conjecture, due to Bourgain and Kenig, that consists in establishing quantitative unique continuation results. Previous results in the two-dimensional setting by Kenig, Sylvestre, Wang in 2014 and more recently by Logunov, Malinnikova, Nadirashvili, Nazarov in 2020 will be recalled and explained. Then, the goal of the talk will be to prove that the qualitative and quantitative Landis conjecture hold for the Laplace operator, perturbed by lower order terms for real-valued solutions in the plane. Even if the strategy of the proof mainly follows the one of Logunov, Malinnikova, Nadirashvili, Nazarov, some new difficulties appear such as new weak quantitative maximum principles, generalization of Stoilow factorization theorem, approximate type Poincare lemma in a perforated domain and a Carleman estimate to a non-homogeneous d_z bar equation with some non-local term. This is a joint work with Diego A. Souza (Universidad de Sevilla).
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Yvan Martel (Université Versailles Saint-Quentin)
Title: Asymptotic stability of small solitary waves for the one-dimensional cubic-quintic Schrödinger equation with an internal mode.
Abstract: I will present a result concerning the asymptotic stability of small solitary waves for the one-dimensional Schrödinger equation with a cubic-quintic, focusing-focusing nonlinearity. For this model, the linearized operator around a small solitary wave has an internal mode whose contribution to the dynamics is handled by checking a nonlinear condition related to the Fermi golden rule.
Reference: Yvan Martel, Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation, arXiv:2312.11016
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Eliot Pacherie (CNRS, CY Cergy University)
Title: Some examples of non unique minimizing travelling waves for a Schrödinger equation.
Abstract: We consider nonlinear Schrödinger equations with a nontrivial condition at infinity for a large class of nonlinearity. It has been shown by Mihai Maris that these equations admit travelling wave solutions for any subsonic speed, and they are constructed as the solutions of a minimizing problem. We are interested in the following question : do we always have uniqueness of the solution of this minimizing problem, up to the natural invariances of the problem ? In this talk, we will show how to construct a specific nonlinearity for which this is not the case. This is a current project in collaboration with Mihai Maris.
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Ewelina Zatorska (University of Warwick)
Title: Analysis of the dissipative Aw-Rascle model.
Abstract: I will introduce and discuss a generalization of the one-dimensional Aw-Rascle model of vehicular traffic, which has recently been proposed as a model for crowd dynamics.
Mathematically, this system lies between the compressible Euler and compressible Navier-Stokes equations, featuring density-modulated dissipation. In one spatial dimension, the same system models the flow of rigid spheres of radius 1 surrounded by a viscous lubricant. At the level of classical solutions, the system is equivalent to the pressureless Navier-Stokes equations with the singular viscosity coefficient $\frac{\epsilon}{1-\rho}$. The first part of my talk will address the questions of existence, uniqueness, and the singular limit of weak and duality solutions as $\epsilon\to 0$. I will then explain the differences and new challenges that arise in the analysis of this system in the multi-dimensional case. Here, we are able to prove the existence and weak-strong uniqueness of measure-valued solutions, as well as the ill-posedness of the model in the class of weak solutions.
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