09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
9:00 - 10:30 (1h30)
Patrick Gérard -------------------- Integrable equations of Benjamin-Ono type
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.
›10:30 (30min)
10:30 - 11:00 (30min)
Pause café
›11:00 (1h)
Pierre Germain ----------------------- Asymptotic stability of solitons in 1D dispersive problems.
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.
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.
11:00 - 12:00 (1h)
Pierre Germain ----------------------- Asymptotic stability of solitons in 1D dispersive problems.
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.
›12:15 (1h45)
12:15 - 14:00 (1h45)
Déjeuner
›15:00 (1h)
Ewelina Zatorska ----------------------- Analysis of the dissipative Aw-Rascle model
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
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
15:00 - 16:00 (1h)
Ewelina Zatorska ----------------------- Analysis of the dissipative Aw-Rascle model
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
›16:00 (30min)
16:00 - 16:30 (30min)
Pause café
›16:30 (1h)
Kevin Le Balc'h ----------------------- Quantitative unique continuation for real-valued solutions to second order elliptic equations in the plane.
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 m
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 m
16:30 - 17:30 (1h)
Kevin Le Balc'h ----------------------- Quantitative unique continuation for real-valued solutions to second order elliptic equations in the plane.
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 m
›17:30 (1h)
Grégory Faye ----------------------- The local limit theorem for complex valued sequences: the parabolic case.
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.
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.
17:30 - 18:30 (1h)
Grégory Faye ----------------------- The local limit theorem for complex valued sequences: the parabolic case.
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.
›18:30 (1h)
18:30 - 19:30 (1h)
Apéritif spécial 50 ans
›19:30 (1h30)
19:30 - 21:00 (1h30)
Dîner
