Nonlinear phenomena in dispersive equations

Europe/Paris
Salle de Réunion - Bâtiment M2 (Laboratoire Paul Painlevé)

Description

Aim and scope

The study of dispersive equations is a major research topic for the analysis of nonlinear PDEs. The main challenge is the understanding of the mechanisms that govern the evolution of solutions of nonlinear equations arising in physics. The questions are multiple and the techniques and methods that are used are very different.

The objective of this conference is to bring together researchers from various thematic communities, all of whom are confronted with the analysis of dispersive PDEs. For this purpose, the program will consist of 14 lectures and 2 mini-courses over 4 days, letting time for scientific exchanges between the participants.

As an illustration of the plurality of topics covered,  two mini-courses will be given: one by Yvan Martel from École Polytechnique, concerning the interactions of solitary waves for the nonlinear Schrödinger equation, and the other one by Valeria Banica from Université Pierre-et-Marie-Curie, concerning the dynamics of the vortex filaments in an incompressible non-viscous fluid.

Program

Confirmed speakers

The conference will start on May 22 at 2 pm. The last talk will finish on May 25 at 1 pm.

Registration

Registration is free but mandatory. After filling the Registration form, you will receive a confirmation e-mail. If you do not, please contact the organizers.

Partners

Participants
• Abdelwahab Bensouilah
• André De Laire
• Antoine Benoit
• Chenmin Sun
• Clément Evain
• Corentin Audiard
• David Chiron
• Didier Smets
• François Genoud
• Gabriel Rivière
• Giuseppe NEGRO
• Guillaume Dujardin
• Hatem Zaag
• Jacek JENDREJ
• Jean-Baptiste Casteras
• Laurent Thomann
• Lifeng Zhao
• Louis Jeanjean
• Luc Molinet
• Moudhaffar Bouthelja
• Ngoc Uyen Cong Nguyen
• Nikolay Tzvetkov
• Oana Pocovnicu
• Oussama Landoulsi
• Philippe Gravejat
• Pierre Mennuni
• Sahbi Keraani
• Stefan Le Coz
• Stephan De Bièvre
• Stephane Randoux
• Thomas Rey
• Tien Vinh NGUYEN
• Tristan Robert
• Valeria Banica
• Van Duong Dinh
• Vianney Combet
• Yvan MARTEL
Contact
• Tuesday, 22 May
• 12:45 14:00
Lunch 1h 15m RU Barrois

RU Barrois

• 14:00 14:20
Welcome & Coffee break 20m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

• 14:20 15:55
Mini-course by Yvan MARTEL: Construction and interaction of solitons for NLS equations (Part 1) Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

• 14:20
Construction and interaction of solitons for NLS equations (Part 1) 1h 35m
We will review some results on the construction and interaction of solitary waves for nonlinear Schrödinger equations with power nonlinearity. After discussing briefly the well-known question of stability of single solitary waves, we will present a short proof of existence of multi-solitary waves in the case of weak interactions. Then, in the sub-critical and super-critical cases, we will show the existence of multi-solitary waves with logarithmic distance in time (case of strong interaction).
Speaker: Yvan Martel
• 15:55 16:20
Coffee break 25m Salle Kampé de Fériet - Bâtiment M2

Salle Kampé de Fériet - Bâtiment M2

Laboratoire Paul Painlevé

• 16:20 17:10
Smooth branch of travelling waves in the Gross-Pitaevskii equation for small speed 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

We shall consider the Gross-Pitaevskii equation in the plane. This model is known to have a branch of travelling waves (the Jones-Roberts branch). Variational methods have already been used to yield existence results for this branch. Up to now, the question of smooth dependency with respect to the speed was not rigorously proved. We shall present a result showing the existence of a smooth branch for small speed. This is a joint work with Eliot Pacherie.
Speaker: David Chiron
• 17:10 18:00
On the Lowest Landau Level equation 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

We study the Lowest Landau Level equation with time evolution. This model is used in the description of fast rotating Bose-Einstein condensates. Using argument coming from the theory of the holomorphic functions, we provide a classification of the stationnary solutions. We also prove some stability results. This is a work in collaboration with Patrick Gérard (Paris-Sud) and Pierre Germain (Courant Institute).
Speaker: Laurent Thomann
• Wednesday, 23 May
• 09:00 10:35
Mini-course by Yvan MARTEL: Construction and interaction of solitons for NLS equations (Part 2) Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

• 09:00
Construction and interaction of solitons for NLS equations (Part 2) 1h 35m
We will review some results on the construction and interaction of solitary waves for nonlinear Schrödinger equations with power nonlinearity. After discussing briefly the well-known question of stability of single solitary waves, we will present a short proof of existence of multi-solitary waves in the case of weak interactions. Then, in the sub-critical and super-critical cases, we will show the existence of multi-solitary waves with logarithmic distance in time (case of strong interaction).
Speaker: Yvan Martel
• 10:35 11:00
Coffee break 25m Salle Kampé de Fériet - Bâtiment M2

Salle Kampé de Fériet - Bâtiment M2

Laboratoire Paul Painlevé

• 11:00 11:50
Schrödinger equations with full or partial harmonic potentials, existence and stability results 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

Speaker: Louis Jeanjean
• 11:50 12:40
Long time regularity of the 2D Euler-Poisson system for electrons with vorticity 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

The Euler-Poisson system for electrons is one of the simplest two-fluid models used to describe the dynamics of a plasma. From the point of view of analysis, it can be reformulated as a system consisting of a quasilinear hyperbolic PDE coupled with a transport-type PDE. In this talk, we will discuss the long time existence for the two-dimensional Euler-Poisson system, with a particular attention to the dependence of the time of existence on the size of the vorticity. This talk is based on joint work with A. Ionescu (Princeton).
Speaker: Oana Pocovnicu
• 12:40 14:30
Lunch 1h 50m RU Barrois

RU Barrois

• 14:30 15:20
A rigidity result for the Camassa-Holm equation 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

The Camassa-Holm equation possesses peaked solitary waves called peakons. We prove a Liouville property for uniformly almost localized (up to translations) $H^1$-global solutions of the Camassa-Holm equation with a momentum density that is a non negative finite measure. More precisely, we show that such solution has to be a peakon. As a consequence, we prove that peakons are asymptotically stable in the class of $H^1$-functions with a momentum density that is a non negative finite measure.
Speaker: Luc Molinet
• 15:20 16:10
Stability of multi-solitons for the derivative nonlinear Schrödinger equation 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

The nonlinear Schrödinger equation with derivative cubic nonlinearity (dNLS) is a model quasilinear dispersive equation. It admits a family of solitons, which are orbitally stable in the energy space. After a review of the many interesting properties of dNLS, we will present a result of orbital stability of multi-solitons configurations in the energy space, and some ingredients of the proof.
Speaker: Stefan Le Coz
• 16:10 16:40
Coffee break 30m Salle Kampé de Fériet - Bâtiment M2

Salle Kampé de Fériet - Bâtiment M2

Laboratoire Paul Painlevé

• 16:40 17:30
A sharpened Strichartz inequality for the wave equation 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

In 2004, Foschi found the best constant, and the extremizing functions, for the Strichartz inequality for the wave equation with data in the Sobolev space $\dot{H}^{1/2} \times \dot{H}^{-1/2} (\mathbf{R}^3)$. We refine this inequality, by adding a term proportional to the distance of the initial data from the set of extremizers. Foschi also formulated a conjecture, concerning the extremizers to this Strichartz inequality in all spatial dimensions $d\ge 2$. We disprove such conjecture for even $d$, but we provide evidence to support it for odd $d$. The proofs use the conformal compactification of the Minkowski space-time given by the Penrose transform.
Speaker: Giuseppe Negro
• Thursday, 24 May
• 09:00 10:35
Mini-course by Valeria BANICA: Dynamics of vortex filaments (Part 1) Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

• 09:00
Dynamics of vortex filaments (Part 1) 1h 35m
In this lectures I shall present first the known models for dynamics of vortex filaments. Then I shall focus on the binormal flow model and on its link with the 1-D cubic nonlinear Schrödinger equation. Finally I shall describe several frameworks of formation of singularities in finite time, both at the level of the binormal flow and at the level of the Schrödinger equation.
Speaker: Valeria Banica
• 10:35 11:00
Coffee break 25m Salle Kampé de Fériet - Bâtiment M2

Salle Kampé de Fériet - Bâtiment M2

Laboratoire Paul Painlevé

• 11:00 11:50
The Sine-Gordon regime of the Landau-Lifshitz equation 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

The Landau-Lifshitz equation gives account of the dynamics of magnetization in ferromagnetic materials. The goal of this talk is to describe a long-wave regime for this equation in which it behaves as the Sine-Gordon equation. This is joint work with André de Laire (University of Lille).
Speaker: Philippe Gravejat
• 11:50 12:40
Normal form approach to well-posedness of nonlinear dispersive PDEs 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

Harmonic analysis has played a crucial role in the well-posedness theory of nonlinear dispersive PDEs such as the nonlinear Schrödinger equations (NLS). In this talk, we present an alternative method to prove well-posedness of nonlinear dispersive PDEs which avoids a heavy machinery from harmonic analysis. As a primary example, we study the Cauchy problem for the one-dimensional NLS on the real line. We implement an infinite iteration of normal form reductions (namely, integration by parts in time) and reformulate the equation in terms of an infinite series of multilinear terms of arbitrarily large degrees. By establishing a simple trilinear estimate and applying it in an iterative manner, we establish enhanced uniqueness of NLS in almost critical spaces.
• 12:40 14:30
Lunch 1h 50m RU Barrois

RU Barrois

• 14:30 15:20
Solving the 4NLS with white noise initial data 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

We will consider the fourth order Nonlinear Schrödinger equation, posed on the circle, with initial data distributed according to the white noise. This problem is well posed for smooth initial data. It is therefore natural to consider the sequence of smooth solutions with data distributed according regularisations (by convolution) of the white noise. We show that a renormalisation of this sequence converges to a unique limit. The limit has the white noise as an invariant measure. The proof shares some features with the modified scattering theory which received a lot of attention in the PDE community. As a consequence the solution has a more intricate singular part compared to the large body of literature on probabilistic well-posedness for dispersive PDE's. This is a joint work with Tadahiro Oh and Yuzhao Wang.
Speaker: Nikolay Tzvetkov
• 15:20 16:10
Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

Many evolution PDEs admit special solutions, called solitons, whose shape does not change in time. A multi-soliton is a solution which is close to a superposition of a finite number K of solitons placed at a large distance from each other. I am interested in describing multi-soliton dynamics for generalized Korteweg-de Vries equations. I will present a general method of formally predicting the time evolution of the centers and velocities of each soliton. Then I will discuss in detail the case K = 2, in particular in the regime of strong interactions, which occurs when the velocities of both solitons converge to the same value for large times. Under the additional assumption that the solitons are linearly unstable, one can show that the formal method correctly predicts the distance between the solitons for large times. I will outline this proof.
Speaker: Jacek Jendrej
• 16:10 16:40
Coffee break 30m Salle Kampé de Fériet - Bâtiment M2

Salle Kampé de Fériet - Bâtiment M2

Laboratoire Paul Painlevé

• 16:40 17:30
Blow-up solution for the Complex Ginzburg-Landau equation in some critical case 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

We construct a solution for the Complex Ginzburg-Landau (CGL) equation in some critical case, which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution.
Speaker: Hatem Zaag
• 19:00 23:00
Conference dinner 4h
• Friday, 25 May
• 09:00 10:35
Mini-course by Valeria BANICA: Dynamics of vortex filaments (Part 2) Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

• 09:00
Dynamics of vortex filaments (Part 2) 1h 35m
In this lectures I shall present first the known models for dynamics of vortex filaments. Then I shall focus on the binormal flow model and on its link with the 1-D cubic nonlinear Schrödinger equation. Finally I shall describe several frameworks of formation of singularities in finite time, both at the level of the binormal flow and at the level of the Schrödinger equation.
Speaker: Valeria Banica
• 10:35 11:00
Coffee break 25m Salle Kampé de Fériet - Bâtiment M2

Salle Kampé de Fériet - Bâtiment M2

Laboratoire Paul Painlevé

• 11:00 11:50
Spectral stability of inviscid columnar vortices 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but the only analytical results available so far provide necessary conditions for instability under either planar or axisymmetric perturbations. In this talk I will discuss a recent work with Thierry Gallay in which we show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particular symmetry.
Speaker: Didier Smets
• 11:50 12:40
Minimal mass blow-up solutions of the L^2 critical NLS with inverse-square potential 50m Salle de Réunion - Bâtiment M2

Salle de Réunion - Bâtiment M2

Laboratoire Paul Painlevé

Speaker: François Genoud
• 12:40 14:30
Lunch 1h 50m RU Barrois