Nonlinear phenomena in dispersive equations

22 mai 2018, 12:00 → 25 mai 2018, 14:30 Europe/Paris

Salle de Réunion - Bâtiment M2 (Laboratoire Paul Painlevé)

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

• Valeria Banica Université Pierre-et-Marie-Curie
• David Chiron Université de Nice
• François Genoud École Polytechnique Fédérale de Lausanne
• Philippe Gravejat Université de Cergy-Pontoise
• Louis Jeanjean Université de Franche-Comté
• Jacek Jendrej Université Paris 13
• Stefan Le Coz Institut de Mathématiques de Toulouse
• Yvan Martel École Polytechnique
• Luc Molinet Université de Tours
• Giuseppe Negro Université Paris 13
• Tadahiro Oh University of Edinburgh
• Oana Pocovnicu Heriot-Watt University
• Didier Smets Université Pierre-et-Marie-Curie
• Laurent Thomann Université de Lorraine
• Nikolay Tzvetkov Université de Cergy-Pontoise
• Hatem Zaag Université Paris 13

 

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

 

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mardi 22 mai

12:45 → 14:00 Lunch

14:00 → 14:20 Welcome & Coffee break

14:20 → 15:55 Mini-course by Yvan MARTEL: Construction and interaction of solitons for NLS equations (Part 1)

slides Mini-course notes

14:20 Construction and interaction of solitons for NLS equations (Part 1)

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).

Orateur: Yvan Martel

15:55 → 16:20 Coffee break

16:20 → 17:10 Smooth branch of travelling waves in the Gross-Pitaevskii equation for small speed

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.

Orateur: David Chiron

17:10 → 18:00 On the Lowest Landau Level equation

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).

Orateur: Laurent Thomann

mercredi 23 mai

09:00 → 10:35 Mini-course by Yvan MARTEL: Construction and interaction of solitons for NLS equations (Part 2)

slides Mini-course notes

09:00 Construction and interaction of solitons for NLS equations (Part 2)

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).

Orateur: Yvan Martel

10:35 → 11:00 Coffee break

11:00 → 11:50 Schrödinger equations with full or partial harmonic potentials, existence and stability results

Orateur: Louis Jeanjean

summary Abstract-Jeanjean.pdf

11:50 → 12:40 Long time regularity of the 2D Euler-Poisson system for electrons with vorticity

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).

Orateur: Oana Pocovnicu

12:40 → 14:30 Lunch

14:30 → 15:20 A rigidity result for the Camassa-Holm equation

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.

Orateur: Luc Molinet

15:20 → 16:10 Stability of multi-solitons for the derivative nonlinear Schrödinger equation

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.

Orateur: Stefan Le Coz

16:10 → 16:40 Coffee break

16:40 → 17:30 A sharpened Strichartz inequality for the wave equation

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.

Orateur: Giuseppe Negro

jeudi 24 mai

09:00 → 10:35 Mini-course by Valeria BANICA: Dynamics of vortex filaments (Part 1)

09:00 Dynamics of vortex filaments (Part 1)

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.

Orateur: Valeria Banica

10:35 → 11:00 Coffee break

11:00 → 11:50 The Sine-Gordon regime of the Landau-Lifshitz equation

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).

Orateur: Philippe Gravejat

11:50 → 12:40 Normal form approach to well-posedness of nonlinear dispersive PDEs

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.

Orateur: Tadahiro Oh

12:40 → 14:30 Lunch

14:30 → 15:20 Solving the 4NLS with white noise initial data

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.

Orateur: Nikolay Tzvetkov

15:20 → 16:10 Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations

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.

Orateur: Jacek Jendrej

16:10 → 16:40 Coffee break

16:40 → 17:30 Blow-up solution for the Complex Ginzburg-Landau equation in some critical case

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.

Orateur: Hatem Zaag

19:00 → 23:00 Conference dinner

vendredi 25 mai

09:00 → 10:35 Mini-course by Valeria BANICA: Dynamics of vortex filaments (Part 2)

09:00 Dynamics of vortex filaments (Part 2)

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.

Orateur: Valeria Banica

10:35 → 11:00 Coffee break

11:00 → 11:50 Spectral stability of inviscid columnar vortices

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.

Orateur: Didier Smets

11:50 → 12:40 Minimal mass blow-up solutions of the L^2 critical NLS with inverse-square potential

Orateur: François Genoud

summary Abstract-Genoud.pdf

12:40 → 14:30 Lunch