Name: Nonlinear phenomena in dispersive equations
Start: 2018-05-22T12:00:00+02:00
End: 2018-05-25T14:30:00+02:00
Location: Laboratoire Paul Painlevé

Nonlinear phenomena in dispersive equations

de mardi 22 mai 2018 (12:00) à vendredi 25 mai 2018 (14:30)

lundi 21 mai 2018
mardi 22 mai 2018

12:45 Lunch
Lunch
12:45 - 14:00

14:00 Welcome & Coffee break
Welcome & Coffee break
14:00 - 14:20
Room: Salle de Réunion - Bâtiment M2
14:20 Construction and interaction of solitons for NLS equations (Part 1)
Construction and interaction of solitons for NLS equations (Part 1)
14:20 - 15:55
Room: Salle de Réunion - Bâtiment M2
Contributions

14:20 Construction and interaction of solitons for NLS equations (Part 1) - Yvan Martel
15:55 Coffee break
Coffee break
15:55 - 16:20
Room: Salle Kampé de Fériet - Bâtiment M2
16:20 Smooth branch of travelling waves in the Gross-Pitaevskii equation for small speed - David Chiron
Smooth branch of travelling waves in the Gross-Pitaevskii equation for small speed
- David Chiron
16:20 - 17:10
Room: Salle de Réunion - Bâtiment M2 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.
17:10 On the Lowest Landau Level equation - Laurent Thomann
On the Lowest Landau Level equation
- Laurent Thomann
17:10 - 18:00
Room: Salle de Réunion - Bâtiment M2 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).
mercredi 23 mai 2018
09:00 Construction and interaction of solitons for NLS equations (Part 2)
Construction and interaction of solitons for NLS equations (Part 2)
09:00 - 10:35
Room: Salle de Réunion - Bâtiment M2
Contributions

09:00 Construction and interaction of solitons for NLS equations (Part 2) - Yvan Martel
10:35 Coffee break
Coffee break
10:35 - 11:00
Room: Salle Kampé de Fériet - Bâtiment M2
11:00 Schrödinger equations with full or partial harmonic potentials, existence and stability results - Louis Jeanjean
Schrödinger equations with full or partial harmonic potentials, existence and stability results
- Louis Jeanjean
11:00 - 11:50
Room: Salle de Réunion - Bâtiment M2
11:50 Long time regularity of the 2D Euler-Poisson system for electrons with vorticity - Oana Pocovnicu
Long time regularity of the 2D Euler-Poisson system for electrons with vorticity
- Oana Pocovnicu
11:50 - 12:40
Room: Salle de Réunion - Bâtiment M2 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).
12:40 Lunch
Lunch
12:40 - 14:30
14:30 A rigidity result for the Camassa-Holm equation - Luc Molinet
A rigidity result for the Camassa-Holm equation
- Luc Molinet
14:30 - 15:20
Room: Salle de Réunion - Bâtiment M2 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.
15:20 Stability of multi-solitons for the derivative nonlinear Schrödinger equation - Stefan Le Coz
Stability of multi-solitons for the derivative nonlinear Schrödinger equation
- Stefan Le Coz
15:20 - 16:10
Room: Salle de Réunion - Bâtiment M2 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.
16:10 Coffee break
Coffee break
16:10 - 16:40
Room: Salle Kampé de Fériet - Bâtiment M2
16:40 A sharpened Strichartz inequality for the wave equation - Giuseppe Negro
A sharpened Strichartz inequality for the wave equation
- Giuseppe Negro
16:40 - 17:30
Room: Salle de Réunion - Bâtiment M2 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.
jeudi 24 mai 2018
09:00 Dynamics of vortex filaments (Part 1)
Dynamics of vortex filaments (Part 1)
09:00 - 10:35
Room: Salle de Réunion - Bâtiment M2
Contributions

09:00 Dynamics of vortex filaments (Part 1) - Valeria Banica
10:35 Coffee break
Coffee break
10:35 - 11:00
Room: Salle Kampé de Fériet - Bâtiment M2
11:00 The Sine-Gordon regime of the Landau-Lifshitz equation - Philippe Gravejat
The Sine-Gordon regime of the Landau-Lifshitz equation
- Philippe Gravejat
11:00 - 11:50
Room: Salle de Réunion - Bâtiment M2 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).
11:50 Normal form approach to well-posedness of nonlinear dispersive PDEs - Tadahiro Oh
Normal form approach to well-posedness of nonlinear dispersive PDEs
- Tadahiro Oh
11:50 - 12:40
Room: Salle de Réunion - Bâtiment M2 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 Lunch
Lunch
12:40 - 14:30
14:30 Solving the 4NLS with white noise initial data - Nikolay Tzvetkov
Solving the 4NLS with white noise initial data
- Nikolay Tzvetkov
14:30 - 15:20
Room: Salle de Réunion - Bâtiment M2 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.
15:20 Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations - Jacek Jendrej
Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations
- Jacek Jendrej
15:20 - 16:10
Room: Salle de Réunion - Bâtiment M2 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.
16:10 Coffee break
Coffee break
16:10 - 16:40
Room: Salle Kampé de Fériet - Bâtiment M2
16:40 Blow-up solution for the Complex Ginzburg-Landau equation in some critical case - Hatem Zaag
Blow-up solution for the Complex Ginzburg-Landau equation in some critical case
- Hatem Zaag
16:40 - 17:30
Room: Salle de Réunion - Bâtiment M2 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.
19:00 Conference dinner
Conference dinner
19:00 - 23:00
vendredi 25 mai 2018
09:00 Dynamics of vortex filaments (Part 2)
Dynamics of vortex filaments (Part 2)
09:00 - 10:35
Room: Salle de Réunion - Bâtiment M2
Contributions

09:00 Dynamics of vortex filaments (Part 2) - Valeria Banica
10:35 Coffee break
Coffee break
10:35 - 11:00
Room: Salle Kampé de Fériet - Bâtiment M2
11:00 Spectral stability of inviscid columnar vortices - Didier Smets
Spectral stability of inviscid columnar vortices
- Didier Smets
11:00 - 11:50
Room: Salle de Réunion - Bâtiment M2 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.
11:50 Minimal mass blow-up solutions of the L^2 critical NLS with inverse-square potential - François Genoud
Minimal mass blow-up solutions of the L^2 critical NLS with inverse-square potential
- François Genoud
11:50 - 12:40
Room: Salle de Réunion - Bâtiment M2
12:40 Lunch
Lunch
12:40 - 14:30