To be updated until april 30th

Léo Bigorgne

## Asymptotics properties of the small data solutions of the Vlasov-Maxwell system

Léo Bigorgne
leo.bigorgne@u-psud.fr
Département de Mathématiques Bâtiment 307 Faculté des Sciences d'Orsay Université Paris-Sud F-91405 Orsay Cedex
The Vlasov-Maxwell system is a classical model in plasma physics. Glassey and Strauss proved global existence for the small data solutions of this system under a compact support assumption on the initial data. I will present how vector field methods can be applied to revisit this problem. In particular, it allows to remove all compact support assumptions on the initial data and obtain sharp asymptotics on the solutions. We will also discuss the null structure of the system which constitutes a crucial element of the proof.

Michaël Brunengo

## Deformation of an elastic material paired with a tree structure

Michaël Brunengo
Michael.BRUNENGO@univ-cotedazur.fr
Laboratoire J.A. Dieudonné, Parc Valrose, 28 Avenue Valrose, 06108 Nice Cedex 02, 06000 Nice
In order to model effects of automated treatment of respiratory physiotherapy with focused pulses, we study deformation of an elastic material under oscillating constraints on its boundaries. Moreover the system is linked to a symmetrical and dichotomous tree built as series of cylinders, idealizing bronchial tree. To do so, under infinitesimal strain theory, we consider that a change a volume on an area of the material creates airflow. Then we force it to flow in the tree and to go through hydrodynamic resistance. This coupling adds friction to the system and gives information on total airflow created at the top of the tree, i.e. at the mouth, under pressure on boundaries.

David Chiron

## Branches of traveling waves for the Nonlinear Schrödinger equation

David Chiron
david.chiron@univ-cotedazur.fr
Laboratoyr J.A. Dieudonné, Université Côte d'Azur, Parc Valrose 06108 NICE Cedex 02, France
We consider the cubic Nonlinear Schrödinger equation in the plane with condition of modulus one at infinity. This model possesses traveling waves. We shall present two types of results of existence of (smooth) branches of traveling waves: a theoretical one obtained in collaboration with E. Pacherie for small speeds and numerical results obtained in collaboration with C. Sheid on the excited states for this model.

Charles Collot

## On singularity formation for the unsteady Prandtl's system

Charles Collot
cc5786@nyu.edu
Courant Institute, New York University
Prandtl's equations arise in the description of boundary layers in fluid dynamics. Solutions might form singularities in finite time, with the first reliable numerical studies performed by Van Dommelen and Shen in the early eighties, and a rigorous proof done later in the nineties in the seminal work of E and Engquist in two dimensions. This singularity formation is intimately linked with a phenomenon: the separation of the boundary layer. The precise structure of the singularity has however not been confirmed yet mathematically. This talk will first describe the dynamics of the inviscid model, for which we explain how the Van Dommelen and Shen singularity appears generically. Then, for the full viscous model, the second part of the talk will focus on the obtention of detailed asymptotics for the solution at a relevant particular location. This is a collaboration with T.-E. Ghoul, S. Ibrahim and N. Masmoudi.

Pacherie Eliot

## Smooth branch of travelling waves for the Gross-Pitaevskii equation in dimension 2 for small speed

Pacherie Eliot
Eliot.PACHERIE@univ-cotedazur.fr
Université Côte d'Azur, CNRS, LJAD, France

We construct a smooth branch of travelling wave solutions for the 2 dimensional Gross-Pitaevskii equations for small speed. These travelling waves exhibit two vortices far away from each other. We also compute the leading order term of the derivatives with respect to the speed. We construct these solutions by an implicit function type argument. In collaboration with David Chiron

Philippe Gravejat

## Two asymptotic regimes of the Landau-Lifshitz equation

Philippe Gravejat
philippe.gravejat@u-cergy.fr
Cergy-Pontoise University, AGM Research Center in Mathematics (UMR 8088), F-95302 Cergy-Pontoise Cedex, France
The Landau-Lifshitz equation gives account of the dynamics of magnetization in ferromagnets. The goal of this talk is to describe the rigorous derivation of two aymptotic regimes of this equation corresponding to the Sine-Gordon equation and the cubic Schrödinger equation. This talk is based on two papers in collaboration with André de Laire (University of Lille).

Benoit Grebert

## Long time behavior of the solutions of NLW on the d-dimensional torus

Benoit Grebert
benoit.grebert@univ-nantes.fr
LMJL Univerrsité de Nantes UMR CNRS 6629, 2 rue de la Houssinière 44300 Nantes, Fra,nce

We consider the non linear wave equation (NLW) on the d-dimensional torus

$$u_{tt} - \Delta u + \mu + f(u) =0\quad x \in \mathbb T^d$$

where $f=\partial_u F$ is analytic on a neighborhood of the origin and which is at least of order 2 at the origin. Let $u(t)$ be a solution corresponding to a small initial datum $u(0)\in H^s(\mathbb T^d)$. We prove that we control $[u(t)]_s$ that mix the $H^s$ norm of the $\varepsilon^{-\beta(r)}$ lower Fourier modes of the solution $u$ and the energy norm of the remaining higher modes during long times of order $\varepsilon^{-r}$.

Our general strategy applies to any Hamiltonian PDEs whose linear frequencies satisfy only a first Melnikov condition. In particular it also applies to the Hamiltonian Boussinesq $abcd$ system and the Whitham-Boussinesq system in water waves theory. Joint work with Joackim Bernier and Erwan Faou.

Billel Guelmame

## A Hamiltonian regularisations of barotropic Euler equations

Billel Guelmame
billel.guelmame@unice.fr
Université Côte d'Azur, LJAD, 28 Avenue Valrose, 06108 Nice Cedex 02, 06000 Nice

The inviscid Burgers, Euler and Saint-Venant equations are nonlinear hyperbolic PDEs modeling fluid flows and surface water waves propagating in shallow water. These equations, prominent in physics, are the subject of numerous mathematical and numerical investigations. It is well-known that these equations develop shocks in finite time, even for regular initial conditions. These shocks are problematic, in particular, for numerical simulations. Therefore, several techniques have been proposed to regularized these equations. Adding viscosity or/and dispersion into the equations can avoid the formation of shocks. Here, we study a regularization of barotropic Euler equations, which conserves the energy, and generalize the conservative regularization of the Saint-Venant equations proposed by Clamond and Dutykh in 2017.

Mohammed Lemou

## Nonlinear stability and instability results for gravitational kinetic models

Mohammed Lemou
mohammed.lemou@univ-rennes1.fr
CNRS & université de Rennes 1
The orbital nonlinear stability of steady states solutions to the gravitational Vlasov-Poisson system, which are decreasing functions of the energy, has been proved in 2012. However, this result is partially based on compactness arguments and does not provide a complete quantitative information on the perturbation. In this talk, I will start by presenting a quantitative version of this nonlinear stability result. In particular a refined functional inequality on extended rearrangements of functions is proved, which is then combined with a Poincaré-like inequality. Another advantage of this approach is its applicability to other systems like the so-called Hamiltonian Mean Field (HMF), where the space domain is bounded and where the decreasing property of the steady states is no more sufficient to guarantee their stability. In fact, an additional explicit criteria is needed for HMF, under which the non-linear stability is proved. In a last part of this talk, we show that this criteria is sharp by proving a nonlinear instability result for HMF when the criteria is not satisfied. To this aim, we use an iterative procedure that was introduced by Grenier in a different context.

Gabriel Rivière

## Equidistribution of toral eigenfunctions along hypersurfaces

Gabriel Rivière
gabriel.riviere@univ-nantes.fr
Université de Nantes, Laboratoire de mathématiques Jean Leray, 2 rue de la Houssinière, BP92208, 44322 Nantes Cedex 3, France
I will discuss asymptotic properties of Laplace eigenfunctions on the flat torus in the high frequency limit. I will present results showing equidistribution of these eigenfunctions along hypersurfaces with nonvanishing curvature. This is a joint work with Hamid Hezari (U.C. Irvine).

Lionel Rosier

## Control of nonlinear parabolic PDEs

Lionel Rosier
lionel.rosier@mines-paristech.fr
Centre automatique et systèmes, Mines de Paris

It is by now well known that the use of Carleman estimates allows to establish the controllability to trajectories of nonlinear parabolic equations. However, by this approach, it is not clear how to decide whether a given function is indeed reachable. That issue has obtained very recently almost sharp results in the linear case. In this talk, we investigate the set of reachable states for a nonlinear heat equation in dimension one. The nonlinear part is assumed to be an analytic function of the spatial variable $x$, the unknown $y$, and its derivative $y_x$.
By investigating carefully a nonlinear Cauchy problem in x in some space of Gevrey functions, and the relationship between the jet of space derivatives and the jet of time derivatives, we derive an exact controllability result for small initial and final data that can be extended as analytic functions on some ball of the complex plane. This is a joint work with Camille Laurent (Sorbonne Université). It time allows, works in progress about the reachable states for KdV and for ZK will be outlined.

Simona Rota Nodari

## Uniqueness and non-degeneracy for a class of semilinear elliptic equations

Simona Rota_Nodari
simona.rota-nodari@u-bourgogne.fr
Institut de Mathématiques de Bourgogne, Université de Bourgogne, UFR Sciences et Techniques, Faculté des Sciences Mirande, 9 avenue Alain Savary, 21078 Dijon Cedex, France

In this talk, I will present a result on the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Next, I will illustrate this result with two examples: a nonlinear Schrödinger equation for a nucleon and a Schrödinger equation with a double power non-linearity. This talk is based on joint works with Mathieu Lewin.

Martin Vogel

## Resonances of random quantum systems

Martin Vogel
vogel@math.unistra.fr
IRMA, Université de Strasbourg
The resonances of Schrödinger operators can be used to describe the large time behaviour of a wave scattered by a potential. In this context, the resonances which are the closest from the real axis are the most relevant. The distribution of resonances for potentials which decay rapidly at infinity has been studied a lot. On the other hand, for random potentials, there are very few known results. In this talk, I will discuss some recent results concerning the distribution of resonances for some random Schrödinger operators (joint work with F. Klopp).