Journées Jeunes EDPistes en France 2024

Liste des contributions

1. Resonances as a computational tool

Katharina Schratz

20/03/2024 14:00

A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient...

2. Étude statistique de la formation des vagues extrêmes

Ricardo Grande Izquierdo

20/03/2024 14:50

Nous nous intéresserons à la formation de vagues extrêmes, en haute mer, en adoptant un point de vue probabiliste. Nous identifierons en premier lieu le premier terme du développement asymptotique de la probabilité d'occurrence d'une telle vague lorsque la hauteur de la vague tend vers l'infini. Si une vague extrême survient, quelle est la donnée initiale la plus probable qui l'a produite ?...

3. Nonlinear Schrödinger equations on compact surfaces

Nicolas Camps

20/03/2024 15:20

This talk is devoted to the general study of the long-time dynamics of solutions to nonlinear Schrödinger equations (NLS) on compact surfaces. In this context, weak dispersion and nonlinear resonances can cause energy cascades from low to high frequency scales of oscillations. Meanwhile, one can use the Galerkin approximation and extend methods from the study of finite-dimensional Hamiltonian...

4. The logarithmic Bramson correction for Fisher-KPP equations on the lattice Z

Mingmin Zhang

20/03/2024 16:20

In the talk, I will present the logarithmic Bramson correction for Fisher-KPP equations on the lattice Z, that is the level sets of solutions with step-like initial conditions are located at position c∗t − (3/(2λ∗))ln t + O(1) as t → +∞ for some explicit positive constants c∗ and λ∗. This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments.

5. Asymptotic analysis and simulation of collisional Vlasov-Poisson models

Alain Blaustein

20/03/2024 16:50

This presentation focuses on collisional Vlasov-Poisson systems. These kinetic models are of primary interest, as they encode the multiple scales that arise in a plasma, ranging from fluid-like behavior when collisions dominate to wave interactions in weakly collisional regimes. We present quantitative results that capture the scales of both the continuous model and its discretized...

6. On the hydrostatic limit of the Euler-Boussinesq equations

Lucas Ertzbischoff

20/03/2024 17:20

I will talk about the hydrostatic approximation of the 2d Euler-Boussinesq system, describing the evolution of an inviscid stratified fluid where the vertical length scale is much smaller than the horizontal one. Even though of importance in oceanography, the justification of the hydrostatic limit in this context has remained an open problem. I will discuss some recent results showing that...

7. Coût d'observabilité en temps petit de l'équation de la chaleur 1D

Jeremi Darde

21/03/2024 09:00

L'estimation du coût d'observabilité en temps petit de l'équation de la chaleur 1D (et, par dualité, celle du coût de contrôle en temps petit de la même équation), est une longue histoire qui commence dans les années 80, et n'est toujours pas terminée.

Dans cet exposé, j'expliquerai comment, dans un travail avec Sylvain Ervedoza (2019), nous avons amélioré l'estimation par au-dessus de ce...

8. The convergence problem in mean-field control theory and related PDEs over the space of probability measures

Samuel Daudin (Université Côte d'Azur)

21/03/2024 09:50

The goal of this talk is to discuss recent progress in the convergence problem in mean-field control theory.

We are interested in control problems involving a large number of (controlled) interacting particles subject to independent noises of Brownian type. When the number of particles tends to infinity, the problem simplifies into a control problem of mean-field type, set on the space of...

9. Existence of strong solutions for a compressible fluid-solid interaction system with Navier slip boundary conditions

Imene Djebour

21/03/2024 10:55

We consider a fluid-structure interaction system coupling a viscous fluid governed by the compressible Navier-Stokes equations and a rigid body immersed in the fluid and modeled by the Newton's law. In this work, we consider the Navier slip boundary conditions. Our aim is to show the local in time existence and uniqueness of the strong solution to the corresponding problem. The main step of...

10. A finite-difference based variational approach for solving Hamilton-Jacobi equations in high-dimensional domains

Carlos Esteve-Yagüe

21/03/2024 11:25

It is well-known that the value function associated to a given optimal control problem or differential game can be characterised as the viscosity solution of an associated Hamilton-Jacobi equation. Numerical methods based on finite-differences are guaranteed to approximate the viscosity solution, provided the numerical scheme has the correct monotonicity. However, these grid-based methods...

11. Regularity issue for the system describing elastic structure interacting with Navier-Stokes equations

Pei Su

21/03/2024 14:00

We are interested in the interaction of a viscous incompressible fluid with an elastic structure, where the structure is located on a part of the fluid boundary. It reacts to the surface forces induced by the fluid and deforms the reference domain $\Omega$ to $\Omega_\eta$. The fluid equations are coupled with the structure via the kinematic condition and the action-reaction principle on the...

12. Two formulations for acoustic and surface waves in a free-surface, vertically stratified ocean

Juliette Dubois

21/03/2024 14:30

I will present two formulations for a linear model describing the propagation of acoustic and surface gravity waves in a free-surface, stratified ocean. The first formulation, already studied in previous works, is obtained by the linearization of the compressible Euler equations, written in Lagrangian coordinates. The second formulation uses a new variable which can be understood as a...

13. Solutions de Yudovich non-bornées pour les équations d'Euler 2D

Dimitri Cobb

21/03/2024 15:00

Dans cet exposé, nous étudierons les solutions non-bornées des équations d'Euler incompressibles en deux dimensions d'espace. Ces solutions trouvent leur interêt dans le fait que les espaces habituels de solutions (p. ex. basés sur une condition d'énergie finie comme $L^2$) ne respectent pas certaines des symétries du problème : l'invariance de Galilée et l'invariance d'échelle. Par ailleurs,...

14. On collapses of single-signed point-vortices with the boundary

Dragos Iftimie (Institut Camille Jordan - Université Lyon 1)

21/03/2024 16:00

We consider the point-vortex system in a domain and assume the masses to be single-signed. We investigate the possibility of collapse with the boundary. We prove that such collapses are not possible in the case of the disk and of the half-plane. For general domains, we give a necessary condition for a collapse with the boundary to occur. This is joint work with M. Donati and L. Godard-Cadillac.

15. Nonlinear waves on lattices and continuum limit

Quentin Chauleur

21/03/2024 16:50

In this talk we will be interested in the dynamics of dispersive PDEs on infinite lattices. In particular, we will highlight how the dispersive properties of the solutions, which are weaker than the one on the continuous setting, can be used in order to study the continuum limit of such systems as the step size of the grid tends to zero. We will also provide some perspectives on the subject.

16. Uniform in time propagation of chaos for the 2D vortex model

Pierre Le Bris

21/03/2024 17:20

We are interested in a system of particles in singular mean-field interaction and wish to prove that, as the number of particles goes to infinity, two given particles within that system become « more and more » independent, a phenomenon known as propagation of chaos. The interaction we will focus on comes from the Biot-Savart kernel, for which the nonlinear limit of the particle system...

17. Burnett's conjecture in General Relativity

Cécile Huneau (CNRS et Ecole Polytechnique)

22/03/2024 09:00

In this work, I will present a work in collaboration with Jonathan Luk where we prove that weak limits of solutions to Einstein vacuum equations, in some setting, converge to solutions to Einstein equations coupled to a massless Vlasov field. The proof uses the microlocal deffect measures of Tartar and Gérard, and compensated compactness.

18. Strictly-Correlated Electrons from the viewpoint of optimal transport and their dissociation at infinity

Rodrigue Lelotte

22/03/2024 09:50

The Strictly-Correlated Electrons (SCE) is a formalism of Density-Functional Theory (DFT) used to approximate ground-state energies of strongly-correlated quantum systems. From a mathematical viewpoint, it is obtained as the semi-classical limit of the Levy-Lieb functional, which is one of the central objects to DFT. I will present this problem, which arises as a multimarginal optimal...

19. Zero resonant states for the Schrödinger operator

Viviana Grasselli

22/03/2024 10:55

The Schrödinger operator on the whole space R^d gives rise to a dispersive equation, meaning that the mass of the solution spreads towards infinity, and these dispersive properties are tightly linked to its spectrum. Resonances can be seen as a generalisation of eigenvalues: they are complex numbers for which the eigenvalue equation admits a non L^2 solution. Their dynamical interpretation is...

20. The Robin Harmonic Measure on non smooth domains

Marco Michetti

22/03/2024 11:25

We analyze the boundary behavior of solutions to elliptic PDE with prescribed Robin data in rough domains. In particular, we construct a ``Robin elliptic measure" and demonstrate the suprising fact that this measure is (quantitatively) mutually absolutely continuous with respect to surface measure on a wide class of domains that includes the complement of certain fractals.