Séminaire EDP et Applications | Nancy

More than 50 years ago, Moore predicted that the number of transistors on a microchip doubles every two years. This exponential growth is approaching its physical limit, highlighting the need for alternative computing paradigms. One promising avenue is neuromorphic computing, which aims to emulate the structure and function of the human brain. A key enabling technology is the memristor, a nonlinear resistor with memory. Memristors are capable of mimicking the dynamic conductance behavior of biological synapses, making them well-suited for implementing energy-efficient neural networks.

This talk focuses on the mathematical analysis of three-species drift-diffusion equations for memristors. We investigate the existence and boundedness of global-in-time weak solutions. The mathematical difficulties originate from the three-species situation and the different types of boundary conditions. These issues are addressed by combining free energy estimates with local and global compactness arguments. Additionally, we analyze memristor models coupled with electrical networks. One-dimensional numerical simulations capture the characteristic hysteresis behavior in the current-voltage curves, which are a fingerprint for memristive devices.

In this talk, I will present recent results on unique continuation for semilinear wave and Schrödinger equations with analytic nonlinearities. After recalling some motivation and classical results on the topic, I will describe a new method, introduced in a joint work with Camille Laurent (CNRS, LMR), that relies on analyticity-in-time regularization in finite time for solutions vanishing on a subset satisfying the Geometric Control Condition (GCC). The proof combines tools from control theory with ideas of Hale and Raugel on the regularity of attractors in dynamical systems. In a more recent work, we refined this approach and applied it to Schrödinger equations on compact manifolds, showing that the GCC suffices for unique continuation, thus answering in the affirmative an open problem posed by Dehman, Gérard, and Lebeau (2006) for the nonlinear case. The method is abstract and can also be applied to study similar questions for other PDEs.

Attention, ce séminaire aura lieu en salle Döblin.

Water-wave models play a crucial role in understanding, predicting, and controlling the dynamics of surface water waves across various real-world scenarios, including oceanic waves, waves in lakes and rivers, and those affecting man-made structures. These models integrate mathematical, physical, and numerical frameworks with wide-ranging applications in environmental science, engineering, and maritime industries. In this talk, we will explore key mathematical results for several water-wave models, highlighting their relevance to real-world applications.

We survey interesting properties of some nonlocal operators that have no analogue for linear second order elliptic PDE.

Je présenterai un travail en collaboration avec Thibault Lacombe, où
nous démontrons que les solutions Lipschitz $u$ de $\mathrm{div}\,
G(\nabla u)=0$ dans un domaine du plan sont $C^1$, pour des champs de
vecteurs strictement monotones $G$ dans $C^0(\mathbb R^2;\mathbb R^2)$
satisfaisant une condition d’ellipticité très générale. Lorsque le champ
de vecteurs $G$ est le gradient d’une fonction strictement convexe,
notre résultat généralise des résultats de De Silva et Savin (Duke Math.
J. 2010). Lorsque $G$ n’est pas un gradient, l’hypothèse d’ellipticité
doit être interprétée correctement, et nous produisons un exemple qui
montre l’effet non trivial de la partie antisymétrique de $\nabla G$.

Il s’agit d’un travail en collaboration avec Guedes-Bonthonneau, Chhaibi, Rivière et To.

Sur une surface riemannienne, nous construisons une mesure de Yang-Mills sur les connexions distributionnelles, avec le processus d’holonomie correspondant. Les ingrédients de notre construction sont: une nouvelle jauge de Morse et l’analyse d’équations de transport pour les flots de gradient en très faible régularité. Nous montrons que notre mesure retrouve les formules de la théorie de Yang-Mills en dimension 2, telles qu’on les trouve dans les travaux de Witten, Driver, Sengupta et Lévy. Enfin, nous expliquons en quel sens notre mesure converge vers le volume symplectique d’Atiyah–Bott–Goldman sur l’espace de modules des connexions plates dans la limite semi-classique.

It is well established that the L_1 optimal transport can be employed to characterize solutions of the Beckmann problem, where one looks for a vector-valued measure of minimal total variation with the constraint that fixes its divergence as the difference of two probabilities distributions. In turn, the Beckmann problem underlies optimal design of heat conductors.

I will talk about a second-order counterpart of this theory, where in the Beckmann problem we have a constraint on the double divergence. In 2D, this problem enjoys the interpretation of optimally designing a ceiling using a grillage structure. I will show that its solutions can be characterized through a new formulation where we look for a probability that dominates the data in the sense of convex order while attaining minimal variance. Afterwards, equivalent optimal transport formulations can be proposed for efficient numerical treatment.

Work in collaboration with Guy Bouchitté.

Dans le cadre de coefficients réguliers et sous une hypothèse de contrôle géométrique que je rappellerai, l’une des méthodes les plus modernes pour démontrer l’observabilité des ondes repose sur la considération de paquets d’ondes dont la fréquence typique tend vers l’infini, sur la compréhension du transport des mesures semiclassiques le long des rayons de l’optique géométrique, ainsi que sur un argument de prolongement unique, le tout enrobé dans un raisonnement par contradiction. J’exposerai cette recette subtile et montrerai comment elle permet de généraliser le résultat d’observabilité à des cas où les coefficients présentent une régularité plus faible, jusqu’à des coefficients de classe C^1. Dans ce cas, le champ de vecteurs hamiltonien qui gouverne les rayons n’est plus que C^0. L’unicité des rayons est alors perdue. Une régularité encore moindre compromettrait l’existence même des rayons.

In the derivation of the wave kinetic equation coming from the Schrödinger equation, a key feature is the invariance of the Schrödinger equation under the action of U(1). This allows quasi-resonances of the equation to drive the effective dynamics of the statistical evolution of solutions to the Schrödinger equation. In this talk, I will give an example of an equation that does not have the same invariance as the Schrödinger equation, and I will show that in this example, exact resonances (always) take precedence over quasi-resonances, so that the effective dynamics of the statistical evolution of the solutions are not kinetic. However, these dynamics are not linear (let alone trivial). I will present the problem and the ideas involved in deriving the effective dynamics and some elements of proof: in particular, I will describe the representation of solutions of the initial equation in diagrammatic form. This talk is based on a joint work with Annalaura Stingo (X) and Arthur Touati (Bordeaux).

In this talk, I will discuss how a general bilinear finite-dimensional closed quantum system with dispersed parameters can be steered between eigenstates. We show that, under suitable conditions on the separation of spectral gaps and the boundedness of parameter dispersion, rotating wave and adiabatic approximations can be employed in cascade to achieve population inversion between arbitrary eigenstates.

Le séminaire aura lieu en visio-conférence dans la salle de conférence.

This talk deals with the stability analysis of discrete shock profiles
for systems of conservation laws. These profiles correspond to
approximations of shocks of systems of conservation laws by
conservative finite difference schemes. Discontinuous solutions
appear naturally in the study of systems of conservation laws, which
can model many physical situations, such as gas dynamics. Existence
and stability of discrete shock profiles for each stable shock of the
approximated system of conservation laws is seen as an
improved consistency condition and implies that the finite difference
scheme should be able to approach discontinuities fairly precisely.

The aim of the talk is to review some stability results regarding
discrete shock profiles and to present a recent effort to extend them.
More precisely, most results known up until recently are focused on
the stability of discrete shock profiles associated with shocks
of small amplitude. The talk will focus on a nonlinear orbital
stability result for discrete shock profiles in quite a general
setting, where the smallness assumption on the shock’s amplitude is
replaced by a spectral stability assumption on the linear operator
obtained by linearizing the numerical scheme about the discrete shock
profile. This nonlinear orbital stability result relies on a precise
description of the Green’s function of the linearization about
discrete profiles.