PDE and applications seminar | Metz

We consider linear waves on a bounded domain where one part of the boundary is governed by a coupled lower-dimensional wave equation (i.e., dynamic Ventcel/Wentzell boundary condition) and is subject to viscous damping. The other (possibly empty) part is left at rest. When the dynamic boundary geometrically controls the domain, we show that the total energy of classical solutions decays like 1/t. The proof relies on an analysis of high-frequency quasimodes, suitable boundary estimates obtained in different microlocal regimes, and a special decoupling argument. Optimality is assessed via an appropriate quasimode construction.

Ongoing work with Nicolas Vanspranghe (Inria team DISCO, L2S –CentraleSupélec).

Résumé à venir

Bohr’s correspondence principle asserts that the predictions of classical and quantum mechanics coincide in the limit of large quantum numbers. This connection becomes especially striking when one studies Schrödinger-type equations for initial data minimising the uncertainty principle, known as “gaussian coherent states”, in the semiclassical limit. More precisely, in the context of quantum mechanics, one can derive a complete asymptotic expansion in the semiclassical parameter for solutions of such equations. The aim of this talk is to explain how to obtain a similar result in the framework of Quantum Field Theory for a certain class on analytic interactions, with a particular stress on spatially cutoff $P(\phi)_2$ interactions.

Résume à venir

Résumé à venir

Inspiré par la physique des films de savon, le problème de Plateau est un problème classique des mathématiques qui consiste à minimiser l’aire d’une surface s’appuyant sur un bord. Il admet en fait de nombreuses formulations, certaines ne permettant pas toujours de représenter fidèlement les films de savon. Dans cet exposé, on s’intéresse à un modèle issu des travaux d’Almgren et de David, particulièrement bien adapté pour décrire ces films. Il s’agit du «problème de Plateau glissant», où l’on minimise les déformations continues d’une surface sans la détacher du bord. Bien qu’assez intuitive, c’est l’une des formulations qui résiste le mieux aux mathématiciens car on ne connait toujours pas de théorème d’existence général. On présentera un résultat d’existence dans un cas particulier, obtenu avec G. David.

In this talk, I will present an asymptotic expansion for the minimal Dirichlet energy of  $\mathbb{S}^2$-valued maps outside a finite number of particles of size $\rho$ in $\mathbb{R}^3$ under general anchoring conditions at the particle boundaries as $\rho\to 0$. The first two terms of this expansion consist of the minimal energy after zooming in at scale $\rho$ around each particle and a Coulomb-like interaction that agrees with the electrostatics analogy depending on the far-field behavior of the corresponding single-particle minimizer. This approximation is commonly used in the physics literature for colloid interactions in nematic liquid crystals and for the first time a precise estimate of the energy error introduced by that linearization is derived, by developing new tools that address the lack of convergence rate when zooming in at scale $\rho$.

The talk is based on joint work with L. Bronsard, X. Lamy and R. Venkatraman.

The objective of this work is to study the stability of two systems of type Rao-Nakra sandwich beam in the whole line $R$ with a frictional damping or an infinite memory acting on the Euler-Bernoulli equation. When the speeds of propagation of the two wave equations are equal, we show that the solutions do not converge to zero when time goes to infinity. In the reverse situation, we prove some $L^2 (R)$-norm and $L^1 (R)$-norm decay estimates of solutions and theirs higher order derivatives with respect to the space variable. Thanks to interpolation inequalities and Carlson inequality, these $L^2 (R)$-norm and $L^1 (R)$-norm decay estimates lead to similar ones in the $L^q (R)$-norm, for any $q\in [1,+\infty]$. In our both $L^2 (R)$-norm and $L^1 (R)$-norm decay estimates, we specify the decay rates in terms of the regularity of the initial data and the nature of the control. Applications to some Cauchy Timoshenko type systems will be also given. The proof is based on the energy method combined with the Fourier analysis (by using the transformation in the Fourier space and well chosen multipliers).

A part of these results was obtained in collaboration with Salim Messaoudi (University of Sharjah, UAE).

For the details, see the following papers:

A. Guesmia, Some $L^q (R)$-norm decay estimates ($q\in[1,+\infty]$) for two Cauchy systems of type Rao-Nakra sandwich beam with a frictional damping or an infinite memory, J. Appl. Anal. Comp., 12 (2022), 2511-2540.
A. Guesmia, On the stability of a linear Cauchy Rao-Nakra sandwich beam under frictional dampings, Taiwanese J. Math., 27 (2023), 799-811.
A. Guesmia and S. Messaoudi, Some $L^2 (R)$-norm and $L^1 (R)$-norm decay estimates for Cauchy Timoshenko type systems with a frictional damping or an infinite memory, J. Math. Anal. Appl., 527 (2023), 127385.

Kato’s well known distributional inequality for the magnetic Laplacian holds equally in the more general setting of non-relativistic quantum electrodynamics (QED), where the wave function is vector-valued and the vector potential is quantized. We give two new applications of this result: First, we show that eigenstates satisfy a subsolution estimate. Second, for general states, with energy distribution strictly below the ionization threshold, we give a short proof of pointwise exponential decay in the electronic configuration.

Joint work with Valentin Kußmaul

The main objective of this work is to study the stability of a linear one-dimensional thermoelastic Bresse system in a bounded domain, where the coupling is given through the first component of the Bresse model with the heat conduction of Gurtin-Pipkin type. Two kinds of coupling are considered; the first coupling is of order one with respect to space variable, and the second one is of order zero. We state the well-posedness and show the polynomial and strong stability of the systems for regular and weak solutions, respectively, where the polynomial decay rates depend on the smoothness of the initial data. Moreover, in case of coupling of order one, we prove the equivalence between the exponential stability and some new conditions on the parameters of the system. However, when the coupling is of order zero, we prove the non-exponential stability independently of the parameters of the system. Applications to the corresponding particular Timoshenko models are also given, where we prove that both couplings lead to the exponential stability if and only if some conditions on the parameters of the systems are satisfied, and both couplings guarantee the polynomial and strong stability for regular and weak solutions, respectively, independently of the parameters of the systems. The proof of the well-posedness result is based on the semigroups theory, whereas a combination of the energy method and the frequency domain approach is used for the proof of the stability results.

For the details, see the following paper:

A. Guesmia, Well-posedness and stability results for thermoelastic Bresse and Timoshenko type systems with Gurtin-Pipkin’s law through the vertical displacements, SeMa J., (2023), 1-49.

Finite-time stabilization of linear systems using either additive or multiplicative controls is studied. A necessary condition is formulated in terms of a weak observability condition and is used to construct the set of initial states from which, the system can reach its equilibrium in finite time. Sufficient conditions are then provided to guarantee finite-time stabilization. The approach mainly relies on Lyapunov-like functions. Applications to finite-dimensional systems and partial differential equations are also presented.

Durant ce séminaire nous étudierons la théorie de la diffusion pour une famille d’opérateurs de Schrödinger. Ces opérateurs possèdent des spectres présentant un changement de multiplicité et donc des seuils plongés. Certains opérateurs possèdent également des résonances aux seuils. Nous construirons alors une ${\rm C}^*-$algèbre à laquelle appartient les opérateurs d’onde. L’étude du quotient de cette algèbre par l’idéal des opérateurs compacts mène directement à l’existence de théorèmes d’indice en théorie de la diffusion. Ces théorèmes peuvent alors s’interpréter comme des théorèmes de Levinson en présence de seuils plongés et de discontinuités de la matrice de diffusion. La dépendance de ces résultats en fonction de certains paramètres sera également discutée. Aucun prérequis ${\rm C}^*-$algébrique n’est nécessaire pour cette présentation.

Dans cet exposé nous commencerons par expliquer le cas linéaire et comment la méthode de pénalisation utilisée notamment par Donati en 1982 permet de montrer l’existence d’une solution à un problème d’obstacle dans le cadre variationnel usuel et l’inégalité de Lewy-Stampacchia associée. Nous aborderons ensuite le cas d’équations paraboliques quasi-linéaires et les difficultés supplémentaires liées à la perte de la monotonie de l’opérateur. Avec une modification ad-hoc de l’opérateur, un résultat de densité et un lemme d’intégration par parties à la Mignot-Bamberger-Alt-Luckhaus nous démontrerons une extension des résultats de Donati pour une classe plus générale d’équations et toujours dans le cadre variationnel.
Enfin, si le temps le permet, nous discuterons de la généralisation aux cas de donnée dans $L^1$, hors du cadre variationnel, avec l’utilisation de la notion de solutions entropiques pour le problème d’obstacle et de la notion de solutions renormalisées pour l’inégalité de Lewy-Stampacchia associée.

In this talk I present a joint paper with Umberto De Maio (Università degli Studi di Napoli “Federico II”, Italy) and Catalin Lefter (Al.I.Cuza University and Octav Mayer Institute of Mathematics, Iasi, Romania).

This paper is devoted to studying the null internal controllability of a Kirchhoff-Love thin plate with a middle surface having a comb-like shaped structure with a large number of thin fingers described by a small positive parameter $\varepsilon$. It is often impossible to directly approach such a problem numerically, due to the large number of thin fingers. So an asymptotic analysis is needed. In this paper, we first prove that the problem is null controllable at each level $\varepsilon$. We then prove that the sequence of the respective controls with minimal $L^2$ norm converges, as $\varepsilon$ vanishes, to a limit control function ensuring the optimal null controllability of a degenerate limit problem set in a domain without fingers.