Séminaire d'analyse (archives)

Dans cet exposé, nous étudions le spectre du Laplacien magnétique dans des domaines extérieurs du plan, avec conditions aux limites de Neumann et champ magnétique uniforme. Pour le complémentaire d'un disque, nous établissons des asymptotiques précises des premières valeurs propres dans la limite de champ magnétique faible. Pour des domaines étoilés plus généraux, nous obtenons une borne supérieure asymptotique sur la plus petite valeur propre, faisant intervenir un moment géométrique de la frontière — résultat optimal dans le cas du disque. De plus, nous montrons que pour des champs magnétiques modérés, le complémentaire du disque réalise un maximum local pour la plus petite valeur propre sous contrainte de moment.

Travail en collaboration avec Vladimir Lotoreichik (Prague) et Mikael Sundqvist (Lund).

The inverse spectral problem asks to what extent one can recover the geometry of a manifold from knowledge of either its Laplace spectrum or dynamical counterparts, e.g., the (marked) length spectrum. While counterexamples do exist in general, there are certain symmetry and nondegeneracy conditions under which spectral uniqueness holds. Perhaps the most tantalizing unsolved case is that of strictly convex planar domains, known as Birkhoff billiard tables. It turns out that there is a deep relationship between the Laplace and length spectra, which is encoded in the Poisson relation. In this talk, I will describe my work on both Laplace and length spectral invariants as well as limitations in using the Poisson relation for inverse problems.

Dans le domaine des interactions fluide/solide, une question cruciale est la description de l'écoulement d'un fluide confiné entre deux surfaces solides en mouvement. Dans ce contexte, il est d'usage de considérer le problème de Stokes stationnaire de la mécanique de fluide complété par des conditions aux bords reproduisant le déplacement des deux surfaces et il s'agit d'obtenir un développement asymptotique de la solution quand la distance entre les deux surfaces tend vers 0.

Dans cet exposé, je présenterai une méthode variationnelle permettant de discuter l'asymptotique de la solution du problème de Stokes dans les cas bidimensionnels et tridimensionnels. Nous verrons que ces différents cas recèlent des difficultés différentes inhérentes aux propriétés des champs de vecteurs à divergence nulle dans chaque dimension. Le contenu de ce travail est le fruit de collaborations avec E. Bocchi, D. Bonheure, C. Patriarca et G. Sperone.

In this talk we consider the four-waves spatially homogeneous kinetic equation arising in weak wave turbulence theory from the microscopic Fermi-Pasta-Ulam-Tsingou (FPUT) oscillator chains. This equation is sometimes referred to as the Phonon Boltzmann Equation. I will discuss the global existence and stability of solutions of the kinetic equation near the Rayleigh-Jeans (RJ) thermodynamic equilibrium solutions. This is a joint work with Pierre Germain (Imperial College London) and Joonhyun La (KIAS).

In this talk, we are interested in a transmission problem between a dielectric and a metamaterial. The question we consider is the following: does the limiting amplitude principle hold in such a medium? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation.

An answer is proposed here in the case of a plane interface between a metamaterial represented by the Drude model and the vacuum, which fill respectively complementary half-spaces. In this context, we reformulate the time-dependent Maxwell’s equations as a conservative Schrödinger equation and perform its complete spectral analysis. This permits a quasi-explicit representation of the solution via the ”generalized diagonalization” of the associated unbounded self-adjoint operator. As an application of this study, we show finally that the limiting amplitude principle holds except for a particular fequency characterized by a ratio of permittivities and permeabilities equal to −1 across the interface. This frequency is a resonance of the system and the response to this excitation blows up linearly in time.

Joint work with Christophe Hazard (CNRS, Poems team) and Patrick Joly (INRIA, Poems team)

In this presentation, we will construct regular solutions to linear and nonlinear elliptic-parabolic equations in which the natural direction of parabolicity reverses along a critical line. To prevent the emergence of singularities, we will impose orthogonality conditions on the source terms, and follow them during the execution of the nonlinear schemes.

This is a joint work with Anne-Laure Dalibard and Jean Rax, motivated by recirculation problems in boundary layer theory for fluid mechanics, and based on the preprint https://arxiv.org/abs/2203.11067

Splitting schemes are a natural and easy to implement approach to integrate numerically in time differential equations. However, high order splitting methods suffer in general from an order reduction phenomena when applied to the integration of partial differential equations with non-periodic boundary conditions. In this talk, inspired by recent corrector techniques for the second order Strang splitting method, we present a new splitting method of order three for a class of semilinear parabolic problems that avoids order reduction. We prove the third order convergence of the method in a simplified linear setting and confirm the result by numerical experiments. Moreover, we observe numerically that the high order convergence persists for an order four variant of a splitting method, and also for a nonlinear source term.

Je vous présenterai un résultat récent, en collaboration avec Benoît Grébert et Tristan Robert, concernant l'existence de tores invariants de dimension infinie pour les équations de Schrödinger non-linéaires sur le cercle (sans paramètres externes). En particulier, nous montrons l'existence de solutions presque-périodiques qui ne sont pas quasi-périodiques. Notre preuve repose notamment sur la théorie KAM et des effets régularisants dispersifs que je décrirai.