Séminaire de EDP - Physique Mathématique

TBA

In this talk, we will present the study of a model for particle suspensions in a non-Newtonian Ostwald-DeWaele fluid with a potentially degenerate viscosity coefficient.

The analysis of problems associated with such systems is a very active research topic, the source of many recent results, particularly in the case of particles that sediment in a Newtonian fluid (see, for example, the works of D. Cobb, R. Höfer, A. Mecherbet, R. Schubert, F. Sueur). We will consider the case where particles are suspended in a non-Newtonian fluid. From a mathematical point of view, this is characterized by a Stokes-Transport equation with the particularity that the Stokes equation is nonlinear, the viscosity term being expressed as a p-Laplacian for the symmetrized gradient. After briefly contextualizing the problem, we will present our result: the existence of global weak energy solutions.

In order to show that we do define an active scalar equation, i.e. one in which the relative density of the particles suspended in the fluid gives meaning to a solution of the system through an inverse mapping, it is then necessary to use monotonicity methods in conjunction with techniques derived from DiPerna-Lions theory to establish the existence of suitable weak solutions. We will therefore present the main ideas for establishing the existence of such ones.

This talk is concerned with asymptotic persistence, extinction and spreading properties for structured population models resulting in non-cooperative Fisher-KPP systems with space-time periodic coefficients, motivated by a wide class of models in population biology. Results are formulated in terms of a family of generalized principal eigenvalues associated with the linearized problem. When the maximal generalized principal eigenvalue is negative, all solutions to the Cauchy problem become locally uniformly positive in long-time, at least one space-time periodic uniformly positive entire solution exists, and solutions with compactly supported initial condition asymptotically spread in space at a speed given by a Freidlin-Gärtner-type formula. When another, possibly smaller, generalized principal eigenvalue is nonnegative, then on the contrary all solutions to the Cauchy problem vanish uniformly and the zero solution is the unique space-time periodic nonnegative entire solution. When the two generalized principal eigenvalues differ and zero is in between, the long-time behavior depends on the decay at infinity of the initial condition. The proofs rely upon double-sided controls by solutions of cooperative systems. The control from below is new for such systems and makes it possible to shorten the proofs and extend the generality of the system simultaneously.

We will focus on the formation of extreme waves in the open sea, adopting a probabilistic point of view. We will first identify the first term of the asymptotic development of the probability of occurrence of such a wave when the wave height tends to infinity. If an extreme wave occurs, what is the most likely mechanism that produced it? We will answer this question using two toy models. In the case of an integrable system, we will show that a linear superposition mechanism is the most likely. In the case of a strongly resonant system, the main formation mechanism is a nonlinear focusing effect, which induces an increase in the probability of occurrence of large waves.

Despite the supreme importance of fluid flow models, the well-posedness of three-dimensional viscous and inviscid flow equations remains unsolved. Promising efforts have recently evolved around the concept of statistical solutions. In this talk, we present stochastic lattice Boltzmann methods for efficiently approximating statistical solutions to the incompressible Navier–Stokes equations in three spatial dimensions. Space-time adaptive kinetic relaxation frequencies are used to find stable and consistent numerical solutions along the inviscid limit toward the Euler equations. With single level Monte Carlo and stochastic Galerkin methods, we approximate responses, e.g., from initial random perturbations of the flow field. The novel combinations of schemes are implemented in the parallel C++ data structure OpenLB and executed on heterogeneous high-performance computing machinery. Based on exploratory computations, we search for scaling of the energy spectra and structure functions in terms of Kolmogorov’s K41 theory. For the first time, we numerically approximate the limit of statistical solutions of the incompressible Navier–Stokes solutions toward weak-strong unique statistical solutions of the incompressible Euler equations in three dimensions. Applications to wall-bounded turbulence and the potential to provide training data for generative artificial intelligence algorithms are discussed.

L'équation de Gross-Pitaevskii décrit le mouvement de superfluides, et possède entre autres des solutions stationnaires en forme de vortex. Si deux vortex sont présents, ils se déplacent ensemble à une vitesse constante.

Dans cet exposé, on montrera un résultat de stabilité orbitale dans un espace métrique pour cette paire. On expliquera comment adapter le schéma de preuve de stabilité à un tel espace, et pourquoi on ne peut pas prouver le résultat dans un espace plus simple. Ce travail a été fait en collaboration avec Philippe Gravejat et Frédéric Valet.

Dans le sillage d'une éolienne ou d'un hélicoptère se créent naturellement des filaments de tourbillon en forme d'hélice. Le mouvement des filaments de tourbillon fait l'objet d'une conjecture importante : lorsque le diamètre du filament tend vers 0 (en conservant son intensité), son mouvement devrait suivre en première approximation le flot par courbure binormale. Cette conjecture n'est prouvée que pour les filaments rectilignes et pour les anneaux de tourbillon. Nous montrons, dans le contexte des équations d'Euler 3D incompressibles en symétrie hélicoïdale que les filaments hélicoïdaux suffisamment concentrés suivent également le flot par courbure binormal.

The slides are in english but the talk will be in french.

Je discuterai un travail récent avec Yann Chaubet et Daniel Han-Kwan (Nantes). Nous nous sommes intéressés à la dynamique en temps long de l'équation de Vlasov non-linéaire sur une variété à courbure négative lorsque le noyau d'interaction est lisse. J'expliquerai que, pour des petites données initiales lisses et supportées loin de la section nulle, les solutions de cette équation convergent à vitesse exponentielle vers un état d'équilibre du problème linéaire. Pour obtenir un tel résultat, on fait appel à des outils d'analyse microlocale développés initialement dans le contexte de l'étude des systèmes dynamiques chaotiques (Baladi, Dyatlov, Faure, Sjöstrand, Tsujii, Zworski).

The Benjamin-Ono (BO) equation is a nonlocal asymptotic model for the unidirectional propagation of weakly nonlinear, long internal waves in a two-layer fluid. The equation was introduced formally by Benjamin in the '60s and has been a source of active research since. For instance, the study of the long-time behavior of solutions, stability of traveling waves, and the low regularity well-posedness of the initial value problem. However, despite the rich theory for the BO equation, it is still an open question whether its solutions are close to the ones of the original physical system.

In this talk, I will explain the main steps involved in the rigorous derivation of the BO equation.

On montrera comment résoudre le problème de F. John sur les objets flottants dans le cas d'un objet fixe. Il s'agit de comprendre comment des vagues linéaires se comportent en présence d'un objet partiellement immergé. La difficulté principale vient du fait que le domaine du fluide présente des singularités aux points de contact entre l'objet et la surface de l'eau.

We are interested here in questions related to the maximal regularity of solutions of elliptic problems div $(A abla\, u) = f$ in $\Omega$ with Dirichlet boundary condition. For the last 40 years, many works have been concerned with questions when $A$ is a matrix or a function and when $\Omega$ is a Lipschitz domain. Some of them contain incorrect results that are corrected in the present work.

We give here new proofs and some complements for the case of the Laplacian, the Bilaplacian and the operator $\mathrm{div}\, (A abla)$, when ${\bf A}$ is a matrix or a function. And we extend this study to obtain other regularity results for domains having an adequate regularity. We give also new results for the {Dirichlet-to-Neumann operator for Laplacian and Bilaplacian.

Using the duality method, we can then revisit the work of Lions-Magenes, concerning the so-called very weak solutions, when the data are less regular.

Thanks to the interpolation theory, it permits us to extend the classes of solutions and then to obtain new results of regularity.

We are interested in standing waves for the nonlinear Schrodinger equation with double power nonlinearities, whose typical example is the cubic-quintic nonlinearity in $R^3$.

The cubic term is focusing and the quintic term can be chosen to be both focusing and defocusing.

I will introduce my recent results on the existence, uniqueness and non-degeneracy of ground state solutions based on the variational method and the shooting method

In this talk, we present results on the eigenvalue distribution for perturbed magnetic Dirac operators in two dimensions. We derive third-order asymptotic formulas that incorporate a geometric property of the perturbation's support. Notably, our approach allows us to consider some perturbations that do not necessarily have fixed sign, which is one the main novelties of our work.

This is part of a joint work together with Vincent Bruneau.

Dans cet exposé nous discuterons des résonances pour un graphe quantique dont sa partie compacte est attachée en un sommet à une arête infinie. Les conditions de transmission à ce sommet dépendent d’un petit paramètre et nous démontrons sous certaines hypothèses sur la géométrie du graphe l’existence d’une famille de résonances dont la partie imaginaire tend vers l’infini.

Ce travail est motivé par une question issue de la physique expérimentale où de telles familles de résonances ont été observées. Je montrerai comment avec des outils mathématiques élémentaires il est possible de montrer l’existence et la localisation de ces résonances.

Il s’agit d’un travail interdisciplinaire en collaboration avec Maxime Ingremeau, Ulrich Kuhl, Olivier Legrand, Junjie Lu (Univ. Nice).

We give a light talk on optimality of shapes in geometry and physics. First, we recollect classical geometric results that the disk has the largest area (respectively, the smallest perimeter) among all domains of a given perimeter (respectively, area). Second, we recall that the circular drum has the lowest fundamental tone among all drums of a given area or perimeter and reinterpret the result in a quantum-mechanical language of nanostructures. In parallel, we discuss the analogous optimality of square among all rectangles in geometry and physics. As the main body of the talk, we present our recent attempts to prove the same spectral-geometric properties in relativistic quantum mechanics, where the mathematical model is a matrix-differential (Dirac) operator with complex (infinite-mass) boundary conditions. It is frustrating that such an illusively simple and expected result remains unproved and apparently out of the reach of current mathematical tools.

We consider the standard Ginzburg-Landau system for N-dimensional maps defined in the unit ball for some parameter eps>0. For a boundary data corresponding to a vortex of topological degree one, the aim is to prove the (radial) symmetry of the ground state of the system. We show this conjecture in any dimension N≥7 and for every eps>0, and then, we also prove it in dimension N=4,5,6 provided that the admissible maps are curl-free. This is part of joint works with L. Nguyen, M. Rus, V. Slastikov and A. Zarnescu.

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