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Lions_Magenes

Appel à posters



Un appel est lancé aux jeunes doctorants et docteurs. Il leur est proposé de présenter leurs résultats sous forme d'un poster lors des Journées Lions Magenes. Il s'agit là d'un excellent moyen de se faire connaitre.

Pour proposer un poster, écrire à en précisant l'auteur, le titre, et en donnant un résumé en anglais (ainsi qu'en français pour les francophones) d'une dizaine de lignes avant le 2 décembre 2011. Chaque poster devra être préparé par son auteur, sous forme d'une feuille de papier au format A0 (84 x 120 centimètres). L'auteur l'apportera et l'affichera à la place qui lui sera désignée. Le texte pourra être écrit en français ou (de préférence) en anglais ; dans les deux cas de façon pédagogique et en caractères assez gros pour que les titres et sous-titres puissent être lus à une distance de deux mètres. L'auteur sera invité à se tenir devant son poster pendant certaines des pauses café pour discuter de ses résultats avec les participants intéressés.


Posters acceptés


Giacomo Canevari (LJLL - Université Pierre et Marie Curie)

Solution to a phase field model and convergence to the Caginalp system

A diffusion model of phase field type is considered. This model involves as variables the thermal displacement w (basically the time integration of temperature) and the order parameter u, which are assumed to satisfy the system of two equations w_tt - a Delta w_t - b Delta + u_t = f in Omega x [0, T] , and u_t - Delta u + c(u) + g(u) \ni w_t in Omega x [0, T] . Here, c(u) is a non-smooth, possibly multivalued maximal monotone graph, while g is a smooth anti-monotone function; we work in a smooth bounded domain Omega of R^3. The system turns out to be a generalization of the well-known Caginalp phase field model for phase transitions, with a diffusive term for the thermal displacement in the balance equation. In a joint paper with P. Colli, the existence and uniqueness of a weak solution to the initial-boundary value problem is proved, along with various regularity results, ensuring that the solution is strong and with bounded components. In our presentation we will report on these points and also illustrate the asymptotic behaviour of the solutions as b tends to zero, by stating the convergence to the Caginalp phase field system and providing error estimates for the difference of the solutions.


Nicolas Carreno (LJLL - Université Pierre et Marie Curie)

Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain

We deal with the local null controllability of the N-dimensional Navier-Stokes system with internal controls having one vanishing component. It is known that this can be done when the adherence of the control domain intersects the boundary of the total volume. The novelty here is that we remove this geometric assumption. We follow a classical strategy: we deduce a suitable Carleman inequality for the linearized system with a right-hand side, which leads to a null controllability result. Then, we deduce the result for the Navier-Stokes system by means of an inverse mapping theorem.


Paul Cazeaux (LJLL - Université Pierre et Marie Curie)

An homogenization approach to modeling the lungs' parenchyma

We study a nonlocal linear two-scale system describing the mechanical displacement of a periodically perforated elastic medium, with holes filled with gas and coupled to a dyadic resistive tree allowing the gas to escape. This constitutes a simplified model of mechanics of the lungs in which the 300 millions tiny alveoli are fed with air through the bronchial tree. Because of the fractal properties of the bronchial tree, we study the asymptotic behavior of this system as the number of generations of the tree tends to infinity, and the ratio between the macroscopic scale and the characteristic size of the alveoli tends to zero. We show that the solutions of this system converge using two-scale homogenization techniques. In the limit, the homogenized medium exhibits viscoelastic properties with a special nonlocal viscous term describing the interaction of the structure with the bronchial tree. We also describe numerical simulations of the respiration process using our homogenized model, computed with the finite element software FreeFem++.

Nous étudions un système linéaire non-local, à double échelle, qui décrit le déplacement mécanique d'un milieu perforé dont les trous sont remplis de gaz et couplés à un arbre dyadique qui permet à ce gaz de s'échapper. Ceci constitue un modèle simplifié pour la mécanique du poumon, où 300 millions d'alvéoles minuscules sont alimentées en air par l'arbre bronchique. Motivés par la structure fractale de l'arbre bronchique, nous étudions le comportement asymptotique de ce système lorsque le nombre de générations de l'arbre tend vers l'infini, en même temps que le rapport entre l'échelle macroscopique et la taille caractéristique des alvéoles tend vers zéro. Nous montrons que les solutions de ce système convergent en utilisant la technique de l'homogénéisation double échelle. À la limite, le milieu homogénéisé présente des caractéristiques viscoélastiques avec un terme non-local original qui décrit l'interaction entre la structure et l'arbre bronchique. Nous décrivons aussi les simulations numériques de la respiration, calculées avec le logiciel éléments finis FreeFem++.


Lucas Chesnel (Ensta)

Radiation condition for a non-smooth interface between a dielectric and a metamaterial

We study a problem of electromagnetism in harmonic regime set in a bounded domain constituted of a classical dielectric and a negative metamaterial. At a given frequency, this metamaterial is modeled by a homogeneous medium with real strictly negative permittivity and permeability. Here, we do not study the case of the whole Maxwell system but instead focus on an academic version in 2-D: (1) find u in H^1_0(\Omega) such that div(\epsilon\nabla u)=f in H^{-1}(\Omega). When the interface between the dielectric and the metamaterial has a corner, according to the ratio of the values of \epsilon (contrast), problem (1) can be ill-posed (not Fredholm) in H^1_0(\Omega). This is due to the degeneration of the two dual singularities associated with the corner which then behave like r^{i \eta} = exp{i \eta ln r}, where \eta is a real number. This apparition of propagative singularities is very similar to the apparition of propagative modes in an unbounded waveguide for the classical Helmholtz equation with Dirichlet boundary condition, the contrast playing the role of the frequency. In this work, we derive for our problem a functional framework by adding to H^1_0(\Omega) one of these propagative singularities. Well-posedness is then obtained by imposing a radiation condition, justified by means of a limiting absorption principle, at the corner between the two media. In this poster, we also point out some original questions which appear when one studies problem (1) set in a domain a with a slightly rounded corner.

Nous nous intéressons à un problème d'électromagnétisme en régime harmonique dans un milieu borné composé d'un diélectrique classique et d'un métamatériau négatif. A fréquence donnée, ce métamatériau est modélisé par un milieu homogène dont les permittivité et perméabilité sont réelles strictement négatives. Dans ce poster, nous étudions une version scalaire 2-D du problème de Maxwell : (1) trouver u dans H^1_0(\Omega) tel que div (\ epsilon\ nabla u) = f dans H^{-1} (\ Omega ). Lorsque l'interface entre le diélectrique et le métamatériau présente un coin, selon le rapport des valeurs de \epsilon (contraste), le problème (1) peut être mal posé (non Fredholm) dans H^1_0 (\ Omega). Cela s'explique par la dégénérescence des deux singularités duales associées au coin, qui se comportent alors en r^{i\ eta} = exp{i\eta ln r}, où \eta est un nombre réel. Cette apparition de singularités propagatives est très similaire à l'apparition de modes propagatifs dans un guide d'ondes non borné pour l'équation de Helmholtz classique avec condition de Dirichlet, le contraste jouant le rôle de la fréquence. Dans ce travail, nous définissons pour notre problème un cadre fonctionnel en ajoutant à H^1_0 (\ Omega) l'une de ces singularités propagatives. Le caractère bien posé du problème est alors obtenu en imposant une condition de radiation, justifiée au moyen du principe d'absorption limite, au voisinage du coin entre les deux matériaux. Nous mettons également en évidence quelques questions originales qui apparaissent lorsqu'on s'intéresse au problème (1) dans un domaine avec un coin légèrement arrondi.


Jean-Paul Daniel (LJLL - Université Pierre et Marie Curie)

A game interpretation for fully nonlinear parabolic and elliptic equations with a Neumann condition

In this work, we give a deterministic-control-based interpretation for a large class of fully nonlinear parabolic PDEs with a continuous Neumann condition in a smooth bounded domain. We construct a family of two-person games depending on a small parameter epsilon which extends the one proposed by Kohn and Serfaty in the whole space (2010). These new games allow us to deal with the Neumann condition by introducing some specic rules near the boundary. The value function associated to a game satisfies a numerical scheme by the dynamic programming principle. We show that the value function converges, in the viscosity sense, to the solution of the PDE as epsilon tends to zero.
Moreover, our construction allows us to treat both oblique and mixed boundary conditions as well as the elliptic case.

L'objectif de notre travail est de donner une interprétation par un problème de contrôle déterministe d'une large classe d'EDP paraboliques complètement non linéaires avec condition de Neumann dans un domaine borné et régulier. Pour cela, nous construisons une famille de jeux à deux personnes dépendant d'un paramètre epsilon qui étend celle introduite par Kohn et Serfaty dans RN (2010). Ces nouveaux jeux permettent de traiter la condition de Neumann par l'introduction de règles spécifiques près de la frontière. La fonction valeur associée à un jeu satisfait un schéma numérique par le principe de la programmation dynamique. Nous montrons que la fonction valeur converge, au sens des solutions de viscosité, vers la solution de l'EDP lorsque epsilon tend vers zéro.
De plus, cette construction permet de traiter les conditions aux bords de type oblique et mixte et aussi de résoudre le cas elliptique.


Olaf Torné (Ecole Centrale Paris)

A one dimensional problem related to the symmetry of minimisers for the Sobolev trace constant in a ball

The symmetry of minimisers for the best constant in the trace inequality in the ball B_r, namely the infimum of the ratio |u|^p_{W^{1,p}(B_r} /over |u|^{p}_{L^q(\partial B_r)}, has been studied by various authors. Partial results are known which imply radial symmetry of minimisers, or lack thereof, depending on the values of trace exponent q and radius r of the ball. In this work we consider a one dimensional analogue of the trace inequality and the corresponding minimisation problem for the best constant. We describe the exact values of q and r for which minimisers are symmetric. We also consider the behaviour of minimisers as the symmetry breaking threshold for q and r is breached, and show a case in which both symmetric and nonsymmetric minimisers coexist.


Magali Tournus (LJLL - Université Pierre et Marie Curie)

Convergence to the stationary state of a simplified model of a kidney concentrating urine mechanism

We study a nonlinear stationary system with specific boundary conditions describing the transport of solutes dissolved in a fluid circulating in a countercurrent tubular architecture, which constitutes a simplified model of a kidney concentrating urine mechanism. We introduce a dynamic system, a PDE which we study through contraction properties, and which relaxes toward the unique solution of the stationary model. A study of the linearized stationary operator enables us, using eigenelements, to further show that under certain conditions regarding the nonlinearity, the relaxation is exponential. We also describe a finite volume scheme which allows us to efficiently approach the numerical solution to the stationary system and to illustrate how the countercurrent arrangement of tubes enhances the axial concentration gradient, thereby favoring the production of highly concentrated urine.


Salvatore Tringali (LJLL - Université Pierre et Marie Curie)

A variational approach to spectral theory

We propose and develop an inherently variational approach to the foundation of the spectral theory of (associative) normed algebras as first introduced, in essence, by Israil M. Gelfand in 1941. Our basic motivation is to abstract the classical theory for the twofold objective of making it portable to different scenarios and somehow more flexible -- mainly for the purpose of enlarging the domain of applicability of familiar variational techniques (such as reduced basis methods) to spectral problems. The outcome is what we call a topological spectral theory, as it brings the usual notion of spectrum, and related ones, from the context of normed and operator algebras into the broader setting of topological vector spaces (with a special focus on normed spaces), substantially exploiting a weak formulation by bilinear mappings. This provides, at the same time, both a generalisation and a sharpening of the classical theory: the former since Gelfand spectra, along with a significant amount of their basic properties, can be recovered as a specialisation of this theory concerning topological spectra in the particular case of Banach algebras; the latter for topological spectra yield a refinement of the classical notion of spectrum to the extent of saving properties, such as closeness and compactness, which are otherwise lost in the classical framework as far as arbitrary (i.e. possibly incomplete) normed algebras are involved.


Bang Cong Vu (LJLL - Université Pierre et Marie Curie)

A splitting algorithm for dual monotone inclusions involving cocoercive operators

We propose a general framework for solving composite monotone inclusions involving maximally monotone operators and cococercive operators. The convergence of the algorithms is proved in real Hilbert spaces. Connections with existing methods are obtained. Reference: B. C. Vu, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comput. Math., to appear.

Nous proposons un cadre général pour résoudre des inclusions composites pour les opérateurs maximalement monotones et cocoercifs. La convergence des algorithmes est démontrée dans des espaces hilbertiens réels. Des liens avec des méthodes existantes sont obtenus. Référence : B. C. Vu, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comput. Math., à paraître.


Soleiman Yousef (LJLL - Université Pierre et Marie Curie)

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

We derive a posteriori error estimates for the time-dependent two-phase Stefan problem. We first introduce a regularized version of the problem which is subsequently discretized by the vertex-centered finite volume method. The Newton method is used to solve the system of nonlinear algebraic equations arising at each discrete time. Our estimates yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L2 space-time error of the temperature. Moreover, they allow to distinguish the space, time, regularization, and linearization errors. Hence, it is possible to equilibrate the space and time errors, to provide a criterion on the choice of our regularization parameter, and to stop the iterations of the Newton algorithm when the corresponding error component is dominated by the others. An adaptive algorithm is proposed, which ensures computational savings and performs a local mesh refinement while equilibrating the space and time errors. Numerical results illustrate the performance of our estimate and the efficiency of the adaptive algorithm.