The seminar of the team EDPs² is under the responsibility of Jimmy Garnier.
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Year 2010

Friday 17th December 2010 at 14h Matthieu Hillairet (Cérémade à Dauphine),
Explosion des solutions regulières de l'équation des ondes énergie-critique

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Dans cette présentation, on s'intéressera aux propriétés qualitatives des solutions régulières de l'équation des ondes semilineaire H^1-critique. Il est connu, notamment depuis les résultats obtenus par C. Kenig et F. Merle [Acta Mathematica, 2008], que la famille des minimiseurs de l'injection de H^1 dans L^{2^*} joue un role particulier dans la caractérisation des données initiales dont les solutions fortes associées explosent en temps fini. Je présenterai un résultat obtenu en collaboration avec P. Raphael sur le comportement des solutions de l'équation des ondes H^1-critique au voisinage de ces minimiseurs en dimension 4.

Tuesday 7th December 2010 at 10h30 Francesco Ghiraldin (Pise),
A Variational Approximation of a Mumford-Shah functional in codimension 2

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After a brief introduction of the concepts of Distributional Jacobians, we will define a Mumford-Shah energy for vector valued maps that generalizes the classical one. We will then introduce a family of approximating energy and prove a Gamma-convergence result, in the spirit of the previous works by Ambrosio and Tortorelli.

Tuesday 30th November 2010 at 09h Thanasis Stylianou (Université de Cologne),
Elliptic systems arising from mixed finite element methods in domains with corners

Friday 26th November 2010 at 14h JERAA (Rhône Alpes - Auvergne),

Friday 19th November 2010 at 15h Quansen Jiu (School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.R.China),
Stability of Rarefaction Waves to the 1D Compressible Navier-Stokes Equations with Density-dependent Viscosity

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In this talk, we will present some recent results about the asymptotic stability of rarefaction waves for the compressible isentropic Navier-Stokes equations with density-dependent viscosity. Both cases will be dicussed. One is that the rarefaction waves do not include vacuum. The other is that the rarefaction waves contact with vacuum. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations. This is joint with Yi Wang and Zhouping Xin.

Friday 19th November 2010 at 14h Jing Li (Institute of Applied Mathematics Academy of Mathematics and System Sciences Chinese Academy of Sciences, Beijing, China),
Blowup Criterion for the Compressible Flows with Vacuum States

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We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional compressible Navier-Stokes equations, which will happen, for example, if the initial density is compactly supported cite{X1}. More precisely, if a solution of the compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce's criterion for 3-dimensional incompressible Euler equations (cite{po}). Moreover, our method can be generalized to the full Compressible Navier-Stokes system which improve the previous results. In addition, initial vacuum states are allowed in our cases.

Friday 12th November 2010 at 15h Xiangdi Huang (University of Science and Technology of China, Hefei, AnHui Province, China),
A Multi-Fluid Compressible System as the Limit of Weak-Solutions of the Isentropic Compressible Navier-Stokes Equations

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This talk mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance [M., Eyrolles (1975)]. More precisely, we show that some built approximate finite-energy weak solutions of the isentropic compressible Navier-Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model provided the initial density sequence is uniformly bounded with a corrresponding Young measure which is a linear convex combination of m Dirac measures.

Friday 12th November 2010 at 14h Li Mingjie (Institute of Applied Mathematics Academy of Mathematics and System Sciences Chinese Academy of Sciences, Beijing, China),
Zero dissipation limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations

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It is well-known that one-dimensional isentropic gas dynamics has two elementary waves, i.e., shock wave and rarefaction wave. Among the two waves, only the rarefaction wave can be connected with vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we can construct a sequence of solutions to one-dimensional compressible isentropic Navier-Stokes equations which converge to the above rarefaction wave with vacuum as the viscosity tends to zero. Moreover, the uniform convergence rate is obtained. The proof consists of a scaling argument and elementary energy analysis, based on the underlying rarefaction wave structures.

Friday 24th September 2010 at 14h Gabriel Peyré (CEREMADE, Université Paris-Dauphine),
Parcimonie, problèmes inverses et échantillonnage compressé

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Compressed sensing (CS) is a new strategy to sample complicated data such as audio signals or natural images. Instead of performing a pointwise evaluation using localized sensors, signals are projected on a small number of delocalized random vectors. This talk is intended to give an overview of this emerging technology. It will cover both theoritical guarantees and practical applications in image processing and numerical analysis. The initial theory of CS was jointly developed by Donoho [1] and Candès, Romberg and Tao [2]. It makes use of the sparsity of signals to minimize the number of random measurements. Natural images are for instance well approximated using a few number of wavelets, and this sparsity is at the heart of the non-linear reconstruction process. I will discuss the extend to which the current theory captures the practical success of CS. I will pay a particular attention to the worse case analysis of the recovery, and perform a non-asymptotic evaluation of the performances [3]. To obtain better recovery guarantees, I propose a probabilistic analysis of the recovery of the sparsity support of the signal, which leads to constants that are explicit and small [4]. CS ideas have the potential to revolutionize other fields beyond signal processing. In particular, the resolution of large scale problems in numerical analysis could beneficiate from random projections. This performs a dimensionality reduction while simplifying the structure of the problem if the projection is well designed. As a proof of concept, I will present a new compressive wave equation solver, that use projections on random Laplacian eigenvectors [5]. [1] D. Donoho, Compressed sensing, IEEE Trans. Info. Theory, vol. 52, no. 4, pp. 1289-1306, 2006. [2] E. Candès, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Info. Theory, vol. 52, no. 2, pp. 489-509, 2006. [3] C. Dossal, G. Peyré and J. Fadili, A Numerical Exploration of Compressed Sampling Recovery, Linear Algebra and its Applications, Vol. 432(7), p.1663-1679, 2010. [4] C. Dossal, M.L. Chabanol, G. Peyré and J. Fadili, Sparse Support Identi

Friday 10th September 2010 at 14h Mehmet ERSOY (LAMA, Université de Savoie),
Modélisation, analyse mathématique et numérique de divers écoulements compressibles ou incompressibles en couche mince

Thursday 9th September 2010 at 14h Eleuterio TORO (Laboratory of Applied Mathematics, Department of Civil and Environmental Engineering, University of Trento, ITALY),
ADER high-order schemes for evolutionary PDEs

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The ADER approach (Toro et al. 2001 and many others) allows the construction of non-linear one step fully discrete numerical schemes of arbitrary order of accuracy in space and time, for solving evolutionary partial differential equations. The ADER approach operates in the frameworks of finite volume and DG finite element methods and is applicable to multidimensional problems on unstructured meshes. The schemes have two basic ingredients: (a) a non-linear spatial reconstruction operator and (b) the solution of a generalized (or high-order) Riemann problem that links spatial data distribution and time evolution. After describing the main ideas of the methodology I will also show some applications involving hyperbolic and parabolic equations.

Friday 2nd July 2010 at 14h Timack Ngom (LAMA, Université de Savoie),

Friday 4th June 2010 at 14h Xiangdi Huang (University of Science and Technology of China, Hefei, AnHui Province, China.),
Serrin Type Criterion for the Three-Dimensional Compressible Flows with vaccum

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We extend the well-known Serrin's blowup criterion for the three-dimensional incompressible Navier-Stokes equations to the 3D compressible Navier-Stokes equations with vacuum. In other words, in addition to Serrin's condition on the velocity, the L^1(0,T;L^{infty}) norm of the divergence of the velocity is also needed to control the possible breakdown of strong (or smooth) solutions for the three-dimensional compressible Navier-Stokes equations. Moreover, under some additional constraint on the viscosity coefficients, either the L^1(0,T;L^{infty}) norm of the divergence of the velocity or the upper bound of the density will be enough to guarantee the global existence of classical (or strong) solutions.``

Friday 28th May 2010 at 14h François Jouve (Jussieu),
Optimisation de formes par la méthode des courbes de niveau, et applications à la simulation de l'endommagement

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Nous présentons les méthodes d'optimisation de structure par la méthode des courbes de niveaux (level set). Nous montrons ensuite comment le modèle de Francfort-Marigo pour l'endommagement peut se traiter numériquement de façon efficace par ce type de méthode dès lors que l'on a calculé la dérivé de forme pour un problème à deux matériaux.

Friday 23rd April 2010 at 14h Ilaria Fragala (Université de Pise),
The optimal compliance problem in thinning domains

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We consider the variational problem which consists in minimizing the compliance of a prescribed amount of elastic material, placed into a given design region, and sumbitted to an exterior balanced load. We discuss the asymptotic analysis of this problem when the design region is either a cylinder of infinitesimal height (case of thin plates) or a cylinder of infinitesimal cross section (case of thin rods). The results are contained in some recent papers in collaboration with Guy Bouchitte' and Pierre Seppecher.

Friday 2nd April 2010 at 14h Michael Renardy (Department of Mathematics at Virginia Tech, Blacksburg, USA),
Ill-posedness of the hydrostatic Euler and Navier-Stokes equations

Friday 26th March 2010 at 14h Luigi Manca (Université de Toulon),
Existence et unicité d'une mesure invariante pour les EDP stochastiques

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Dans le cas des EDP stochastique, les solutions sont définies sur un espace de dimension infinie et les techniques utilisées pour des équations stochastiques ordinaires - fonction de Lyapunov, hypoellipticité, compacité du semi groupe de transition etc.- ne peuvent pas être appliquées ou nécessitent d'être adaptées. Dans cet exposé j'illustrerai des méthodes utilisées pour l'étude des mesures invariantes pour les EDP stochastiques et leurs applications à des cas spécifiques: dynamique de populations, équation de Burgers, équations de Navier-Stokes etc.

Friday 19th March 2010 at 14h Olivier Goubet (L.A.M.F.A. Université de Picardie Jules Verne),
Décroissance des solutions d'équations d'ondes hydrodynamiques avec viscosité non locale

Thursday 4th March 2010 at 11h30 Manuel Luna-Laynez (Université de Séville),
Some numerical results for control problems in the coefficients.

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We present some numerical methods to solve control problems in the coefficients where the cost functional may depend on the gradient of the state non linearly. The main difficulty comes from the fact that the relaxed functional cost is not explicitly known. We prove some convergence results just using an upper or a lower approximation of this relaxed functional.

Thursday 4th March 2010 at 10h30 Juan Casado-Díaz (Université de Séville),
Control problems in the coefficients with a nonlinear cost in the gradient

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We consider a control problem in the coefficients for an elliptic linear equation where the cost functional is non-linear in the gradient of the function state. The control variables are the coefficients of the diffusion matrix. This type of problems arises in Optimal Design of Composite Materials. It is well known that they have not a solution in general. Here we use the homogenization method to obtain a relaxed formulation.

Friday 29th January 2010 at 14h Céline LABART (Université Pierre et Marie Curie (Paris), laboratoire de Probabilités et Modèles Aléatoires),
Analyse de discrétisation des EDSR et simulation

Friday 22nd January 2010 at 14h JAMES Nicolas. (Université de Clermont-Ferrand),
Méthodes multi-niveaux sur grilles décalées. Application à la simulation numérique d'écoulements autour d'obstacles

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La simulation numérique des écoulements turbulents est délicate. En effet, lorsque le pas d'espace du maillage est plus grand que l'échelle dissipative, le maillage ne permet pas la représentation des plus petites échelles de l'écoulement réel. L'énergie transférée depuis les grandes échelles vers les petites échelles, par l'action des termes d'interaction non linéaires, n'est pas dissipée correctement. On constate alors une augmentation anormale de l'énergie au niveau des échelles qui correspondent à la taille de la maille de calcul. En conséquence, la réalisation d'une simulation numérique directe (résolution de toutes les échelles physiques sans modélisation de la turbulence) pour des écoulements caractérisés par un nombre de Reynolds élevé est très coûteuse en ressources informatiques. Plusieurs méthodes ont été développées pour permettre la simulation numérique de tels écoulements. La méthode multi-niveaux que nous proposons consiste à appliquer un traitement spécifique à chaque échelle, en considérant les propriétés physiques de l'écoulement. La décomposition des échelles du champ de vitesse est utilisée pour imposer une décroissance correcte du spectre d'énergie. La dynamique des grandes échelles est améliorée par le contrôle de l'accumulation de l'énergie sur les modes élevés.

Friday 8th January 2010 at 14h Emmanuel Russ (Université Aix Marseille III),
Opérateur divergence et inégalités de Poincaré dans un domaine arbitraire

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On donne une condition géométrique nécessaire et suffisante sur un domaine borné arbitraire pour que l'opérateur divergence possède un inverse à droite continu dans des espaces de Lebesgue et de Sobolev à poids. On relie aussi cette question à des inégalités de Poincaré. On retrouve en particulier des résultats connus lorsque le domaine est lipschitzien ou plus généralement est un domaine de John.

The seminar of the team EDPs² is under the responsibility of Jimmy Garnier.
Settings: See with increasing date . Hide abstracts
Other years: 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023, all years together.