It is well known that for general evolution problems it is not necessarily possible to infer linear stability from spectra. Known counterexamples include hyperbolic PDEs. A possible way out of this is to investigate criteria in addition to the spectrum which would imply stability. Such criteria are typically based on a WKB type approximation for short wave disturbances. In recent work by Shvydkoy, such criteria, originally developed for the Euler equations, are generalized to a class of equations he calls ``advective.'' It is proved that creeping flows of nonlinear viscoelastic fluids of Maxwell type fall into this category. Shvydkoy's results are for problems with periodic boundary conditions. If homogeneous Dirichlet conditions are imposed on the boundary, it can be shown that wall modes are spectrally determined, and stability can still be decided on the basis of Shvydkoy's criterion. In addition to the spectrum of the linearized operator, this involves determining the stability of a variable coefficient ODE system along each streamline of the base flow. It is also proved that linear stability implies nonlinear stability for small perturbations.
It is a well accepted point of view that the flow of amorphous media is realized via local plastic events that correspond to small rearrangements in the disordered structure. When such materials are actively deformed, the local plastic events will organize into avalanches, that span the whole system in the limit of small strain rates. In this talk I will describe how this cooperative behavior influences diffusion in the sheared material and I will show a direct relation between the diffusion coefficient and the dynamical susceptibility. Considering experiments this means that the measure of the often more easily accessible diffusion coefficient of tracer particles in a sheared disordered material can provide detailed inside into its microscopic rheology.
Dans ce travail commun avec D. Cohen-Steiner et F. Chazal, nous introduisons et étudions les mesures de bord d'un compact de l'espace Euclidien, qui sont étroitement reliées aux mesure de courbure introduites par Federer -- une notion courbure extrinsèque généralisée à une classe assez large de compacts de l'espace Euclidien. Notre but original est de faire de l'inférence géométrique, c'est-à-dire d'estimer des propriétés géométriques d'un 'objet' qu'on ne connaît qu'à travers un échantillon fini. Notre résultat principal est un théorème de stabilité qui permet d'utiliser les mesures de bord dans ce cadre: la mesure de bord d'un compact change peu lorsque celui-ci est remplacé par une approximation Hausdorff --- sans aucune hypothèse de régularité sur aucun des deux compacts. Ce théorème est quantitatif et optimal en un certain sens. En corollaire, on montre qu'il est possible d'approcher les mesures de courbure de Federer d'un compact (dans la classe considérée par Federer) à partir d'un échantillon fini suffisamment Hausdorff-proche. Les aspects algorithmiques du calcul seront brièvement discutés.
Des événements pluvieux sur des surfaces agricoles peuvent conduire à du ruissellement de surface. Ce ruissellement peut occasionner des effets indésirables. Au niveau du champ, le ruissellement peut être à l'origine de l'érosion du sol et du transport de polluants. En aval des champs, les constructions humaines peuvent-être dégradées. Afin de prévenir ces effets néfastes, il existe des moyens permettant de contrôler les écoulements d'eau tels que l'utilisation de bandes enherbées. Pour cela, nous devons prévoir les flux en eau à l'aide de simulations numériques. Ce type de problème est modélisé à l'aide du système de Saint-Venant. Nous utilisons un schéma volume fini équilibré basé sur la méthode de reconstruction hydrostatique, couplé avec un traitement semi-implicite du terme de friction. Nous avons effectué des validations de FullSWOF_2D (code de calcul en C++) sur des solutions analytiques, ainsi que sur des mesures expérimentales (INRA d'Orléans) et des mesures de terrain en Afrique (IRD).
Nous définissons d'une manière intrinsèque pour le système des équations de Navier-Stokes compressibles une classe spécifique des solutions faibles re-normalisées et convenables. Ces solutions vérifient en plus de l'équation de continuité et de l'équation du mouvement une inégalité d'entropie introduite par plusieurs auteurs. Nous démontrons l'existence de ces solutions puis étudions quelques propriétés, en particulier l'unicité forte-faible.
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.
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.
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.
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.
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.
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.
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
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.