CNRS Mathematics Laboratory

Le Bourget-du-Lac, France

CNRS Mathematics Laboratory

Le Bourget-du-Lac, France
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Bresch D.,CNRS Mathematics Laboratory | Noble P.,Camille Jordan Institute
Indiana University Mathematics Journal | Year: 2011

The purpose of this paper is to derive rigorously the so-called viscous shallow-water equations given for instance in [18, pp. 958-959]. Such a system of equations is similar to compressible Navier-Stokes equations for a barotropic fluid with a non-constant viscosity. To do that, we consider a layer of incompressible and Newtonian fluid which is relatively thin, assuming no surface tension at the free surface. The motion of the fluid is described by 3d Navier- Stokes equations with constant viscosity and free surface. We prove that for a set of suitable initial data (asymptotically close to "shallowwater initial data" close to rest state), the Cauchy problem for these equations is well posed, and the solution converges to the solution of viscous shallow-water equations. More precisely, we build the solution of the full problem as a perturbation of the strong solution of the viscous shallow-water equations. The method of proof is based on a Lagrangian change of variable which fixes the fluid domain, and we have to prove the well-posedness in thin domains: in particular, we have to pay special attention to constants in classical Sobolev inequalities and regularity in the Stokes problem.


Kerautret B.,French National Center for Scientific Research | Lachaud J.-O.,CNRS Mathematics Laboratory
IEEE Transactions on Pattern Analysis and Machine Intelligence | Year: 2012

The automatic detection of noisy or damaged parts along digital contours is a difficult problem since it is hard to distinguish between information and perturbation without further a priori hypotheses. However, solving this issue has a great impact on numerous applications, including image segmentation, geometric estimators, contour reconstruction, shape matching, or image edition. We propose an original strategy to detect what the relevant scales are at which each point of the digital contours should be considered. It relies on theoretical results of asymptotic discrete geometry. A direct consequence is the automatic detection of the noisy or damaged parts of the contour, together with its quantitative evaluation (or noise level). Apart from a given maximal observation scale, the proposed approach does not require any parameter tuning and is easy to implement. We demonstrate its effectiveness on several datasets. We present different direct applications of this local measure to contour smoothing and geometric estimators whose algorithms initially required a noise/scale parameter to tune: They show the pertinence of the proposed measure for digital shape analysis and reconstruction. © 2012 IEEE.


Clamond D.,University of Nice Sophia Antipolis | Dutykh D.,CNRS Mathematics Laboratory
Physica D: Nonlinear Phenomena | Year: 2012

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a 'relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving a variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact traveling wave solutions are constructed when possible. © 2011 Elsevier B.V. All rights reserved.


Bucur D.,CNRS Mathematics Laboratory
Archive for Rational Mechanics and Analysis | Year: 2012

For every k ∈ ℕ, we prove the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure. Moreover, we prove that every minimizer is bounded and has a finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem. © 2012 Springer-Verlag.


Olivier J.,CNRS Mathematics Laboratory
Zeitschrift fur Angewandte Mathematik und Physik | Year: 2010

In this article, we rigourously prove several asymptotical results for the flow curves of the Hébraud-Lequeux model, a rheological model which describes the behaviour of soft glassy fluids. This model has a control parameter α which governs the behaviour of the fluid at low shear rate. More precisely, we consider τ(γ̇) the stress in a block that is sheared at a constant rate γ̇ and we prove that the system exhibits a transition in its behaviour at low shear rate when α goes through a critical value. The study is complicated by the fact that one of the parameter is only given implicitly and also we have to study two variable function in the neighbourhood of singularities. © 2009 Birkhäuser Verlag Basel/Switzerland.


Bucur D.,CNRS Mathematics Laboratory | Giacomini A.,University of Brescia
Archive for Rational Mechanics and Analysis | Year: 2010

The isoperimetric inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions was recently proved by Daners in the context of Lipschitz sets. This paper introduces a new approach to the isoperimetric inequality, based on the theory of special functions of bounded variation (SBV). We extend the notion of the first eigenvalue λ1 for general domains with finite volume (possibly unbounded and with irregular boundary), and we prove that the balls are the unique minimizers of λ1 among domains with prescribed volume. © 2010 Springer-Verlag.


Bucur D.,CNRS Mathematics Laboratory | Feireisl E.,Academy of Sciences of the Czech Republic | Necasova S.,Academy of Sciences of the Czech Republic
Archive for Rational Mechanics and Analysis | Year: 2010

We consider a family of solutions to the evolutionary Navier-Stokes system supplemented with the complete slip boundary conditions on domains with rough boundaries. We give a complete description of the asymptotic limit by means of Γ-convergence arguments, and identify a general class of boundary conditions. © 2009 Springer-Verlag.


Lepigre R.,CNRS Mathematics Laboratory
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2016

We present a new type system with support for proofs of programs in a call-by-value language with control operators. The proof mechanism relies on observational equivalence of (untyped) programs. It appears in two type constructors, which are used for specifying program properties and for encoding dependent products.The main challenge arises from the lack of expressiveness of dependent products due to the value restriction. To circumvent this limitation we relax the syntactic restriction and only require equivalence to a value.The consistency of the system is obtained semantically by constructing a classical realizability model in three layers (values, stacks and terms). © Springer-Verlag Berlin Heidelberg 2016.


Fedele F.,Georgia Institute of Technology | Dutykh D.,CNRS Mathematics Laboratory
Journal of Fluid Mechanics | Year: 2012

Dyachenko & Zakharov (J. Expl Theor. Phys. Lett., vol. 93, 2011, pp. 701-705) recently derived a compact form of the well-known Zakharov integro-differential equation for the third-order Hamiltonian dynamics of a potential flow of an incompressible, infinitely deep fluid with a free surface. Special travelling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. In particular, unstable travelling waves with wedge-type singularities, namely peakons, are numerically discovered. To gain insight into the properties of these singular solutions, we also consider the academic case of a perturbed version of the compact equation, for which analytical peakons with exponential shape are derived. Finally, by means of an accurate Fourier-type spectral scheme it is found that smooth solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation. © 2012 Cambridge University Press.


Vuillon L.,CNRS Mathematics Laboratory | Lesieur C.,Ecole Normale Superieure de Lyon
Current Opinion in Structural Biology | Year: 2015

To fulfill the biological activities in living organisms, proteins are endowed with dynamics, robustness and adaptability. The three properties co-exist because they allow global changes in structure to arise from local perturbations (dynamics). Robustness refers to the ability of the protein to incur such changes without suffering loss of function; adaptability is the emergence of a new biological activity. Since loss of function may jeopardize the survival of the organism and lead to disease, adaptability may occur through the combination of two local perturbations that together rescue the initial function. The review highlights the relevancy of computational network analysis to understand how a local change produces global changes. © 2015.

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