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25 results on '"'Simion Stoilow' Institute of Mathematics (IMAR)"'

1. Model of two-temperature convective transfer in porous media

Author

Isabelle Gruais, Dan Poliševski, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 'Simion Stoilow' Institute of Mathematics ( IMAR ), Romanian Academy of Sciences, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 'Simion Stoilow' Institute of Mathematics (IMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Convection, Convective heat transfer, Stokes-Boussinesq system, General Mathematics, first-order jump interface, General Physics and Astronomy, Heat transfer coefficient, two-scale convergence, Thermal diffusivity, 01 natural sciences, 35B27, 76M50, 76Rxx, 74F10, 74Q05, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Fluid dynamics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Boussinesq approximation (water waves), Physics, Homogenization, Applied Mathematics, 010102 general mathematics, Mechanics, 010101 applied mathematics, "-caps, heat transfer coecient, Transport phenomena, Porous medium
Abstract

In this paper, we study the asymptotic behaviour of the solution of a convective heat transfer boundary problem in an $$\varepsilon $$ -periodic domain which consists of two interwoven phases, solid and fluid, separated by an interface. The fluid flow and its dependence with respect to the temperature are governed by the Boussinesq approximation of the Stokes equations. The tensors of thermal diffusion of both phases are $$\varepsilon $$ -periodic, as well as the heat transfer coefficient which is used to describe the first-order jump condition on the interface. We find by homogenization that the two-scale limits of the solutions verify the most common system used to describe local thermal non-equilibrium phenomena in porous media (see Nield and Bejan in Convection in porous media, Springer, New York, 1999; Rees and Pop in Transport phenomena in porous media III, Elsevier, Oxford, 2005). Since now, this system was justified only by volume averaging arguments.

Published
2017
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2. Fluid flows through fractured porous media along Beavers-Joseph interfaces

Author

Dan Poliševski, Isabelle Gruais, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), and 'Simion Stoilow' Institute of Mathematics ( IMAR )

Subjects
Homogenization, 35B27, 76M50, 76S05, 76T99, Applied Mathematics, General Mathematics, Fractured porous media, Geometry, Stokes flow, Mechanics, Homogenization (chemistry), Two-scale convergence, Physics::Fluid Dynamics, Permeability (earth sciences), [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Fluid dynamics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, Beavers-Joseph interface, Porous medium, Mathematics
Abstract

We study a fluid flow traversing a porous medium and obeying the Darcy's law in the case when this medium is fractured in blocks by an e-periodic ( e > 0 ) distribution of fissures filled with a Stokes fluid. These two flows are coupled by a Beavers–Joseph type interface condition. The existence and uniqueness of this flow in our e-periodic structure are proved. As the small period of the distribution shrinks to zero, we study the asymptotic behaviour of the flow when the permeability and the entire contribution on the interface of the Beavers–Joseph transfer coefficients are of unity order. We find the homogenized problem verified by the two-scale limits of the coupled velocities and pressures. It is well-posed and provides the corresponding classical homogenized problem.

Published
2014
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3. The effective permeability of fractured porous media subject to the Beavers-Joseph contact law

Author

Florentina-Alina Stanescu, Isabelle Gruais, Dan Poliševski, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 'Simion Stoilow' Institute of Mathematics ( IMAR ), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Homogenization, General Mathematics, 010102 general mathematics, Fractured porous media, Subject (philosophy), 01 natural sciences, Two-scale convergence, Stokes flow \sep Beavers-Joseph interface, 010101 applied mathematics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], \MSC 35B27, 76M50, 76S05, 76T99, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Geotechnical engineering, 0101 mathematics, Porous medium, Relative permeability, Geology
Abstract

We are interested in the asymptotic behaviour of a fluid flow contained in a microscopic periodic distribution of fissures perturbating a porous medium where the Darcy law is valid, when the coupling between both systems is modeled by the Beavers-Joseph interface condition. As the small period of the distribution tends to zero, the interface condition is preserved on a microscopic scale under the additional assumption that the permeability coefficients behave like the squared period of the distribution which is also the squared size of the fissures. Moreover, the resulting pressure is purely macroscopic unlike the velocity field which also depends on the microscopic variable.

Published
2014
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4. Asymptotic heat equation for crossing superconductive thin walls

Author

Dan Poliševski, Isabelle Gruais, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 'Simion Stoilow' Institute of Mathematics ( IMAR ), Romanian Academy of Sciences, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), 'Simion Stoilow' Institute of Mathematics (IMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and Gruais, Isabelle

Subjects
Homogenization, Condensed matter physics, Box-type structure, Applied Mathematics, Fine-scale, 010102 general mathematics, Thermodynamics, Binary number, Conductivity, Thermal conduction, Microstructure, Conduction, 01 natural sciences, Homogenization (chemistry), 010101 applied mathematics, Singularity, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 35B27, 35K57, 76R50, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Heat equation, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Electrical conductor, Analysis, Mathematics
Abstract

International audience; This work deals with the homogenization of the nonstationary heat conduction which takes place in a binary three-dimensional medium consisting of an ambiental phase having conductivity of unity order and a set of highly conductive thin walls crossing orthogonally and periodically. This situation covers in fact three types of microstructures, usually called: box-type (or honeycomb), gridwork and layered. The study is based on the energetic procedure of homogenization associated to a control-zone method, specific to the geometry of the microstructure and to the singularity of the conductivity coefficients. In the present case the main result is the system that governs the asymptotic behaviour of the temperature distribution in this binary medium. It displays a significant increase of the conductivity due to the superconductive thin walls, revealing their seemingly paradoxal behaviour of having an everlasting action on the environment, in spite of an obvious vanishing volume. Moreover, the dependence of this behaviour with respect to the relative thicknesses of the walls can be detailed.

Published
2012

5. Homogenizing the Darcy/Stokes coupling

Author

Gruais, Isabelle, Polisevski, Dan, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 'Simion Stoilow' Institute of Mathematics ( IMAR ), Gruais, Isabelle, AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Physics::Fluid Dynamics, Homogenization, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 35B27, 76M50, 76S05, 76T99, Fractured porous media, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Stokes flow, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Beavers-Joseph interface, Two-scale convergence
Abstract

We study a fluid flow traversing a porous medium and obeying the Darcy's law in the case when this medium is fractured by a periodical distribution of fissures filled with a Stokes fluid. These two flows are coupled by a Beavers-Joseph type interface condition. As the small period of the distribution shrinks to zero, the resulting asymptotic behaviour is implicitely described by two underlying macroscopic quantities: the limit of the Stokes velocity and the limit of the Darcy pressure, solutions of a new coupled system obtained by homogenization. The behaviour of the macroscopic two-scale limit of the filtration velocity is given by an explicit two-scale Darcy type law, presenting coupling terms with the gradient of the limit of the Darcy pressure and with the limit of the Stokes velocity.

Published
2011

6. Periodic structures separated by highly conductive thin layers

Author

Gruais, Isabelle, Polisevski, Dan, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 'Simion Stoilow' Institute of Mathematics ( IMAR ), Romanian Academy of Sciences, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 'Simion Stoilow' Institute of Mathematics (IMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Gruais, Isabelle

Subjects
conduction, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], genetic structures, fine-scale substructure, homogenization, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35B27, 35K57,76R50, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

We derive a model for the conduction in an $\veps$-periodic structure containing highly conductive thin layers. The case of plane thin layers is first considered. It is shown that the resulting model displays an increased conductivity along the directions of the layers planes. The more involved case of tubular layers yields a similar result with an increase of the conductivity along the direction of the tubes, while the presence of highly conductive thin layers confined between spheres of $\veps$-order radii does not increase the macroscopic conductivity in any direction. It seems that the presence of $\veps$-periodic highly conductive thin layers determines an increase of the macroscopic conductivity in a certain direction only if these layers cover entirely segment lines of unity order having this direction.

Published
2010

7. $\epsilon$-periodic structures containing highly conductive thin layers

Author

Gruais, Isabelle, Polisevski, Dan, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 'Simion Stoilow' Institute of Mathematics ( IMAR ), Romanian Academy of Sciences, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 'Simion Stoilow' Institute of Mathematics (IMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Guillemer, Marie-Annick

Subjects
[ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35B27, 35J25, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2010

8. Diffusion around a vanishing suspension

Author

Bentalha , Fadila, Gruais , Isabelle, Polisevski , Dan, Laboratoire de Mathématiques, Université Hadj Lakhdar Batna, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 'Simion Stoilow' Institute of Mathematics ( IMAR ), Romanian Academy of Sciences, Université Hadj Lakhdar Batna 1, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 'Simion Stoilow' Institute of Mathematics (IMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Guillemer, Marie-Annick

Subjects
[ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS, 35B27, 35K57, 76T20
Abstract

International audience

Published
2008

9. Homogenization of a diffusion process in a rarefied binary structure

Author

Bentalha, Fadila, Gruais, Isabelle, Polisevski, Dan, Department of Mathematics, Université Hadj Lakhdar Batna, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 'Simion Stoilow' Institute of Mathematics ( IMAR ), Romanian Academy of Sciences, Université Hadj Lakhdar Batna 1, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 'Simion Stoilow' Institute of Mathematics (IMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Diffusion, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 35B27, 35K57, 76R50, fine-scale substructure, FOS: Mathematics, homogenization, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

We study the homogenization of a diffusion process which takes place in a binary structure formed by an ambiental connected phase surrounding a suspension of very small spheres distributed in an $\veps$-periodic network. The asymptotic distribution of the concentration is determined for both phases, as $\veps\to 0$, assuming that the suspension has mass of unity order and vanishing volume. Three cases are distinguished according to the values of a certain limit capacity. When it is positive and finite, the macroscopic system involves a two-concentration system, coupled through a term accounting for the non local effects. In the other two cases, where the capacity is either infinite or going to zero, although the form of the system is much simpler, some peculiar effects still account for the presence of the suspension.

Published
2005

10. Effects of vorticity on the travelling waves of some shallow water two-component systems

Author

Delia Ionescu-Kruse, Denys Dutykh, Laboratoire de Mathématiques (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), 'Simion Stoilow' Institute of Mathematics (IMAR), and Romanian Academy of Sciences

Subjects
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Integrable system, Zakharov-Ito system, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Near and far field, Physics - Classical Physics, Pattern Formation and Solitons (nlin.PS), [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], phase-plane analysis, MSC: 74J30 (primary), 76F10, 35C07, 76B25, 70K05, 35Q35 (secondary)PACS: 47.35.Bb (primary), 47.11.Kb, 47.10.Df (secondary), 01 natural sciences, Camassa-Holm equations, [NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Traveling wave, Discrete Mathematics and Combinatorics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Nonlinear Sciences::Pattern Formation and Solitons, Kaup-Boussinesq equations, Physics, [PHYS.PHYS.PHYS-AO-PH]Physics [physics]/Physics [physics]/Atmospheric and Oceanic Physics [physics.ao-ph], Nonlinear Sciences - Exactly Solvable and Integrable Systems, Applied Mathematics, Mathematical analysis, Fluid Dynamics (physics.flu-dyn), Classical Physics (physics.class-ph), Physics - Fluid Dynamics, Vorticity, Nonlinear Sciences - Pattern Formation and Solitons, waves on shear flow, 010101 applied mathematics, analytical solutions, Physics - Atmospheric and Oceanic Physics, Waves and shallow water, Nonlinear Sciences::Exactly Solvable and Integrable Systems, Phase space, multi-pulsed solutions, solitary waves, Atmospheric and Oceanic Physics (physics.ao-ph), cnoidal waves, Phase plane analysis, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Exactly Solvable and Integrable Systems (nlin.SI), Analysis
Abstract

In the present study we consider three two-component (integrable and non-integrable) systems which describe the propagation of shallow water waves on a constant shear current. Namely, we consider the two-component Camassa-Holm equations, the Zakharov-Ito system and the Kaup--Boussinesq equations all including constant vorticity effects. We analyze both solitary and periodic-type travelling waves using the simple and geometrically intuitive phase space analysis. We get the pulse-type solitary wave solutions and the front solitary wave solutions. For the Zakharov-Ito system we underline the occurrence of the pulse and anti-pulse solutions. The front wave solutions decay algebraically in the far field. For the Kaup-Boussinesq system, interesting analytical multi-pulsed travelling wave solutions are found., Comment: 27 pages, 14 figures, 45 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/

Published
2019
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11. On a Class of Parametric (p, 2)-equations

Author

Vicenţiu D. Rădulescu, Dušan Repovš, Nikolaos S. Papageorgiou, National Technical University of Athens [Athens] (NTUA), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, University of Ljubljana, and Slovenian Research Agency through Grants P1-0292, J1-7025 and J1-6721 (D. Repovs) and the Romanian Research Council through the grant UEFISCDI-PCCA-23/2014 (V. Radulescu)

Subjects
Control and Optimization, nonlinear maximum principle, Multiplicity results, 35J20, 35J60, 58E05, Lambda, 01 natural sciences, Omega, Combinatorics, Mathematics - Analysis of PDEs, near resonance, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], constant sign and nodal solutions, 0101 mathematics, Parametric equation, Eigenvalues and eigenvectors, Parametric statistics, Physics, Applied Mathematics, 010102 general mathematics, udc:517.956.2, Multiplicity (mathematics), local minimizer, 35J20, 35J60, 58E05, 010101 applied mathematics, critical group, Laplace operator, Analysis of PDEs (math.AP)
Abstract

We consider parametric equations driven by the sum of a ▫$p$▫-Laplacian and a Laplace operator (the so-called ▫$(p, 2)$▫-equations). We study the existence and multiplicity of solutions when the parameter ▫$lambda > 0$▫ is near the principal eigenvalue ▫$hat{lambda}_1(p) > 0$▫ of ▫$(-Delta_p,W^{1-p}_0(Omega))$▫. We prove multiplicity results with precise sign information when the near resonance occurs from above and from below of ▫$hat{lambda}_1(p) > 0$▫.

Published
2016
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12. Travelling wave solutions for some two-component shallow water models

Author

Denys Dutykh, Delia Ionescu-Kruse, Laboratoire de Mathématiques (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), 'Simion Stoilow' Institute of Mathematics (IMAR), and Romanian Academy of Sciences

Subjects
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Integrable system, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Type (model theory), phase-plane analysis, 01 natural sciences, Camassa-Holm equations, 010305 fluids & plasmas, [NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], Component (UML), 0103 physical sciences, Traveling wave, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Mathematics, [PHYS.PHYS.PHYS-AO-PH]Physics [physics]/Physics [physics]/Atmospheric and Oceanic Physics [physics.ao-ph], 74J30 (primary), 76B15, 76B25, 70K05 (secondary)47.35.Bb (primary), 47.11.Kb, 47.10.Df (secondary), Applied Mathematics, 010102 general mathematics, Mathematical analysis, Green-Naghdi model, Waves and shallow water, Classical mechanics, solitary waves, cnoidal waves, Phase plane analysis, Serre equations, travelling waves, Analysis
Abstract

21 pages, 13 figures, 37 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/; International audience; In the present study we perform a unified analysis of travelling wave solutions to three different two-component systems which appear in shallow water theory. Namely, we analyse the celebrated Green-Naghdi equations, the integrable two-component Camassa-Holm equations and a new two-component system of Green-Naghdi type. In particular, we are interested in solitary and cnoidal-type solutions, as two most important classes of travelling waves that we encounter in applications. We provide a complete phase-plane analysis of all possible travelling wave solutions which may arise in these models.In particular, we show the existence of new type of solutions.

Published
2016
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13. Asymptotic analysis of a thin rigid stratified elastic plate – viscous fluid interaction problem

Author

Ruxandra Stavre, Grigory Panasenko, Irina Malakhova-Ziablova, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), 'Simion Stoilow' Institute of Mathematics (IMAR), and Romanian Academy of Sciences

Subjects
dimension reduction, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], homogenization, Modulus, Viscous liquid, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Homogenization (chemistry), Physics::Fluid Dynamics, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, Boundary value problem, 0101 mathematics, Variational analysis, Linear elasticity, Mathematics, asymptotic expansion, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Stokes equations, 010101 applied mathematics, interface, Asymptotic expansion, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; A three-dimensional model for the interaction of a thin stratified rigid plate and a viscous fluid layer is considered. This problem depends on a small parameter $\varepsilon$ which is the ratio of the thickness of the plate and that of the fluid layer. The right-hand side functions are 1-periodic with respect to the tangential variables of the plate. The plate’s Young’s modulus is of order $\varepsilon^{-3}$, i.e. it is great, while its density is of order 1. At the solid–fluid interface, the velocity and the normal stress are continuous. The variational analysis of this model (including the existence, uniqueness of the solution and its regularity) is provided. An asymptotic expansion of the solution is constructed and justified. The error estimate is proved for the partial sums of the asymptotic expansion. The limit problem contains a non-standard boundary condition for the Stokes equations. The existence, uniqueness, and regularity of its solution are proved. The asymptotic analysis is applied to the partial asymptotic dimension reduction of the solid phase and the derivation of the asymptotically exact junction conditions between two-dimensional and three-dimensional models of the plate.

Published
2016
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14. Positivity, decay, and extinction for a singular diffusion equation with gradient absorption

Author

Razvan Gabriel Iagar, Philippe Laurençot, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Diffusion equation, viscosity solutions, Diffusion, gradient estimates, 01 natural sciences, Singular diffusion, gradient absorption, Mathematics - Analysis of PDEs, FOS: Mathematics, Initial value problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Absorption (logic), 0101 mathematics, Mathematics, Mathematical physics, extinction, 010102 general mathematics, Mathematical analysis, p-Laplacian, 35B40, 35K67, 35K92, 35K10, 35B33, 49L25, 010101 applied mathematics, Extinction (optical mineralogy), Finite time, Analysis, Analysis of PDEs (math.AP)
Abstract

We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption ∂ t u − Δ p u + | ∇ u | q = 0 in ( 0 , ∞ ) × R N , where N ⩾ 1 , p ∈ ( 1 , 2 ) , and q > 0 . Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as t → ∞ for q > p − N / ( N + 1 ) , optimal decay estimates as t → ∞ for p / 2 ⩽ q ⩽ p − N / ( N + 1 ) , or extinction in finite time for 0 q p / 2 . In addition, we show how the diffusion prevents extinction in finite time in some ranges of exponents where extinction occurs for the non-diffusive Hamilton–Jacobi equation.

Published
2012
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15. Large time behavior for the fast diffusion equation with critical absorption

Author

Said Benachour, Philippe Laurençot, Razvan Gabriel Iagar, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Zero order absorption, 35B33, 35B40, 35B45, 35K67, Diffusion equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, gradient estimates, 01 natural sciences, Upper and lower bounds, large time behavior, 010101 applied mathematics, Mathematics - Analysis of PDEs, fast diffusion, FOS: Mathematics, Initial value problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Absorption (logic), 0101 mathematics, lower bound, critical absorption, Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption$$\partial_{t}u-\Delta u^m+u^q=0 \quad \quad \hbox{in} \(0,\infty)\times\real^N\ ,$$with $m_c:=(N-2)_{+}/N < m < 1$ and $q=m+2/N$. Given an initial condition $u_0$ decaying arbitrarily fast at infinity, we show that the asymptotic behavior of the corresponding solution $u$ is given by a Barenblatt profile with a logarithmic scaling, thereby extending a previous result requiring a specific algebraic lower bound on $u_0$. A by-product of our analysis is the derivation of sharp gradient estimates and a universal lower bound, which have their own interest and hold true for general exponents $q > 1$.

Published
2016
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16. Asymptotic analysis for the Kelvin–Voigt model for a thin laminate

Author

Ruxandra Stavre, Grigory Panasenko, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), 'Simion Stoilow' Institute of Mathematics (IMAR), and Romanian Academy of Sciences

Subjects
Marketing, Strategy and Management, Dimensionality reduction, 010102 general mathematics, Mathematical analysis, Dimensional modeling, 01 natural sciences, Homogenization (chemistry), 010101 applied mathematics, Kelvin–Voigt material, Solid mechanics, Media Technology, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], General Materials Science, 0101 mathematics, Standard linear solid model, Asymptotic expansion, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

A two dimensional Kelvin–Voigt model of a visco-elastic thin stratified strip with Neumann condition at the lateral boundary is considered. The dimension reduction combined with the homogenization procedure allows us to construct a complete asymptotic expansion of the solution and to justify the limit one dimensional model containing the long-fading memory term while the initial model corresponds to the short memory.

Published
2015
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17. Bergman and Calderón projectors for Dirac operators

Author

Colin Guillarmou, Jinsung Park, Sergiu Moroianu, Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), and Guillarmou, Colin

Subjects
Spinor, 010102 general mathematics, Holomorphic function, Order (ring theory), Boundary (topology), Riemannian manifold, Dirac operator, 01 natural sciences, 58J32, 35P25, symbols.namesake, Differential geometry, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Quantum mechanics, 0103 physical sciences, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, Geometry and Topology, 0101 mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Mathematical physics, Spin-½, Mathematics
Abstract

For a Dirac operator \(D_{\bar{g}}\) over a spin compact Riemannian manifold with boundary \((\bar{X},\bar{g})\), we give a new construction of the Calderon projector on \(\partial\bar{X}\) and of the associated Bergman projector on the space of L2 harmonic spinors on \(\bar{X}\), and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of \(D_{\bar{g}}\) and the scattering theory for the Dirac operator associated with the complete conformal metric \(g=\bar{g}/\rho^{2}\) where ρ is a smooth function on \(\bar{X}\) which equals the distance to the boundary near \(\partial\bar{X}\). We show that \(\frac{1}{2}(\operatorname{Id}+\tilde{S}(0))\) is the orthogonal Calderon projector, where \(\tilde{S}(\lambda)\) is the holomorphic family in {ℜ(λ)≥0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.

Published
2014

18. Asymptotic behavior for a singular diffusion equation with gradient absorption

Author

Philippe Laurençot, Razvan Gabriel Iagar, 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
singular diffusion, bounded measures, Diffusion equation, Applied Mathematics, Mathematical analysis, p-Laplacian, large time behavior, gradient absorption, Mathematics - Analysis of PDEs, Singular solution, 35K67, 35K92, 35B40, 35D40, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], very singular solutions, Absorption (logic), Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

We study the large time behavior of non-negative solutions to the singular diffusion equation with gradient absorption ∂ t u − Δ p u + | ∇ u | q = 0 in ( 0 , ∞ ) × R N , for p c : = 2 N / ( N + 1 ) p 2 and p / 2 q q ⁎ : = p − N / ( N + 1 ) . We prove that there exists a unique very singular solution of the equation, which has self-similar form and we show the convergence of general solutions with suitable initial data towards this unique very singular solution.

Published
2014
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19. Eternal solutions to a singular diffusion equation with critical gradient absorption

Author

Philippe Laurençot, Razvan Gabriel Iagar, 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Diffusion equation, critical exponent, Eternal solution, $p$-Laplacian, General Physics and Astronomy, Type (model theory), self-similar solution, 01 natural sciences, gradient absorption, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Beta (velocity), Absorption (logic), Nabla symbol, 0101 mathematics, Mathematical Physics, Mathematical physics, Mathematics, singular diffusion, Applied Mathematics, 010102 general mathematics, Zero (complex analysis), Order (ring theory), uniqueness, Statistical and Nonlinear Physics, 010101 applied mathematics, p-Laplacian, 35K67, 35K92, 34B40, 34C11, 35B33, Analysis of PDEs (math.AP)
Abstract

The existence of nonnegative radially symmetric eternal solutions of exponential self-similar type $u(t,x)=e^{-p\beta t/(2-p)} f_\beta(|x|e^{-\beta t};\beta)$ is investigated for the singular diffusion equation with critical gradient absorption \begin{equation*} \partial_{t} u-\Delta_{p} u+|\nabla u|^{p/2}=0 \quad \;\;\hbox{in}\;\; (0,\infty)\times\real^N \end{equation*} where $2N/(N+1) < p < 2$. Such solutions are shown to exist only if the parameter $\beta$ ranges in a bounded interval $(0,\beta_*]$ which is in sharp contrast with well-known singular diffusion equations such as $\partial_{t}\phi-\Delta_{p} \phi=0$ when $p=2N/(N+1)$ or the porous medium equation $\partial_{t}\phi-\Delta\phi^m=0$ when $m=(N-2)/N$. Moreover, the profile $f(r;\beta)$ decays to zero as $r\to\infty$ in a faster way for $\beta=\beta_*$ than for $\beta\in (0,\beta_*)$ but the algebraic leading order is the same in both cases. In fact, for large $r$, $f(r;\beta_*)$ decays as $r^{-p/(2-p)}$ while $f(r;\beta)$ behaves as $(\log r)^{2/(2-p)} r^{-p/(2-p)}$ when $\beta\in (0,\beta_*)$.

Published
2013
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20. Asymptotic expansion of the solution to the Stokes flow problem in a thin cylindrical elastic tube

Author

Panasenko, Grigory, Stavre, Ruxandra, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), 'Simion Stoilow' Institute of Mathematics (IMAR), and Romanian Academy of Sciences

Subjects
Physics::Fluid Dynamics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

International audience; The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.

Published
2012
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21. Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow

Author

Panasenko, Grigory, Stavre, Ruxandra, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, and Panasenko, Grigory

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2012

22. Inverse problem for the heat equation and the Schrodinger equation on a tree

Author

Liviu I. Ignat, Ademir F. Pazoto, Lionel Rosier, Rosier, Lionel, 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Basque Center for Applied Mathematics (BCAM), Basque Center for Applied Mathematics, Instituto de Matemática da Universidade Federal do Rio de Janeiro (IM / UFRJ), Universidade Federal do Rio de Janeiro (UFRJ), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Mathematics::Analysis of PDEs, Boundary (topology), Carleman estimate, 01 natural sciences, Stability (probability), Theoretical Computer Science, Schrödinger equation, symbols.namesake, Tree (descriptive set theory), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Mathematical Physics, Mathematics, Applied Mathematics, Heat equation, 010102 general mathematics, Mathematical analysis, Inverse problem, Lipschitz continuity, Schrodinger equation, Computer Science Applications, 010101 applied mathematics, Signal Processing, network, symbols, Schrödinger's cat, Tree
Abstract

International audience; In this paper we establish global Carleman estimates for the heat and Schrodinger equations on a network. The heat equation is considered on a general tree and the Schrodinger equation on a star-shaped tree. The Carleman inequalities are used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the heat (resp. the Schrodinger) equation from boundary measurements.

Published
2012

23. A Viscous Fluid Flow through a Thin Channel with Mixed Rigid-Elastic Boundary: Variational and Asymptotic Analysis

Author

Ruxandra Stavre, R. Fares, Grigory Panasenko, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), 'Simion Stoilow' Institute of Mathematics (IMAR), and Romanian Academy of Sciences

Subjects
Asymptotic analysis, Article Subject, lcsh:Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), Stokes flow, lcsh:QA1-939, 01 natural sciences, 010101 applied mathematics, Boundary layer, Asymptotology, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], A priori and a posteriori, Uniqueness, 0101 mathematics, Asymptotic expansion, Analysis, Mathematics
Abstract

We study the nonsteady Stokes flow in a thin tube structure composed by two thin rectangles with lateral elastic boundaries which are connected by a domain with rigid boundaries. After a variational approach of the problem which gives us existence, uniqueness, regularity results, and somea prioriestimates, we construct an asymptotic solution. The existence of a junction region between the two rectangles imposes to consider, as part of the asymptotic solution, some boundary layer correctors that correspond to this region. We present and solve the problems for all the terms of the asymptotic expansion. For two different cases, we describe the order of steps of the algorithm of solving the problem and we construct the main term of the asymptotic expansion. By means of thea prioriestimates, we justify our asymptotic construction, by obtaining a small error between the exact and the asymptotic solutions.

Published
2012
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24. Existence and uniqueness of very singular solutions for a fast diffusion equation with gradient absorption

Author

Razvan Gabriel Iagar, Philippe Laurençot, 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
singular diffusion, Diffusion equation, self-similar solutions, General Mathematics, p-Laplacian, 010102 general mathematics, Mathematics::Analysis of PDEs, uniqueness, 01 natural sciences, 010101 applied mathematics, gradient absorption, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, Absorption (logic), Nabla symbol, very singular solution, 0101 mathematics, 35K67, 35K92, 34B40, 34C11, 35B33, Mathematical physics, Mathematics, Analysis of PDEs (math.AP)
Abstract

Existence and uniqueness of radially symmetric self-similar very singular solutions are proved for the singular diffusion equation with gradient absorption \begin{equation*} \partial_t u -\Delta_{p}u+|\nabla u|^q=0, \ \hbox{in} \ (0,\infty)\times\real^N, \end{equation*} where $2N/(N+1)

Published
2011
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25. Homogenization of a highly rarefied binary structure of finite diffusivity

Author

Gruais, Isabelle, Polisevski, Dan, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Multiple scale method, Homogenization, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Binary number, 35B27, Auxiliary function, 16. Peace & justice, Thermal diffusivity, Critical value, 01 natural sciences, Homogenization (chemistry), 010101 applied mathematics, Diffusion, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Small particles, 0101 mathematics, Binary structure, Analysis, Mathematics
Abstract

International audience; The homogenization of a binary structure made of very small particles of general shape is performed when the diffusivity is finite and when the size of the particles has a critical value with respect to a rarefaction number. The limit problem involves an auxiliary function which can be interpreted as the solution of a problem of cellular type, thus filling the gap with classical methods of multiple scales.

Published
2007
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