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Liste des exposés

I. Nouvelles tendances en mécanique des fluides

II. Problèmes à frontière libre

III. Modèles mathèmatiques et méthodes numériques en mécanique des milieux continus

VI. Analyse et contrôle des EDP

VIII. Analyse non-lisse et optimisation

(cette liste est en processus d'actualisation)

**List des exposés**

1.

BIROUD Kheireddine kh_biroud@yahoo.fr

Ecole préparatoire d'économie de Tlemcen, Algéria

Titre: Existence and nonexistence for semilinear problem with sigular term (details)**Résumé:**

In this work part we consider the following class of elliptic problem \begin{equation*} \left\{ \begin{array}{cc} -\Delta u=\dfrac{u^{q}}{d^2} & \text{in }\Omega , \\ u>0 & \text{in }\Omega , \\ u=0 & \text{on }\partial \Omega ,% \end{array}% \right. \end{equation*}% where \(0

Ecole préparatoire d'économie de Tlemcen, Algéria

Titre: Existence and nonexistence for semilinear problem with sigular term (details)

In this work part we consider the following class of elliptic problem \begin{equation*} \left\{ \begin{array}{cc} -\Delta u=\dfrac{u^{q}}{d^2} & \text{in }\Omega , \\ u>0 & \text{in }\Omega , \\ u=0 & \text{on }\partial \Omega ,% \end{array}% \right. \end{equation*}% where \(0

2.

CAZACU Cristian cristi_cazacu2002@yahoo.com

University Politehnica of Bucharest and Simion Stoilow Institute of Mathematics of the Romanian Academy, Romania

Titre: Controllability results for a Kuramoto-Sivashinsky model on trees (details)**Résumé:**

In this talk we present some recent results on the boundary controllability of the Kuramoto-Sivashinsky equation on star-shaped trees. Roughly speaking, we show that with few exceptions, at any positive time the system is null-controllable when acting with a control on some of the external vertices. This is a work in progress with Liviu Ignat (IMAR, Romania) and Ademir Pazoto (Rio de Janeiro, Brasil).

University Politehnica of Bucharest and Simion Stoilow Institute of Mathematics of the Romanian Academy, Romania

Titre: Controllability results for a Kuramoto-Sivashinsky model on trees (details)

In this talk we present some recent results on the boundary controllability of the Kuramoto-Sivashinsky equation on star-shaped trees. Roughly speaking, we show that with few exceptions, at any positive time the system is null-controllable when acting with a control on some of the external vertices. This is a work in progress with Liviu Ignat (IMAR, Romania) and Ademir Pazoto (Rio de Janeiro, Brasil).

3.

FARCASEANU Maria farcaseanu.maria@yahoo.com

University of Craiova & "Simion Stoilow" Institute of Mathematics of the Romanian Academy, Romania

Titre: On the convergence of the sequence of solutions for a family of eigenvalue problems (details)**Résumé:**

The asymptotic behavior of the sequence \(\{u_n\}\) of positive first eigenfunctions for a class of eigenvalue problems is studied in a bounded domain \(\Omega\subset\RR^N\) with smooth boundary \(\partial \Omega\). After extracting a subsequence, we prove \(u_n\rightarrow\|\delta\|_{L^2(\Omega)}^{-1}\delta\), where \(\delta\) is the distance function to \(\partial\Omega\). Our study complements some earlier results by Payne &Philippin, Bhattacharya, DiBenedetto&Manfredi and Kawohl obtained in relation with the “torsional creep problem".This is a joint work with Mihai Mihailescu and Denisa Stancu-Dumitru.

University of Craiova & "Simion Stoilow" Institute of Mathematics of the Romanian Academy, Romania

Titre: On the convergence of the sequence of solutions for a family of eigenvalue problems (details)

The asymptotic behavior of the sequence \(\{u_n\}\) of positive first eigenfunctions for a class of eigenvalue problems is studied in a bounded domain \(\Omega\subset\RR^N\) with smooth boundary \(\partial \Omega\). After extracting a subsequence, we prove \(u_n\rightarrow\|\delta\|_{L^2(\Omega)}^{-1}\delta\), where \(\delta\) is the distance function to \(\partial\Omega\). Our study complements some earlier results by Payne &Philippin, Bhattacharya, DiBenedetto&Manfredi and Kawohl obtained in relation with the “torsional creep problem".This is a joint work with Mihai Mihailescu and Denisa Stancu-Dumitru.

4.

GAGNON Ludovick gagnon@ann.jussieu.fr

Université Pierre et Marie Curie, France

Titre: Rapid Stabilization of a Schrödinger Equation (details)**Résumé:**

In the recent years, the stabilization of many PDEs was obtained by the backstepping approach. By considering a transformation mapping the equation to stabilize to a stable equation, one reduces the stabilization problem to proving that the transformation is invertible. This strategy is applied on a linear Schrödinger equation with a distributed control. The rapid stabilization of this equation is then proved thanks to the introduction of a uniqueness condition.

Université Pierre et Marie Curie, France

Titre: Rapid Stabilization of a Schrödinger Equation (details)

In the recent years, the stabilization of many PDEs was obtained by the backstepping approach. By considering a transformation mapping the equation to stabilize to a stable equation, one reduces the stabilization problem to proving that the transformation is invertible. This strategy is applied on a linear Schrödinger equation with a distributed control. The rapid stabilization of this equation is then proved thanks to the introduction of a uniqueness condition.

5.

GRECU Andreea grecu_andreea92@yahoo.com

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Romania

Titre: Dispersive and Strichartz Estimates for the Solution of Schrodinger Equation on a Graph with Cycle (details)**Résumé:**

We present an explicit solution of the free linear Schr\"{o}dinger equation on a graph with cycle, more precisely, on a model of infinite length which has a loop coupled at the origin, and then we discuss its dispersive properties and we prove Strichartz estimates.

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Romania

Titre: Dispersive and Strichartz Estimates for the Solution of Schrodinger Equation on a Graph with Cycle (details)

We present an explicit solution of the free linear Schr\"{o}dinger equation on a graph with cycle, more precisely, on a model of infinite length which has a loop coupled at the origin, and then we discuss its dispersive properties and we prove Strichartz estimates.

6.

IAGAR Razvan Gabriel razvan_iagar@hotmail.es

Instituto de Ciencias Matemáticas (ICMAT), Madrid, Spain

Titre: Finite time extinction for diffusive Hamilton-Jacobi equations (details)**Résumé:**

The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption \[ \partial_tu-\Delta_p u+|\nabla u|^{p-1}=0 \quad \hbox{in} \ (0,\infty)\times\real^N\ , \] and fast diffusion \(2N/(N+1)

Instituto de Ciencias Matemáticas (ICMAT), Madrid, Spain

Titre: Finite time extinction for diffusive Hamilton-Jacobi equations (details)

The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption \[ \partial_tu-\Delta_p u+|\nabla u|^{p-1}=0 \quad \hbox{in} \ (0,\infty)\times\real^N\ , \] and fast diffusion \(2N/(N+1)

0\) for any \(x\in\real^N\), the support of \(u(t,x)\) is compact for any \(t>0\); moreover, the solutions remain then localized, that is, there exists \(R>0\) such that \({\rm supp}u(t,x)\subseteq B(0,R)\) for any \(t>0\). Finally, we discuss how extinction takes place in this range.

7.

LAURENT Camille laurent@ann.jussieu.fr

UPMC, Paris 6, France

Titre: Quantitative unique continuation, intensity of waves in the shadow of obstacle and approximate contro (details)**Résumé:**

Unique continuation is very often proved by Carleman estimates or Holmgren theorem. The first one requires the strong geometric assumption of pseudoconvexity of the hypersurface. The second one only requires that the hypersurface is non characteristic, but the coefficients need to be analytic. Motivated by the example of the wave equation, several authors (Tataru, Robbiano-Zuily, Hömander) finally proved in great generality that there could be unique continuation in some intermediate situation where the coefficients are analytic in part of the variables. In particular, for the wave equation, it allowed to prove the unique continuation across any non characteristic hypersurface for non analytic metric. In this talk, after presenting these works, I will describe some recent work where we quantify this unique continuation. This leads to optimal (in general) logarithmic stability estimates. They quantify the penetration into the shadow region and the cost of approximate controllability for waves

UPMC, Paris 6, France

Titre: Quantitative unique continuation, intensity of waves in the shadow of obstacle and approximate contro (details)

Unique continuation is very often proved by Carleman estimates or Holmgren theorem. The first one requires the strong geometric assumption of pseudoconvexity of the hypersurface. The second one only requires that the hypersurface is non characteristic, but the coefficients need to be analytic. Motivated by the example of the wave equation, several authors (Tataru, Robbiano-Zuily, Hömander) finally proved in great generality that there could be unique continuation in some intermediate situation where the coefficients are analytic in part of the variables. In particular, for the wave equation, it allowed to prove the unique continuation across any non characteristic hypersurface for non analytic metric. In this talk, after presenting these works, I will describe some recent work where we quantify this unique continuation. This leads to optimal (in general) logarithmic stability estimates. They quantify the penetration into the shadow region and the cost of approximate controllability for waves

8.

LIARD Thibault liard@ann.jussieu.fr

Laboratoire Jacques Louis Lions, Université Pierre et Marie Curie, France

Titre: A KALMAN RANK CONDITION FOR THE INDIRECT CONTROLLABILITY OF COUPLED SYSTEMS OF LINEAR OPERATOR GROUPS (details)**Résumé:**

We give a necessary and sufficient condition of Kalman type for the indirect controllability of systems of groups of linear operators, under some “regularity and locality” conditions on the control operator which fit very well the case of distributed controls. Moreover, in the case of first order in time systems, when the Kalman rank condition is not satisfied, we characterize exactly the initial conditions that can be controlled.

Laboratoire Jacques Louis Lions, Université Pierre et Marie Curie, France

Titre: A KALMAN RANK CONDITION FOR THE INDIRECT CONTROLLABILITY OF COUPLED SYSTEMS OF LINEAR OPERATOR GROUPS (details)

We give a necessary and sufficient condition of Kalman type for the indirect controllability of systems of groups of linear operators, under some “regularity and locality” conditions on the control operator which fit very well the case of distributed controls. Moreover, in the case of first order in time systems, when the Kalman rank condition is not satisfied, we characterize exactly the initial conditions that can be controlled.

9.

LISSY Pierre lissy@ceremade.dauphine.fr

CEREMADE, Université Paris-Dauphine, FRANCE

Titre: The cost of fast controls for the heat equation (details)**Résumé:**

In this talk, I will present the problem of estimating the growth of fast null controls for the one-dimensional heat equation with Dirichlet boundary conditions and boundary control on one side of the interval. I will explain how to obtain lower bounds and upper bounds on the cost of the control. The lower bound is obtained by estimating the cost of controlling the first eigenfunction and some tools from complex analysis, wheras upper bounds can be obtained thanks to the moment method and the construction of appropriate Gevrey functions of order 2. To conclude, since there is a gap between the lower and upper bound, I will try to explain why I conjecture that the exact behavior is given by the lower bound obtained.

CEREMADE, Université Paris-Dauphine, FRANCE

Titre: The cost of fast controls for the heat equation (details)

In this talk, I will present the problem of estimating the growth of fast null controls for the one-dimensional heat equation with Dirichlet boundary conditions and boundary control on one side of the interval. I will explain how to obtain lower bounds and upper bounds on the cost of the control. The lower bound is obtained by estimating the cost of controlling the first eigenfunction and some tools from complex analysis, wheras upper bounds can be obtained thanks to the moment method and the construction of appropriate Gevrey functions of order 2. To conclude, since there is a gap between the lower and upper bound, I will try to explain why I conjecture that the exact behavior is given by the lower bound obtained.

10.

MARDARE Sorin sorin.mardare@univ-rouen.fr

Université de Rouen, France

Titre: Analyse asymptotique du problème de Neumann dans de longs cylindres (details)**Résumé:**

Nous considérons dans un cylindre le problème de Neumann pour une équation elliptique linéaire et nous étudions le comportement asymptotique de ce problème lorsque la longueur du cylindre tend vers l'infini. Sous de très faibles hypothèses sur les données, nous prouvons que la solution de ce problème converge vers la solution d'un problème de Neumann dans le cylindre infini, la vitesse de convergence étant exponentielle. Nous nous intéressons ensuite au cas des données constantes dans la direction de l'axe du cylindre. Ce travail a été effectué en collaboration avec Michel Chipot.

Université de Rouen, France

Titre: Analyse asymptotique du problème de Neumann dans de longs cylindres (details)

Nous considérons dans un cylindre le problème de Neumann pour une équation elliptique linéaire et nous étudions le comportement asymptotique de ce problème lorsque la longueur du cylindre tend vers l'infini. Sous de très faibles hypothèses sur les données, nous prouvons que la solution de ce problème converge vers la solution d'un problème de Neumann dans le cylindre infini, la vitesse de convergence étant exponentielle. Nous nous intéressons ensuite au cas des données constantes dans la direction de l'axe du cylindre. Ce travail a été effectué en collaboration avec Michel Chipot.

11.

MARICA Aurora-mihaela mauramihaela@yahoo.com

Universitatea Politehnica Bucuresti, Romania

Titre: Wave propagation on irregular grids (details)**Résumé:**

Abstract: In this presentation, we analyze the propagation of discrete waves on grids in the following situations: a) obtained by diffeomorphic transformations mixing convex-concave pieces; b) with pathological accumulations of points in some parts of the domain. c) when methods with several modes (quadratic finite elements or discontinuous Galerkin) are used on non-uniform meshes. We use a multiplier approach.

Universitatea Politehnica Bucuresti, Romania

Titre: Wave propagation on irregular grids (details)

Abstract: In this presentation, we analyze the propagation of discrete waves on grids in the following situations: a) obtained by diffeomorphic transformations mixing convex-concave pieces; b) with pathological accumulations of points in some parts of the domain. c) when methods with several modes (quadratic finite elements or discontinuous Galerkin) are used on non-uniform meshes. We use a multiplier approach.

12.

MARINOSCHI Gabriela gabriela.marinoschi@acad.ro

Institute of Mathematical Statistics and Applied Mathematics, Romania

Titre: Feedback stabilization of system for phase separation (details)**Résumé:**

We are concerned with the internal feedback stabilization of the phase field system of Cahn-Hilliard type, modeling the phase separation in a binary mixture. Under suitable assumptions on an arbitrarily fixed stationary solution, we construct via spectral separation arguments, a feedback controller having the support in an arbitrary open subset of the space domain, such that the closed loop nonlinear system exponentially reach the prescribed stationary solution.

Institute of Mathematical Statistics and Applied Mathematics, Romania

Titre: Feedback stabilization of system for phase separation (details)

We are concerned with the internal feedback stabilization of the phase field system of Cahn-Hilliard type, modeling the phase separation in a binary mixture. Under suitable assumptions on an arbitrarily fixed stationary solution, we construct via spectral separation arguments, a feedback controller having the support in an arbitrary open subset of the space domain, such that the closed loop nonlinear system exponentially reach the prescribed stationary solution.

13.

MIHAILESCU Mihai mmihailes@yahoo.com

University of Craiova & "Simion Stoilow" Institute of Mathematics of the Romanian Academy, Romania

Titre: Classification of isolated singularities for inhomogeneous operators in divergence form (details)**Résumé:**

Consider the equation \({\rm div}\left(\frac{\phi(|\nabla u|)}{|\nabla u|}\nabla u\right)=0\) on the punctured unit ball from \({\mathbb R}^N\) (\(N\geq 2\)), where \(\phi\) is an odd, increasing homeomorphism from \(\mathbb R\) onto \(\mathbb R\) of class \(C^1\). Under reasonable assumptions on \(\phi\) we prove that if \(u\) is a non-negative solution of the equation, then either \(0\) is a removable singularity of \(u\) or \(u\) behaves near \(0\) as the fundamental solution of the equation. In particular, our result complements to the case on inhomogeneous operators in divergence form Bocher's Theorem and some classical results by J. Serrin.

University of Craiova & "Simion Stoilow" Institute of Mathematics of the Romanian Academy, Romania

Titre: Classification of isolated singularities for inhomogeneous operators in divergence form (details)

Consider the equation \({\rm div}\left(\frac{\phi(|\nabla u|)}{|\nabla u|}\nabla u\right)=0\) on the punctured unit ball from \({\mathbb R}^N\) (\(N\geq 2\)), where \(\phi\) is an odd, increasing homeomorphism from \(\mathbb R\) onto \(\mathbb R\) of class \(C^1\). Under reasonable assumptions on \(\phi\) we prove that if \(u\) is a non-negative solution of the equation, then either \(0\) is a removable singularity of \(u\) or \(u\) behaves near \(0\) as the fundamental solution of the equation. In particular, our result complements to the case on inhomogeneous operators in divergence form Bocher's Theorem and some classical results by J. Serrin.

14.

MOYANO Ivan ivan.moyano-garcia@polytechnique.edu

CMLS, Ecole polytechnique, France

Titre: Local exact controllability of a quantum particle in a time-varying 2D disc with radial data (details)**Résumé:**

In this talk we consider the problem of the controllability of the Schrödinger equation via domain deformations, i.e., using the domain as a control. We obtain a result of this kind in the case of the two-dimensional unit disk, for radial data. Our methods are based on a local exact controllability result around a certain trajectory, obtained thanks to the Inverse Mapping theorem.

CMLS, Ecole polytechnique, France

Titre: Local exact controllability of a quantum particle in a time-varying 2D disc with radial data (details)

In this talk we consider the problem of the controllability of the Schrödinger equation via domain deformations, i.e., using the domain as a control. We obtain a result of this kind in the case of the two-dimensional unit disk, for radial data. Our methods are based on a local exact controllability result around a certain trajectory, obtained thanks to the Inverse Mapping theorem.

15.

MUNTEANU Ionut ionut.munteanu@uaic.ro

Bielefeld Universitat, Bielefeld, Germany

Titre: Stabilization of parabolic-type equations (details)**Résumé:**

We design here a finite-dimensional feedback stabilizing Dirichlet boundary controller for the equilibrium solutions to parabolic equations. The feedback is given in a very simple form.

Bielefeld Universitat, Bielefeld, Germany

Titre: Stabilization of parabolic-type equations (details)

We design here a finite-dimensional feedback stabilizing Dirichlet boundary controller for the equilibrium solutions to parabolic equations. The feedback is given in a very simple form.

16.

PIRVU Traian tpirvu@math.mcmaster.ca

McMaster University, Canada

Titre: On a Stochastic Control Problem with Regime Switching (details)**Résumé:**

We study the Merton Stochastic Control Problem with Regime Switching Utilities. The risk preferences are of CRRA type with risk aversion depending on the regime. The value function satisfy a nonlinear PDE. We establish the existence of a unique classical solution for this PDE by using convex duality

McMaster University, Canada

Titre: On a Stochastic Control Problem with Regime Switching (details)

We study the Merton Stochastic Control Problem with Regime Switching Utilities. The risk preferences are of CRRA type with risk aversion depending on the regime. The value function satisfy a nonlinear PDE. We establish the existence of a unique classical solution for this PDE by using convex duality

17.

ROSIER Lionel lionel.rosier@mines-paristech.fr

Mines ParisTech, France

Titre: Controllability of some evolution equations by the flatness approach (details)**Résumé:**

In this talk, we review some recent exact controllability results obtained by the flatness approach, which provides the control and the trajectory explicitly as series, and gives accurate numerical simulations. The heat equation with degenerate/singular coefficients and the Schrodinger equations will be considered.

Mines ParisTech, France

Titre: Controllability of some evolution equations by the flatness approach (details)

In this talk, we review some recent exact controllability results obtained by the flatness approach, which provides the control and the trajectory explicitly as series, and gives accurate numerical simulations. The heat equation with degenerate/singular coefficients and the Schrodinger equations will be considered.

18.

ROVENTA Ionel ionelroventa@yahoo.com

Department of Mathematics, University of Craiova, Romania

Titre: Uniform boundary observability for finite differences discretisation of a clamped beam equation (details)**Résumé:**

We study boundary controllability properties of a finite difference semi-discretizattion of the clamped beam equation. The strategy used is based on a Ingham's type inequality combined to a Rouch\'e eigenvalue localisation argument, asymptotic estimates and discrete multiplier method.

Department of Mathematics, University of Craiova, Romania

Titre: Uniform boundary observability for finite differences discretisation of a clamped beam equation (details)

We study boundary controllability properties of a finite difference semi-discretizattion of the clamped beam equation. The strategy used is based on a Ingham's type inequality combined to a Rouch\'e eigenvalue localisation argument, asymptotic estimates and discrete multiplier method.

19.

SAAOF Abdel-ilah abdelilah.saaof@gmail.com

Faculty of Sciences, Moulay Ismail University, Meknes, Morocco

Titre: Boundary constrained observability for hyperbolic systems (details)**Résumé:**

The purpose of this talk is to introduce the concept of boundary observability with constraints for distributed hyperbolic system evolving in spatial domain \(\Omega\). It consists the reconstruction of the initial conditions, on a subregion \(\Gamma\) of \(\partial\Omega\), knowing that the initial state is between two prescribed functions on \(\Gamma\) also that the initial speed is between two others functions also prescribed on \(\Gamma\). We give some definitions and proprieties of this kind of observability and we describe an approach to solve this problem. The approach is based on the Lagrangian multiplier which gives an algorithm allowing the reconstruction of the initial conditions. This algorithm is then implemented numerically and illustrated through an example.

Faculty of Sciences, Moulay Ismail University, Meknes, Morocco

Titre: Boundary constrained observability for hyperbolic systems (details)

The purpose of this talk is to introduce the concept of boundary observability with constraints for distributed hyperbolic system evolving in spatial domain \(\Omega\). It consists the reconstruction of the initial conditions, on a subregion \(\Gamma\) of \(\partial\Omega\), knowing that the initial state is between two prescribed functions on \(\Gamma\) also that the initial speed is between two others functions also prescribed on \(\Gamma\). We give some definitions and proprieties of this kind of observability and we describe an approach to solve this problem. The approach is based on the Lagrangian multiplier which gives an algorithm allowing the reconstruction of the initial conditions. This algorithm is then implemented numerically and illustrated through an example.

20.

STANCU-DUMITRU Denisa denisa.stancu@yahoo.com

Politehnica University of Bucharest & "Simion Stoilow" Institute of Mathematics of the Romanian Academy, Bucharest, Romania

Titre: A perturbed eigenvalue problem on general domains (details)**Résumé:**

The perturbed problem \(-\Delta u-\Delta_p u=\lambda V(x) u\), where \(p\in (1,N)\setminus\{2\}\) and \(V\) is a weight function that may have singular points is studied in the setting of Orlicz-Sobolev spaces on general open sets. The analysis of this problem leads to the full characterization of the set of parameters \(\lambda\), as being an unbounded open interval, for which the problem possesses nontrivial solutions. This is a joint work with Mihai Mihăilescu.

Politehnica University of Bucharest & "Simion Stoilow" Institute of Mathematics of the Romanian Academy, Bucharest, Romania

Titre: A perturbed eigenvalue problem on general domains (details)

The perturbed problem \(-\Delta u-\Delta_p u=\lambda V(x) u\), where \(p\in (1,N)\setminus\{2\}\) and \(V\) is a weight function that may have singular points is studied in the setting of Orlicz-Sobolev spaces on general open sets. The analysis of this problem leads to the full characterization of the set of parameters \(\lambda\), as being an unbounded open interval, for which the problem possesses nontrivial solutions. This is a joint work with Mihai Mihăilescu.

21.

STANCUT Ionela - Loredana stancutloredana@yahoo.com

University of Craiova, Romania

Titre: Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces (details)**Résumé:**

We consider an eigenvalue problem that involves a nonhomogeneous elliptic operator, variable growth conditions and a potential \(V\) on a bounded domain in \(R^{N}\) (\(N\geq3\)) with smooth boundary. We establish three main results with various assumptions. The first one asserts that any \(\lambda>0\) is an eigenvalue of the problem. The second theorem states the existence of a constant \(\lambda_{*}>0\) such that any \(\lambda\in(0,\lambda_{*}]\) is an eigenvalue, while the third theorem claims the existence of a constant \(\lambda^{*}>0\) such that every \(\lambda\in[\lambda^{*}, \infty)\) is an eigenvalue of the problem. We look for weak solutions in a subspace of the anisotropic Orlicz-Sobolev space.

University of Craiova, Romania

Titre: Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces (details)

We consider an eigenvalue problem that involves a nonhomogeneous elliptic operator, variable growth conditions and a potential \(V\) on a bounded domain in \(R^{N}\) (\(N\geq3\)) with smooth boundary. We establish three main results with various assumptions. The first one asserts that any \(\lambda>0\) is an eigenvalue of the problem. The second theorem states the existence of a constant \(\lambda_{*}>0\) such that any \(\lambda\in(0,\lambda_{*}]\) is an eigenvalue, while the third theorem claims the existence of a constant \(\lambda^{*}>0\) such that every \(\lambda\in[\lambda^{*}, \infty)\) is an eigenvalue of the problem. We look for weak solutions in a subspace of the anisotropic Orlicz-Sobolev space.

22.

TOREBEK Berikbol turebekb85@mail.ru

Institute of Mathematics and Mathematical Modeling, Kazakhstan

Titre: Green function of the Robin and Steklov problems for the Laplace operator (details)**Résumé:**

The paper is devoted to investigation questions about constructing the explicit form of the Green’s function of the Robin and Steklov problems in the unit ball of \(ℝ^2\). In constructing this function we use the representation of the fundamental solution of the Laplace equation in the form of a series. An integral representation of the Green function is obtained and for some values of the parameters the Green function is given in terms of elementary functions.

Institute of Mathematics and Mathematical Modeling, Kazakhstan

Titre: Green function of the Robin and Steklov problems for the Laplace operator (details)

The paper is devoted to investigation questions about constructing the explicit form of the Green’s function of the Robin and Steklov problems in the unit ball of \(ℝ^2\). In constructing this function we use the representation of the fundamental solution of the Laplace equation in the form of a series. An integral representation of the Green function is obtained and for some values of the parameters the Green function is given in terms of elementary functions.

23.

VIOREL Adrian Adrian.Viorel@math.utcluj.ro

Technical University of Cluj-Napoca, Romania

Titre: Metastability for the Radiative Gas Model (details)**Résumé:**

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{amsart} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsrefs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} %\numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Metastability for the Radiative Gas Model} \author{Adrian Viorel} \email{Adrian.Viorel@math.utcluj.ro} \maketitle Studied intialy in connection with the Allen-Chan model, metastable dynamics are characterized by the existence of a slow motion stage during which solutions remain trapped in the neighborhood of a metastable manifold of solutions. More recently, Kim and Tzavaras \cite{KimTzavaras} have shown that metastable phenomena also appear in the viscous Burgers model \begin{equation}\label{Burgers} u_t + u\,u_x =\varepsilon u_{xx}, \end{equation} and, by means of the Cole-Hopf transformation, they were able to provide an explicit description of the metastable manifold consisting of so called diffusive \(N\)-waves. Subsequently, a more detailed analysis, from a dynamical systems point of view, has been carried out in \cite{BeckWayne}. In this contribution, based on joint work with C. Rohde, we investigate metastability in the radiative gas model \cite{LattanzioMarcati}, \cite{LiuTadmor}, \cite{SchochetTadmor92} \begin{equation}\label{radBurgers} u_t + u\,u_x =\varepsilon q_{x},\quad -\varepsilon^2 q_{xx}+q=u_{x}, \end{equation} showing that this nonlocal Burgers equation displays a very similar behavior to \eqref{Burgers}. Actually, we obtain estimates for the \(L^1\)-distance between solutions of \eqref{Burgers} and \eqref{radBurgers} both during the initial transient and the super slow motion stage. \\ \begin{biblist} \bib{BeckWayne}{article}{ author={{M. Beck} and {C. E. Wayne}}, title={Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity}, journal={SIAM J. Appl. Dyn. Syst.}, volume={8}, date={2009}, number={3}, pages={1043--1065}, } \bib{KimTzavaras}{article}{ author={{Y. J. Kim} and {A. E. Tzaavaras}}, title={Diffusive {\(N\)}-waves and metastability in the {B}urgers equation}, journal={SIAM J. Math. Anal.}, volume={33}, date={2001}, number={3}, pages={607--633}, } \bib{LattanzioMarcati}{article}{ author={{C. Lattanzio} and {P. Marcati}}, title={Global well-posedness and relaxation limits of a model for radiating gas}, journal={J. Differential Equations}, volume={190}, date={2003}, number={2}, pages={439--465}, } \bib{LiuTadmor}{article}{ author={{H. Liu} and {E. Tadmor}}, title={Critical thresholds in a convolution model for nonlinear conservation laws}, journal={SIAM J. Math. Anal.}, volume={33}, date={2001}, number={4}, pages={930--945}, } \bib{SchochetTadmor92}{article}{ author={{S. Schochet} and {E. Tdmor}}, title={The regularized {C}hapman-{E}nskog expansion for scalar conservation laws}, journal={Arch. Rational Mech. Anal.}, volume={119}, date={1992}, number={2}, pages={95--107}, } \end{biblist} \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Technical University of Cluj-Napoca, Romania

Titre: Metastability for the Radiative Gas Model (details)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{amsart} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsrefs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} %\numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Metastability for the Radiative Gas Model} \author{Adrian Viorel} \email{Adrian.Viorel@math.utcluj.ro} \maketitle Studied intialy in connection with the Allen-Chan model, metastable dynamics are characterized by the existence of a slow motion stage during which solutions remain trapped in the neighborhood of a metastable manifold of solutions. More recently, Kim and Tzavaras \cite{KimTzavaras} have shown that metastable phenomena also appear in the viscous Burgers model \begin{equation}\label{Burgers} u_t + u\,u_x =\varepsilon u_{xx}, \end{equation} and, by means of the Cole-Hopf transformation, they were able to provide an explicit description of the metastable manifold consisting of so called diffusive \(N\)-waves. Subsequently, a more detailed analysis, from a dynamical systems point of view, has been carried out in \cite{BeckWayne}. In this contribution, based on joint work with C. Rohde, we investigate metastability in the radiative gas model \cite{LattanzioMarcati}, \cite{LiuTadmor}, \cite{SchochetTadmor92} \begin{equation}\label{radBurgers} u_t + u\,u_x =\varepsilon q_{x},\quad -\varepsilon^2 q_{xx}+q=u_{x}, \end{equation} showing that this nonlocal Burgers equation displays a very similar behavior to \eqref{Burgers}. Actually, we obtain estimates for the \(L^1\)-distance between solutions of \eqref{Burgers} and \eqref{radBurgers} both during the initial transient and the super slow motion stage. \\ \begin{biblist} \bib{BeckWayne}{article}{ author={{M. Beck} and {C. E. Wayne}}, title={Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity}, journal={SIAM J. Appl. Dyn. Syst.}, volume={8}, date={2009}, number={3}, pages={1043--1065}, } \bib{KimTzavaras}{article}{ author={{Y. J. Kim} and {A. E. Tzaavaras}}, title={Diffusive {\(N\)}-waves and metastability in the {B}urgers equation}, journal={SIAM J. Math. Anal.}, volume={33}, date={2001}, number={3}, pages={607--633}, } \bib{LattanzioMarcati}{article}{ author={{C. Lattanzio} and {P. Marcati}}, title={Global well-posedness and relaxation limits of a model for radiating gas}, journal={J. Differential Equations}, volume={190}, date={2003}, number={2}, pages={439--465}, } \bib{LiuTadmor}{article}{ author={{H. Liu} and {E. Tadmor}}, title={Critical thresholds in a convolution model for nonlinear conservation laws}, journal={SIAM J. Math. Anal.}, volume={33}, date={2001}, number={4}, pages={930--945}, } \bib{SchochetTadmor92}{article}{ author={{S. Schochet} and {E. Tdmor}}, title={The regularized {C}hapman-{E}nskog expansion for scalar conservation laws}, journal={Arch. Rational Mech. Anal.}, volume={119}, date={1992}, number={2}, pages={95--107}, } \end{biblist} \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

24.

ZOUITEN Hayat zouiten.hayat1991@gmail.com

Moulay Ismail University, Faculty of Sciences, Meknes, Morocco

Titre: Enlarged Observability of the Gradient : A Numerical Approach (details)**Résumé:**

The aim of this paper is to explore the concept of observability with constraints of the gradient for distributed parabolic system evolving in spatial domain \(\Omega\). It consists in the reconstruction of the initial state gradient between two prescribed functions in subregion \(\omega\) of \(\Omega\). We give some definitions and properties of this notion, and then we solve the problem of regional gradient reconstruction using the Hilbert Uniqueness Method (HUM). This approach leads to an algorithm which is performed by numerical example and simulation.

Moulay Ismail University, Faculty of Sciences, Meknes, Morocco

Titre: Enlarged Observability of the Gradient : A Numerical Approach (details)

The aim of this paper is to explore the concept of observability with constraints of the gradient for distributed parabolic system evolving in spatial domain \(\Omega\). It consists in the reconstruction of the initial state gradient between two prescribed functions in subregion \(\omega\) of \(\Omega\). We give some definitions and properties of this notion, and then we solve the problem of regional gradient reconstruction using the Hilbert Uniqueness Method (HUM). This approach leads to an algorithm which is performed by numerical example and simulation.

**Participants**

1.

ARAMA Bianca - Elena bianca.arama@yahoo.com

Universitatea Alexandru Ioan Cuza Iasi, Romania

Universitatea Alexandru Ioan Cuza Iasi, Romania

2.

CATANA Viorel catana_viorel@yahoo.co.uk

University Politehnica of Bucharest, Romania

University Politehnica of Bucharest, Romania

3.

MARINESCU Andreea - Paula andreeapaula.marinescu@gmail.com

Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania

Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania

4.

MICU Sorin sd_micu@yahoo.com

University of Craiova, Romania

University of Craiova, Romania

5.

OUZAHRA Mohamed m.ouzahra@yahoo.fr

Ecole Normale Supérieure, Université Sidi Mohamed Ben Abd Ellah, Maroc

Ecole Normale Supérieure, Université Sidi Mohamed Ben Abd Ellah, Maroc

6.

POPA Catalin George cpopa@uaic.ro

Universitatea "Al. I. Cuza" din Iasi, Romania

Universitatea "Al. I. Cuza" din Iasi, Romania

7.

SIRE Yannick sire@math.jhu.edu

Johns Hopkins University, USA

Johns Hopkins University, USA

8.

ZERRIK El Hassan zerrik3@yahoo.fr

Professor, Morocco

Professor, Morocco