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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.

AMBROSE David dma68@drexel.edu

Drexel University, USA

Titre: Traveling waves in interfacial fluid dynamics with multi-valued height (details)**Résumé:**

We present a formulation for traveling waves in interfacial fluid dynamics which allows for waves with multi-valued height. For 2D flows with surface tension, we use this formulation to prove a global bifurcation theorem. We illustrate this theorem with detailed numerical simulations, which show that all of the predicted terminal behaviors from the global bifurcation theorem can indeed occur. These behaviors include the reconnection of the bifurcation curve to a trivial state, which is a phenomenon typically proved impossible for pure gravity water waves.

Drexel University, USA

Titre: Traveling waves in interfacial fluid dynamics with multi-valued height (details)

We present a formulation for traveling waves in interfacial fluid dynamics which allows for waves with multi-valued height. For 2D flows with surface tension, we use this formulation to prove a global bifurcation theorem. We illustrate this theorem with detailed numerical simulations, which show that all of the predicted terminal behaviors from the global bifurcation theorem can indeed occur. These behaviors include the reconnection of the bifurcation curve to a trivial state, which is a phenomenon typically proved impossible for pure gravity water waves.

2.

CASTRO Angel angel.castro@uam.es

Universidad Autónoma de Marid and ICMAT, Spain

Titre: Mixing solutions for the Muskat problem (details)**Résumé:**

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type \(H^5\) initial data in the fully unstable regime.

Universidad Autónoma de Marid and ICMAT, Spain

Titre: Mixing solutions for the Muskat problem (details)

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type \(H^5\) initial data in the fully unstable regime.

3.

CIOMAGA Adina adina@math.univ-paris-diderot.fr

Universite Paris Diderot, France

Titre: Homogenization of interfaces (details)**Résumé:**

I will present recent results on homogenization of intefaces, in stationary ergodic environments. These problems can be reformulated, using the levelset method, as homogenizations problems for Hamilton-Jacobi equations with non-coercive Hamiltonians. We extend the results obtained in the periodic setting by Cardaliaguet, Lions and Souganidis (2009) and show that although the interfaces may break, there is weak convergence of solutions, determined by the properties of the random media.

Universite Paris Diderot, France

Titre: Homogenization of interfaces (details)

I will present recent results on homogenization of intefaces, in stationary ergodic environments. These problems can be reformulated, using the levelset method, as homogenizations problems for Hamilton-Jacobi equations with non-coercive Hamiltonians. We extend the results obtained in the periodic setting by Cardaliaguet, Lions and Souganidis (2009) and show that although the interfaces may break, there is weak convergence of solutions, determined by the properties of the random media.

4.

DE POYFERRé Thibault tdepoyfe@dma.ens.fr

Ecole Normale Supérieure, France

Titre: Dispersion and low regularity theory for capillary water waves (details)**Résumé:**

The capillary water waves equation describes the motion of a liquid surface subject to surface tension, a dispersive physical phenomenon. A mathematical consequence of this dispersion is the family of Strichartz estimates. I will present a work with Quang Huy Nguyen, in which we prove those estimates at low regularity and use them to solve the Cauchy problem at low regularity, corresponding to a non-Lipschitz velocity field.

Ecole Normale Supérieure, France

Titre: Dispersion and low regularity theory for capillary water waves (details)

The capillary water waves equation describes the motion of a liquid surface subject to surface tension, a dispersive physical phenomenon. A mathematical consequence of this dispersion is the family of Strichartz estimates. I will present a work with Quang Huy Nguyen, in which we prove those estimates at low regularity and use them to solve the Cauchy problem at low regularity, corresponding to a non-Lipschitz velocity field.

5.

DE SILVA Daniela desilva@math.columbia.edu

Columbia University, USA

Titre: The thin free boundary problem (details)**Résumé:**

We present an overview of regularity results for the so-called thin one-phase free boundary problem introduced by Caffarelli-Roquejoffre-Sire as a model of a "non-local" Bernoulli problem. The starting point is the regularity theory for the classical Bernoulli problem, first investigated by Alt-Caffarelli. We also discuss some connections with other thin obstacle-type free boundary problems.

Columbia University, USA

Titre: The thin free boundary problem (details)

We present an overview of regularity results for the so-called thin one-phase free boundary problem introduced by Caffarelli-Roquejoffre-Sire as a model of a "non-local" Bernoulli problem. The starting point is the regularity theory for the classical Bernoulli problem, first investigated by Alt-Caffarelli. We also discuss some connections with other thin obstacle-type free boundary problems.

6.

GIANNOULIS Ioannis giannoul@uoi.gr

University of Ioannina, Greece

Titre: Interaction of modulated gravity water waves of finite depth (details)**Résumé:**

Starting from the Zakharov/Craig-Sulem formulation for the water waves problem of finite depth with and without surface tension (capillary-gravity and gravity waves, respectively), we are interested in the macroscopic manifestation of the interaction of different weakly amplitude-modulated plane waves of the linearized problem when amplitude, macroscopic space and macroscopic time have the same scaling coefficient. Apart from the formal derivation of the corresponding modulation equations, we present results concerning their justification in the case of gravity waves, which are based on recent work of Alvarez-Samaniego and Lannes on the long-time well-posedness of the water waves problem of finite depth.

University of Ioannina, Greece

Titre: Interaction of modulated gravity water waves of finite depth (details)

Starting from the Zakharov/Craig-Sulem formulation for the water waves problem of finite depth with and without surface tension (capillary-gravity and gravity waves, respectively), we are interested in the macroscopic manifestation of the interaction of different weakly amplitude-modulated plane waves of the linearized problem when amplitude, macroscopic space and macroscopic time have the same scaling coefficient. Apart from the formal derivation of the corresponding modulation equations, we present results concerning their justification in the case of gravity waves, which are based on recent work of Alvarez-Samaniego and Lannes on the long-time well-posedness of the water waves problem of finite depth.

7.

NILSSON Dag dag.nilsson@math.lu.se

Lund University, Sweden

Titre: Internal gravity-capillary solitary waves in finite depth (details)**Résumé:**

Internal waves are waves which propagate along the interface of two fluids of different density. In this talk I will present some new results regarding existence of internal solitary waves under the influence of gravity and surface tension. The main idea is to use a spatial dynamics approach and formulate the steady Euler equations as an evolution equation. This equation is then studied by using the center manifold theorem. These techniques have previously been applied succesfully to the surface wave case.

Lund University, Sweden

Titre: Internal gravity-capillary solitary waves in finite depth (details)

Internal waves are waves which propagate along the interface of two fluids of different density. In this talk I will present some new results regarding existence of internal solitary waves under the influence of gravity and surface tension. The main idea is to use a spatial dynamics approach and formulate the steady Euler equations as an evolution equation. This equation is then studied by using the center manifold theorem. These techniques have previously been applied succesfully to the surface wave case.

8.

PARAU Emilian e.parau@uea.ac.uk

University of East Anglia, UK

Titre: Axisymmetric solitary waves on a ferrofluid jet (details)**Résumé:**

Travelling axisymmetric solitary waves on the surface of a cylindrical ferrofluid jet are investigated. An azimuthal magnetic field is generated by an electric current flowing along a stationary metal rod which is mounted along the axis of the moving jet. A numerical method is used to compute fully nonlinear travelling solitary waves and comparisons with weakly nonlinear theories and experiments are presented. The time evolution of the axisymmetric nonlinear waves will be simulated.

University of East Anglia, UK

Titre: Axisymmetric solitary waves on a ferrofluid jet (details)

Travelling axisymmetric solitary waves on the surface of a cylindrical ferrofluid jet are investigated. An azimuthal magnetic field is generated by an electric current flowing along a stationary metal rod which is mounted along the axis of the moving jet. A numerical method is used to compute fully nonlinear travelling solitary waves and comparisons with weakly nonlinear theories and experiments are presented. The time evolution of the axisymmetric nonlinear waves will be simulated.

9.

SAVIN Ovidiu savin@math.columbia.edu

Columbia University, USA

Titre: Obstacle type problems for minimal surfaces (details)**Résumé:**

We describe certain obstacle type problems involving a standard and a nonlocal minimal surface. We discuss optimal regularity of the solution and a characterization of the free boundary.

Columbia University, USA

Titre: Obstacle type problems for minimal surfaces (details)

We describe certain obstacle type problems involving a standard and a nonlocal minimal surface. We discuss optimal regularity of the solution and a characterization of the free boundary.

10.

VARHOLM Kristoffer kristoffer.varholm@math.ntnu.no

Norwegian University of Science and Technology, Norway

Titre: Global bifurcation of gravity water waves with multiple critical layers (details)**Résumé:**

We establish the existence of global curves of steady periodic gravity water waves with an affine vorticity distribution, extending previous results for small-amplitude waves. The formulation used allow for waves with an arbitrary number of critical layers, at least sufficiently close to the bifurcation point. This is a work in progress with Gabriele Brüll.

Norwegian University of Science and Technology, Norway

Titre: Global bifurcation of gravity water waves with multiple critical layers (details)

We establish the existence of global curves of steady periodic gravity water waves with an affine vorticity distribution, extending previous results for small-amplitude waves. The formulation used allow for waves with an arbitrary number of critical layers, at least sufficiently close to the bifurcation point. This is a work in progress with Gabriele Brüll.

11.

VEGA SMIT Mariana mariana.vega-smit@uni-due.de

University of Duisburg-Essen, Germany

Titre: The obstacle problem for the fractional Laplacian with drift (details)**Résumé:**

We present the \(C^{1,\alpha}\) regularity of the regular part of the free boundary in the obstacle problem defined by the fractional Laplacian operator with gradient perturbation, in the subcritical regime \((s\in (1/2,1))\). More specifically, we consider \[ \min\{Lu,u- \varphi \}=0, \] where we denote \(Lu:=(-\Delta)^su+\langle b(x),\nabla u\rangle+c(x)u\). Our proof relies on a new Weiss-type monotonicity formula and an epiperimetric inequality. Both are generalizations of the ideas of G. Weiss, used in the classical obstacle problem for the Laplace operator, to our framework of fractional powers of the Laplace operator with drift.

University of Duisburg-Essen, Germany

Titre: The obstacle problem for the fractional Laplacian with drift (details)

We present the \(C^{1,\alpha}\) regularity of the regular part of the free boundary in the obstacle problem defined by the fractional Laplacian operator with gradient perturbation, in the subcritical regime \((s\in (1/2,1))\). More specifically, we consider \[ \min\{Lu,u- \varphi \}=0, \] where we denote \(Lu:=(-\Delta)^su+\langle b(x),\nabla u\rangle+c(x)u\). Our proof relies on a new Weiss-type monotonicity formula and an epiperimetric inequality. Both are generalizations of the ideas of G. Weiss, used in the classical obstacle problem for the Laplace operator, to our framework of fractional powers of the Laplace operator with drift.

12.

VELICHKOV Bozhidar bozhidar.velichkov@gmail.com

Université Grenoble Alpes, France

Titre: Regularity of the optimal sets for spectral functionals (details)**Résumé:**

We prove that the optimal set for the sum of Dirichlet eigenvalues \(\lambda_1+\dots+\lambda_k\), among all sets of prescribed Lebesgue measure, has a boundary which is \(C^{1,\alpha}\) regular up to a set of small dimension.

Université Grenoble Alpes, France

Titre: Regularity of the optimal sets for spectral functionals (details)

We prove that the optimal set for the sum of Dirichlet eigenvalues \(\lambda_1+\dots+\lambda_k\), among all sets of prescribed Lebesgue measure, has a boundary which is \(C^{1,\alpha}\) regular up to a set of small dimension.

13.

WHEELER Miles mwheeler@cims.nyu.edu

Courant Institute of Mathematical Sciences, USA

Titre: Properties of solitary waves in deep water (details)**Résumé:**

We consider two- and three-dimensional solitary water waves in infinite depth, both with and without surface tension. Under an assumption that the free surface and velocity potential decay algebraically, we show that the velocity potential behaves like a dipole with a nonzero "dipole moment" related to the kinetic energy. This implies that the angular momentum is infinite, and also gives related asymptotics for the free surface: In two dimensions it is positive near infinity while in three dimensions it changes sign. These conclusions complement previous nonexistence results for three-dimensional solitary waves without surface tension.

Courant Institute of Mathematical Sciences, USA

Titre: Properties of solitary waves in deep water (details)

We consider two- and three-dimensional solitary water waves in infinite depth, both with and without surface tension. Under an assumption that the free surface and velocity potential decay algebraically, we show that the velocity potential behaves like a dipole with a nonzero "dipole moment" related to the kinetic energy. This implies that the angular momentum is infinite, and also gives related asymptotics for the free surface: In two dimensions it is positive near infinity while in three dimensions it changes sign. These conclusions complement previous nonexistence results for three-dimensional solitary waves without surface tension.

**Participants**

1.

VARVARUCA Eugen eugen.varvaruca@uaic.ro

University "Alexandru Ioan Cuza" of Iasi, Romania

University "Alexandru Ioan Cuza" of Iasi, Romania