Séminaire Analyse Appliquée (AA)

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Agenda

• Mardi 26 février 11:00-12:00 - Giulia CAVAGNARI - University of Pavia

Problèmes de contrôle optimal non local à champ moyen

Résumé : TBA

Giulia CAVAGNARI

Lieu : CMI, salle de séminaire R164 (1er étage) - I2M - Château-Gombert
39 rue Frédéric Joliot-Curie
13453 MARSEILLE cedex 13

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• Mardi 5 mars 11:00-12:00 - Antonin Monteil - Université catholique de Louvain

Ginzburg-Landau relaxation for harmonic maps valued into manifolds

Résumé : We will look at the classical problem of minimizing the Dirichlet energy of a map $u :\Omega\subset\mathbbR^2\to N$ valued into a compact Riemannian manifold $N$ and subjected to a Dirichlet boundary condition $u=\gamma$ on $\partial\Omega$. It is well known that if $\gamma$ has a non-trivial homotopy class in $N$, then there are no maps in the critical Sobolev space $H^1(\Omega,N)$ such that $u=\gamma$ on $\partial\Omega$. To overcome this obstruction, a way is to rather consider a relaxed version of the Dirichlet energy leading to singular harmonic maps with a finite number of topological singularities in $\Omega$. This was done in the 90’s in a pioneering work by Bethuel-Brezis-Helein in the case $N=\mathbbS^1$, related to the Ginzburg-Landau theory. In general, we will see that minimizing the energy leads at main order to a non-trivial combinatorial problem which consists in finding the energetically best topological decomposition of the boundary map $\gamma$ into minimizing geodesics in $N$. Moreover, we will introduce a renormalized energy whose minimizers correspond to the optimal positions of the singularities in $\Omega$.

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• Mardi 12 mars 11:00-12:00 - Valentina Franceschi - Université de Paris Sud

Valentina Franceschi (TBA)

• Mardi 26 mars 11:00-12:00 - Ivan Moyano - University of Cambridge

(TBA) Ivan Moyano

• Mardi 9 avril 11:00-12:00 - Enrique D Fernández Nieto - Universidad de Sevilla

On the Saint-Venant-Exner model with arbitrarily sloping sediment beds

Résumé : Firstly, in this talk a formal deduction of the Saint-Venant-Exner model through an asymptotic analysis of the Navier-Stokes equations will be presented. A multi-scale analysis is performed in order to take into account that the velocity of the sediment layer is smaller than the one of the fluid layer. Then, a correction of classical models is proposed. The proposed models have an associated energy and they incorporate a necessary modification that must be taken into account in order to be applied to arbitrarily sloping beds. Some of these simplified models correspond to a generalization of classic ones such as Meyer-Peter&Müller and Ashida-Michiue models. Secondly, a bilayer shallow water type model will be presented, that can be considered to describe bedload sediment transport for strong and weak interactions between the fluid and the sediment.
Several numerical tests will be finally presented, to study the evolution of a dune in terms of the repose angle of the material, to see the influence of the proposed definition of the effective shear stress in comparison with the classic one, by comparing the Saint-Venant-Exner model with the bilayer one, and by comparing with experimental data.

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Manifestation scientifique

Nature Séminaire Analyse Appliquée Loïc Le Treust, Morgan Morancey Analyse Appliquée (AA) Hebdomadaire Mardi, 11h-12h CMI, salle de séminaire (accès) -

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Il est donc souhaitable que les exposés ne soient pas spécialisés outre mesure, et bien sur qu’ils soient aussi clairs et pédagogiques que possible.

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