## Séminaire Modélisation, Analyse et Calcul

publié le , mis à jour le

Organisateurs : Nicolas Godet, Grégory Faye & Ariane Trescases

Horaires et lieux habituels : mardi à 11h en amphi Schwartz (Bât. 1R3)

• ### Mardi 14 novembre 2017 11:00-12:00 - Mariana Haragus - Université de France Comté

Bifurcations et stabilité des ondes non linéaires dans l’équation de Lugiato-Lefever

Résumé : Nous étudions l’existence et la stabilité des ondes périodiques pour un modèle non linéaire, l’équation de Lugiato-Lefever, issu de l’optique. En utilisant des méthodes de la théorie des bifurcations, nous étudions les bifurcations de Turing et montrons l’existence de solutions périodiques. Cette approche permet également de conclure sur la stabilité de ces solutions vis-à-vis de perturbations périodiques dont la période est un multiple entier de la période de l’onde. En utilisant ensuite de méthodes de la théorie des opérateurs, nous montrons la stabilité spectrale de ces solutions vis-à-vis de perturbations générales, bornées.

Lieu : Salle MIP

• ### Mardi 21 novembre 2017 11:00-12:00 - Andres Contreras - New Mexico State University

Nearly parallel vortex filaments in 3D Ginzburg-Landau theory

Résumé : In this talk I present the derivation of a reduced model for nearly parallel vortex filaments in Ginzburg-Landau theory. The reduced energy of the filaments had been formally derived in many contexts in soft condensed matter (Aftalion-Riviere, del Pino-Kowalczyk) and it was known that its validity would constitute a fundamental step towards solving several open problems in the field. Our main theorem gives the first rigorous proof of this phenomenon. The analysis relies on an improved characterization of the vorticity region and a more precise asymptotic expansion of the energy, valid in this context. This is joint work with Robert L. Jerrard.

Lieu : Salle MIP

• ### Mardi 28 novembre 2017 11:00-12:00 - Birte Schmidtmann - RWTH Aachen University (Germany)

Reconstructions avec limiteurs de 3ème ordre pour des méthodes de volumes finis

Résumé : We are interested in the numerical solution of hyperbolic conservation laws on the most local
compact stencil consisting of only nearest neighbors. In the Finite Volume setting, in order
to obtain higher order methods, the main challenge is the reconstruction of the interface
values. These are crucial for the de-nition of the numerical
ux functions, also referred to
as the Riemann solver of the scheme.
Often, the functions of interest contain smooth parts as well as discontinuities. Treating
such functions with high-order schemes may lead to undesired oscillations. However, what
is required is a solution with sharp discontinuities while maintaining high-order accuracy
in smooth regions. One possible way of achieving this is the use of limiter functions in the
MUSCL framework which switch the reconstruction to lower order when necessary. Another
possibility is the third-order variant of the WENO family, called WENO3.
In this work, we will recast both methods in the same framework to demonstrate the relation
between Finite Volume limiter functions and the way WENO3 performs limiting. We present
a new limiter function, which contains a decision criterion that is able to distinguish between
discontinuities and smooth extrema. Our newly-developed limiter function does not require
an arti-cial parameter, instead, it uses only information of the initial condition.
We compare our insights with the formulation of the weight-functions in WENO3. The
weights contain a parameter ", which was originally introduced to avoid the division by
zero. However, we will show that " has a signi-cant in
uence on the behavior of the
reconstruction and relating the WENO3 weights to our decision criterion allows us to give
a clarifying interpretation.
In a second part, we will review some well-known Riemann solvers and introduce a family
of incomplete Riemann solvers which avoid solving the eigensystem. Nevertheless, these
solvers still reproduce all waves with less dissipation than other methods such as HLL and
FORCE, requiring only an estimate of the globally fastest wave speeds in both directions.
Therefore, the new family of Riemann solvers is particularly e-cient for large systems of
conservation laws when no explicit expression for the eigensystem is available.

• ### Mardi 5 décembre 2017 11:00-12:00 - Charlotte Perrin - Institut de Mathématiques de Marseille

Lagrangian approach to one-dimensional constrained systems

Résumé : In this talk I will introduce and study two constrained systems which
may appear in fluid mechanics in the modelling of mixtures (constraint
on the maximal volume fraction) or of partially free surface flows
(constraint on the maximal height of the flow). I will develop a
Lagrangian approach, based on one-dimensional optimal transport tools,
which enables to obtain original existence results. I will finally show
that this approach can be also used from a numerical point of view.

Lieu : Salle MIP

• ### Mardi 12 décembre 2017 11:00-12:00 -

Pas de séminaire en raison de l’école d’hiver : Méthodes Déterministes et Stochastiques en Neurosciences

Résumé : Voir plus d’informations sur l’école d’hiver ici :
http://www.cimi.univ-toulouse.fr/mib/fr/Ecole-DSMN
N’oubliez pas de vous inscrire !

• ### Mardi 19 décembre 2017 09:45-10:45 - Radu Ignat - Institut de Mathématiques de Toulouse

Le caractère bien posé d’une EDP stochastique non-locale modélisant l’ondulation de l’aimantation

Résumé : L’ondulation de l’aimantation est une microstructure formée par la distribution des moments magnétiques dans une couche ferromagnétique mince. Cette microstructure est déclenchée par l’orientation aléatoire du réseau polycristallin du matériau. Dans un certain régime asymptotique, le modèle est décrit par une EDP elliptique non-locale et non-linéaire (fortement anisotrope) en dimension 2 avec bruit blanc comme membre de droite. Comme pour les EDP stochastiques singulières, le membre de droite est trop "rugueux" pour la non-linéarité de l’équation. Pour montrer le caractère bien posé de cette équation pour des petites données, nous nous inspirons de l’approche récente des chemins rugueux pour les EDP stochastiques singulières. Pour atteindre ce but, nous développons une théorie de régularité de Schauder pour le symbole non-local (non-standard) $|k_1|^3+k_2^2$. C’est un travail en collaboration avec Felix Otto (Leipzig).

Lieu : Amphi L. Schwartz

• ### Mardi 9 janvier 2018 11:00-12:00 -

Pas de séminaire en raison de la journée nouveaux entrants MIP

• ### Mardi 16 janvier 2018 11:00-12:00 - Jacek Jendrej - Université Paris 13

Two-bubble dynamics for the equivariant wave maps equation

Résumé : I will consider the energy-critical wave maps equation with values in the two-dimensional sphere in the equivariant case, that is for symmetric initial data. It is known that if the initial data has small energy, then the corresponding solution scatters. Moreover, the initial data of any scattering solution has topological degree 0. I try to answer the following question : what are the non-scattering solutions of topological degree 0 and the least possible energy ? According to the Soliton Resolution Conjecture, such "threshold" solutions should decompose asymptotically into a superposition of two ground states at different scales, with no radiation.
It turns out that one can construct non-scattering threshold solutions. I will also describe the dynamical behavior of any threshold solution : the two ground states collide and the solution scatters in one time direction.
Joint work with Andrew Lawrie (MIT).

Lieu : Salle MIP

• ### Mardi 23 janvier 2018 11:00-12:00 - Stefanie Petermichl - Institut de Mathématiques de Toulouse

And old question on quasiconformality, the use of the heat equation and the theory of weights

Résumé : In this lecture we discuss classical singular integral operators such
conjugate functions or Riesz transforms, known for their applications to PDE.
In the first part of the lecture we present the solution to a
regularity problem for the
Beltrami equation in the complex plane through a change of measure (by
a weight). We discuss its link to singular integrals and their sharp
weighted and unweighted norm estimates in Lebesgue spaces.
We then discuss three easily understood technical corner stones that
arose in the theory weights during the last 15 years, that have given
us a different understanding of
operators such as the Hilbert transform. These interpretations all
have an underlying
probabilistic nature and confirm a fruitful link, ideologically known
for some time in one form in some of the works of Bourgain and
Burkholder and Gundy-Varopoulos in another.
The talk is accessible to a general audience.

Lieu : Salle MIP

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