Journée thématique "Recent progress in Partial Differential Equations"
"Recent progress in Partial Differential Equations"
Marseille, 2 décembre 2016
Denis Bonheure (Université Libre de Bruxelles)
Giampiero Palatucci (Università degli Studi di Parma)
Wolfgang Reichel (Karlsruhe Institut für Technologie)
Michael Winkler (Universität Paderborn)
François Hamel, Enea Parini
Le workshop, qui consiste en quatre exposés d'une heure chacun, se
déroulera dans la salle de conférences de la FRUMAM située au 2e étage (plan d'accès).
Le repas du midi sera offert aux participants qui se seront inscrits en ligne (formulaire), dans la limite des disponibilités.
- 9h30-10h30: exposé de Denis Bonheure
- 11h00-12h00: exposé de Michael Winkler
- 14h00-15h00: exposé de Wolfgang Reichel
- 15h30-16h30: exposé de Giampiero Palatucci
Denis Bonheure: "Multi-layer patterns for the stationary Keller-Segel system"
The stationary Keller-Segel system, which is a classical model for
chemotaxis, leads (with some specific choice in the model) to the
-\varepsilon2 \Delta v+v=v^p in \Omega,
v>0 in \Omega,
\partial\nu v=0 on \partial \Omega
and the Keller-Segel equation
-\varepsilon2 \Delta v+v=\lambda e^v in \Omega,
v>0 in \Omega,
\partial\nu v=0 on \partial \Omega.
In this talk, I will show how to build radial multi-layer solutions,
assuming \Omega is a ball, of those equations using either bifurcations,
the Lyapunov-Schmidt method or a variational gluing method. It is a
remarkable fact that even if those solutions are obtained in an
asymptotic regime p\to \infty or \lambda\to 0, the layers do not
accumulate on the boundary of the ball. Instead they solve an optimal
The talk is based on several joint works with Massimo Grossi, Susanna
Terracini, Benedetta Noris, Christophe Troestler and Jean-Baptiste
Giampiero Palatucci: "The obstacle problem for nonlinear integro-differential operators"
We deal with the obstacle problem for a class of nonlinear nonlocal
equations, which include as a particular case some fractional
Laplacian-type equations. We will show the existence and uniqueness of
the solution to the obstacle problem. We will also show that the
regularity of the solutions to the obstacle problem inherits the
regularity of the obstacle, both in the case of boundedness and (Hölder)
continuity, up to the boundary.
Wolfgang Reichel: "Localized time-periodic solutions of nonlinear wave equations"
In this talk I will collect some results obtained recently for
time-periodic solutions of nonlinear wave equations. The model problems
arise from Maxwell's equations in the presence of nonlinear material
responses. The emphasis will be on the aspect of "localization", i.e.,
the effect of having solutions that decay to zero in the (unbounded)
spatial directions since the unboundedness of the underlying spatial
domain poses the main difficulties.
Michael Winkler: "Mathematical challenges arising in the analysis of chemotaxis-fluid interaction"
We consider models for the spatio-temporal evolution of populations of
microorganisms, moving in an incopressible fluid, which are able to
partially orient their motion along gradients of a chemical signal.
According to modeling approaches accounting for the mutual
interaction of the swimming cells and the surrounding fluid, we study
parabolic chemotaxis systems coupled to the (Navier-)Stokes equations
through transport and buoyancy-induced forces. The presentation
discusses mathematical challenges encountered even in the context of
basic issues such as questions concerning global existence and
boundedness, and attempts to illustrate this by reviewing some recent
developments. A particular focus will be on strategies toward achieving a
priori estimates which provide information sufficient not only for the
construction of solutions, but also for some qualitative analysis.
Dernière modification: 21/11/2016