Workshop

14.00 - 14.45: Gilles Carbou
14.45 - 15.30: Renato Lucà

15.30 - 16.00: Coffee-break

16.00 - 16.45: Marco Bravin
16.45 - 17.30: Victor Arnaiz Solorzano

09.30 - 10.15: Rafael Granero-Belinchón
10.15 - 11.00: Mehdi Badsi

11.00 - 11.30: Coffee-break

11.30 - 12.15: Pablo Miranda Rozas
12.15 - 13.00: Sergei Iakunin

13.00 - 14.30: Lunch break

14.30 - 15.15: Vincent Duchêne
15.15 - 16.00: Marco Antonio Fontelos

TITLE: Stability and diffusion of Schrödinger solutions in quantum KAM systems

ABSTRACT: In this talk I will focus on the high-energy concentration and dispersion properties of solutions of the linear Schrödinger equation in quantum KAM (Kolmogorov-Arnold-Moser) Hamiltonian systems. We will look at some spectral stability results and show examples in which the concentration of solutions around invariant tori and the energy diffusion in unstable regions of phase space can coexist.

This talk is partially based on work in progress with Nicolas Camps (Nantes) and Chenmin Sun (CNRS).

TITLE: Plasma sheath solutions for a bi-species Vlasov-Poisson-BGK model

ABSTRACT: The Debye sheath is a positively charged boundary layer that forms when a plasma is in contact with a metallic wall. It has been shown that it is possible to construct sheath type solutions for the Vlasov-Poisson equations (in a bounded domain)  provided the ions distribution function verifies the so called Bohm condition of plasma physics.

When particles experience collision or friction, the role played by the Bohm condition is unclear.  We will show that for a simple Vlasov-Poisson-BGK model it is possible to construct solutions for which the density is non negative and the electric potential is strictly monotone. Our method of proof shows that there is a competition between the Debye length and the collision frequency. We will discuss numerically whether one can enlarge the range of the physical parameters.

TITLE: Well-posedness and regularity for the Stokes system in domains with corners

ABSTRACT: In this talk I will present a recent result where we study the Stokes system in a wedge domain. This system appears naturally in the study of the lubrification approximation to deduce the thin film equations from the Navier-Stokes system with free boundary.

Having in mind this goal we study existence and regularity in weighted Sobolev spaces of solutions to the Stokes system with Navier-Slip boundary conditions in the case the angle is small enough. Moreover, we show estimates uniform in the angle to be able to study the limiting system when the angle tends to zero.

This is a joint work with Gnann, Knüpfe, Masmoudi, Roodenburg, Sauer.

TITLE:  Domain wall dynamics in notched ferromagnetic nanowires

ABSTRACT: Ferromagnetic nanowires have promising applications in data storage. In such devices, the information is encoded by Domain Walls (DW) which are thin zones of magnetization reversal. The magnetization behavior is described by the non linear Landau-Lifschitz model.

In this talk, we investigate the stability of DW configurations. In particular we highlight DW-pinning properties of notches patterned along the wire, and DW-depinning effects of  applied magnetic field.

TITLE: Three-scale singular limits

ABSTRACT: I will discuss a class of problems emerging from a "hyperbolization" strategy for the numerical approximation of weakly dispersive PDEs.

These singular problems feature three scales in the sense that the stiff operator is a combination of a component homogeneous of order 1 and a component homogeneous of order 0, each of them being associated with distinct large parameters.

I will insist on features that depart from the standard two-scale theory, and in particular on the possible dynamical emergence of a new spatial scale.

It is a joint work in progress with Arnaud Duran and Khawla Msheik.

TITLE: Evolution of viscous vortex filaments and desingularization of the Biot-Savart integral

ABSTRACT: We consider a viscous fluid with kinematic viscosity ν and initial data consisting of a smooth closed vortex filament with circulation Γ.
We show that, for short enough time, the solution consists of a deformed Lamb-Oseen vortex whose center (a filament) follows the binormal flow dynamics plus leading order corrections that depend locally on the filament curvature and the nonlocal interactions with distant parts of the filament. In order to achieve this scale separation we require Γ/ν to be sufficiently small.

TITLE: Well-posedness and qualitative properties of gravity internal waves in viscous flow

ABSTRACT: In this talk we will review some recent results regarding the motion of a free boundary between two viscous fluids. In particular, we will establish the well-posedness and certain qualitative properties of the solutions such as decay or instability in certain configurations. Joint work with Francisco Gancedo and Elena Salguero.

TITLE: Reconnection of infinitely thin vortices and its relation to the finite thickness case

ABSTRACT: The reconnection of vortices is an important example of the transition from laminar to turbulent flow. The simplest case is the reconnection of a pair of antiparallel vortices, e.g. condensation trails of an aircraft. During this process, the initially straight vortices undergo sinusoidal deformation called Crow waves and eventually intersect, giving rise to a cascade of coherent structures. Due to the finite thickness of the vortices, the definition of the reconnection time, the reconnection point, and extraction of the centerlines of the vortices after the reconnection are quite challenging tasks. However, the emerging structures, such as the horseshoe and helical waves, are very reminiscent of those observed in the evolution of infinitely thin polygonal vortex subject to the localized induction approximation (LIA). This raises the question: do vortices form a corner singularity during the reconnection? In this talk, we consider a LIA-based model of reconnection of infinitely thin vortices with an interaction term that allows the formation of a corner from initially smooth solution. We demonstrate that the eye-shaped vortex obtained after the reconnection can be recovered with another angles after a time $\pi / 2$ that is independent of the initial angle. Given that for finite thickness vortices it is difficult to extract the vortex centerline, we consider the fluid impulse, an integral quantity that can be defined for both the LIA and the solution of the Navier-Stokes equations. It turns out that this quantity can be very useful for determining the reconnection time and the reconnection points even for the case of vortices with finite thickness.

TITLE: Phase blow up for the cubic NLS and connections with the binormal flow

ABSTRACT: In this talk, we will discuss a suitable family of borderline regularity solutions of the 1d cubic Schrodinger equation exhibiting a phase blow-up. We will also discuss the connection between these solutions and the evolution of curves under the binormal curvature equation. This is a joint work with V. Banica, N. Tzvetkov and L. Vega.

TITLE: Eigenvalue Asymptotics near a flat band in presence of a slowly decaying potential

ABSTRACT: In this talk, we consider a Dirac-type operator in the network $\mathbb{Z}^n$.  We study the distribution of discrete eigenvalues  when a  perturbation by a multiplication operator that decays as $|\mu|^{-\gamma}$, with $\gamma