16th March

15:00h *Nicolas Vauchelet (Universidad Paris 6) **Mathematical study of a cell model for tumor growth : travelling front and incompressible limit *

We consider mathematical models at macroscopic scale to describe tumor growth. In this view, tumor cells are considered as an elastic material subjected to mechanical pressure. Two main classes of model can be encountered: those describing the dynamics of tumor cells density and those describing the dynamic of the tumor thanks to the motion of its domain. These latter models are free boundary problem. We will show that such free boundary problem of Hele-Shaw type can be derived thanks to an incompressible limit from models describing the dynamics of cells density. Moreover, for this model we study the existence of travelling waves, allowing to describe the spread of the tumor.

16:05h *Pierpaolo Esposito (Universidad de Roma Tre) **Equilibria of point-vortices on closed surfaces *

I will discuss the existence of equilibrium configurations for the Hamiltonian point-vortex model on a closed surface. Its topological properties determine the occurrence of three distinct situations, corresponding topologically to the sphere, to the real projective plane and to the remaining cases. As a by-product, new existence results are obtained for the singular mean-field equation with exponential nonlinearity.

Joint work with T. D’Aprile.

17:00h *Oscar Agudelo (University of West Bohemia) **Singularly perturbed Allen-Cahn equation with catenoidal nodal sets *

In this lecture we review some recent results concerning existence and asymptotic behavior of solutions to the singularly perturbed problem

\alpha^2 \Delta u + u(1-u^2)=0, in Omega

where \Omega \subset \mathbb{R}^N is either a smooth bounded domain or the entire space and N\geq 2. We take advantage of the deep connection between the equation above and the theory of minimal surfaces to study asymptotic profiles of the solutions. Particular attention is paid to solutions with catenoidal nodal set.

From 30th March to 2nd April*Cursillo “On vortex dynamics in two-dimensional or three-dimensional incompressible flows”,*

por Evelyne Miot (CNRS and Ecole Polytechnique, investigadora invitada CMM) *– Lecture 1 (Lunes 30 Marzo, 16h-17:30h, Sala Seminario 5to. piso):
Vortices in incompressible fluids. *

In this first lecture we will consider the Euler equations governing the motion of incompressible fluids, in particular in a two-dimensional setting. We will focus on the vortex solutions and present Marchioro and Pulvirenti’s result on the derivation of the point vortex system from the 2D Euler equations. We will also briefly mention the vortex dynamics in other related equations (the Navier-Stokes equation and the Gross-Pitaevskii equation).

*– Lecture 2 (Miércoles 1 Abril, 12h-13:30h, Sala Seminario 5to. piso)*

Convergence of the point vortex system to the 2D Euler equation.

Convergence of the point vortex system to the 2D Euler equation.

This lecture will explore more in detail the connection between the discrete model, described by the point vortex system, and the continuous fluid dynamics given by the Euler equation. We will show how the results by Goodman, Lowengrub and Hou and Schochet ensure that the point vortex system is a good approximation of the Euler equation when the number of vortices is large.

*– Lecture 3 (Jueves 2 Abril, 14:30h-16h, Sala Seminario 7mo. piso):*

Vortex filaments.

Vortex filaments.

We will study the analogous notion of point vortices in three dimensions, namely the vortex filaments. We will explain the formal derivation leading to the binormal curvature flow equation governing the motion of one single vortex filament. We will also relate the binormal curvature flow equation and the cubic 1D Schrödinger equation via the Hasimoto transform. Finally we will present a system of simplified equations proposed by Klein, Majda and Damodaran to describe the interaction of several almost parallel vortex filaments.