**SCHEDULE**

**14:30-15:30 Thierry Gallay (Grenoble)**

Title: *Viscous vortex rings with axial symmetry*

Abstract: A three-dimensional incompressible flow is called a vortex ring if the associated vorticity distribution is concentrated in a solid torus, so that the fluid particles spin around an imaginary line that forms a closed loop. Such flows are ubiquitous in nature, and appear to be very stable. For the Euler equations with axial symmetry, a large family of uniformly translating vortex rings can be constructed by variational techniques. In the viscous case, we show that the Navier-Stokes equations have a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large Reynolds number. The solutions constructed in this way are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogues of the self-similar Lamb-Oseen vortices in two-dimensional flows. This talk is based on a joint work with Vladimir Sverak (Minneapolis).

**15:45-16:45 Matteo Novaga (Pisa)**

Title: *A two phase model with cross and self attractive interactions*

Abstract: I consider a variational model for two interacting phases, subject to cross and self attractive forces. I discuss existence and qualitative properties of minimizers. Depending on the strengths of the forces, different behaviors are possible: phase mixing or phase separation with nested or disjoint phases.

**17:00 – 18:00 Andrea Corli (Ferrara)**

Title: *Traveling waves for collective movements on a network*Abstract: The talk is concerned with the propagation of (semi-) wavefront solutions for scalar parabolic equations, on the real line or on a network. These equations have a possibly degenerate diffusion term and, if they are nonhomogeneous, the process has only one stationary state. They can be interpreted as modeling collective movements (crowd dynamics, for instance). In the case of a single equation on the real line, the results concern the existence, regularity, monotone properties of semi-wavefront solutions; the convergence to wavefront solutions is also discussed. In the case of a network and without source terms, we find necessary and sufficient conditions for the existence of wavefronts, which are essentially of algebraic nature.