María del Mar González (Universidad Autónoma de Madrid). Título: Symmetry and symmetry breaking for the fractional Caffarelli-Kohn-Nirenberg inequality. Abstract: We will consider a fractional version of the classical Caffarelli-Kohn-Nirenberg inequality. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results for the symmetry and symmetry breaking region for the minimizers. In order to get these we reformulate the inequality in cylindrical variables so that we can use the recently developed non-local ODE theory for radial solutions. We also get non-degeneracy of critical points and uniqueness of minimizers in the radial symmetry class.
Monica Musso (University of Bath) Título: Travelling helices and the vortex filament conjecture in the incompressible Euler equations. Abstract: Consider the Euler equations in R^3 expressed in vorticity form. A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be the first rigorous construction of helical filaments, associated to a translating-rotating helix. The solution is defined at all times and does not change form with time. The result generalizes to multiple similar helical filaments travelling and rotating together. This work is in collaboration with J. Dávila, M. del Pino and J. Wei.