A parabolic Ginzburg-Landau equation for vector fields on surfaces: vortices and the gradient flow of the renormalised energy

Giacomo Canevari, Università di Verona

Abstract.

In the PDE community, “Ginzburg-Landau equations” is an umbrella term for a variety of problems which originate as (possibly, simplified) models from the physics or materials science and share some common traits from the mathematical point of view. These kind of problems have been extensively studied since the seminal work by Bethuel, Brezis and Hélein. In this talk, we will discuss a parabolic Ginzburg-Landau equation for vector fields on a smooth, compact, Riemannian surface without boundary. We will show that, in a suitable asymptotic regime, the energy of the solutions concentrates on a finite number of points, the vortices, whose evolution is described by the gradient flow of a suitable scalar function: the so-called renormalised energy. The talk is based on a joint work with Antonio Segatti.