Jerome Wettstein - Half-Harmonic Gradient Flows: Existence, Uniqueness and Bubbling

Abstract: Non-local energies are a useful tool in order to model interactions of points far away from each other, but still contributing to the energy. For instance, the half-harmonic map equation, a prototype/model case for fractional non-linear PDEs, is intimately connected to free-boundary minimal discs and therefore a generalisation of the classical Plateau problem. In addition, in more recent works by Simon Blatt, Armin Schikorra, Philipp Reiterer and Nicole Vorderobermeier, the techniques used in the context of fractional harmonic maps have been adapted to more general examples of non-local energies, such as knot energies. In this talk, the speaker aims to provide a broad overview of some of the most important features of the half-harmonic map equation and his research in the context of the associated gradient flow. In particular, fundamental properties such as existence, regularity, uniqueness and energy concentration of solutions to the half-harmonic gradient flow, as proven by the speaker, are discussed and connections to the local analogue, the harmonic gradient flow, are drawn to motivate the approach and techniques underlying the general setup.