Cusp Formation for a Nonlocal Evolution Equation

Cordoba et al. (Ann Math 162(3): 1377-1389, 2005) introduced a nonlocal active scalar equation as a one-dimensional analogue of the surface-quasigeostrophic equation. It has been conjectured, based on numerical evidence, that the solution forms a cusp-like singularity in finite time. Up until now, no active scalar with nonlocal flux is known for which cusp formation has been rigorously shown. In this paper, we introduce and study a nonlocal active scalar, inspired by the CordobaCordoba- Fontelos equation, and prove that either a cusp-or needle-like singularity forms in finite time.