On sharp fronts and almost-sharp fronts for singular SQG

In this paper we consider a family of active scalars with velocity field given by u = ? ? 1 + ? ? ? ? , for ? ( 0 ) . This equations is more singular version the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to We evolution sharp fronts studying families almost-sharp fronts. These are smooth solutions simple geometry in transition solution occurs tubular neighbourhood (of size ? ). study their and that compatible curves, introduce notion spine obtain improved results, gaining full power compared other curves.