On the evolution of almost-sharp fronts for the surface quasi-geostrophic equation

I will describe recent work with Charles Fefferman on a

construction of families of analytic almost-sharp fronts for SQG. These

are special solutions of SQG which have a very sharp transition in a

very thin layer. One of the main difficulties of the construction is the

fact that there is no formal limit for the family of equations. I will

show how to overcome this difficulty, linking the result to joint work

with C. Fefferman and Kevin Luli on the existence of a "spine" for

almost-sharp fronts. This is a curve, defined for every time slice by a

measure-theoretic construction, that describes the evolution of the

almost-sharp front.