On sharp fronts and almost-sharp fronts for singular SQG

In this paper we consider a family of active scalars with a velocity field given by u =^{Λ - 1 + α}∇^⊥ θ, for α ∈ (0 , 1). This family of equations is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to α = 0.

We consider the evolution of sharp fronts by studying families of almost-sharp fronts. These are smooth solutions with simple geometry in which a sharp transition in the solution occurs in a tubular neighbourhood (of size δ). We study their evolution and that of compatible curves, and introduce the notion of a spine for which we obtain improved evolution results, gaining a full power (of δ) compared to other compatible curves.