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 $u = Λ^{-1+α}\nabla^{\perp} θ$, for $α\in (0,1)$. 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.

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