Local Existence of Analytic Sharp Fronts for Singular SQG

In this paper, we prove local existence and uniqueness of analytic sharp-front solutions to a generalised SQG equation by the use of an abstract Cauchy--Kowalevskaya theorem. Here, the velocity is determined by $u = |\nabla|^{-2\beta}\nabla^\perp\theta$ which (for $1<\beta\leq 2$) is more singular than in SQG. This is achieved despite the appearance of pseudodifferential operators of order higher than one in our equation, by recasting our equation in a suitable integral form. We also provide a full proof of the abstract version of the Cauchy--Kowalevskaya theorem we use.