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 = Lambda(-1+alpha)del(perpendicular to)theta, for alpha is an element of (0, 1). This family of equations is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to alpha = 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 delta). 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 delta) compared to other compatible curves. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates a family of active scalars with a velocity field, which is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation. By studying families of almost-sharp fronts, the evolution of sharp fronts and compatible curves are analyzed, introducing the concept of a spine for improved evolution results.