On Sharp Fronts and Almost-Sharp Fronts for singular SQG

Abstract

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.

Global solutions for a family of GSQG front equations

“…Similar methods lead to contour dynamics equations for the boundary of SQG and GSQG patches. Local-in-time patch solutions of the SQG and GSQG equations are analyzed in [7,8,14,26,27,38,39,41,44], but a proof of whether the patch boundaries form singularities in finite time or stay globally smooth is an open question. There is, however, numerical evidence suggesting finite-time singularity formation in SQG and GSQG patches [16,47,48] and a proof of finite-time singularity formation for GSQG patches in the presence of a boundary for α strictly less than 2 and sufficiently close to 2 [27,40].…”

Temperature patches for the subcritical Boussinesq-Navier-Stokes System with no diffusion

“…By a 'temperature patch solution', we mean a solution of (1.1) where θ is an indicator function for each time t ≥ 0. These are similar to the sharp front solutions of the SQG equation and their variants which have been and are extensively studied [47], [46], [20], [19], [18], [21], [13], [5], [7], [38], [39], [34], [35], [32], [33], and the more classical theory of vortex patches for the 2D Euler equation as laid out in [41], and also [10], [11], and [37]. The Boussinesq system also supports initial data of 'Yudovich' type.…”