Local Existence of Analytic Sharp Fronts for Singular SQG

Abstract

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 = |∇| −2β ∇ ⊥ θ which (for 1 < β ≤ 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.

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.…”