Well-posedness for SQG sharp fronts with unbounded curvature

Abstract

Patch solutions for the surface quasigeostrophic (SQG) equation model sharp temperature fronts in atmospheric and oceanic flows. We establish local well-posedness for SQG sharp fronts of low Sobolev regularity, H 2+s for arbitrarily small s > 0, allowing for fronts with unbounded curvature.

Long time interface dynamics for gravity Stokes flow

“…Equipped with this regularity and using standard energy arguments (see [17] section 6 for details), we conclude the continuity in time of h in H…”

“…One of the classical methods to deal with free boundary problems is to exploit potential theory in order to reformulate the problem into a new contour dynamics equation, which will be typically nonlocal and strongly non-linear. Let us mention that this kind of approach has been extensively and successfully used in other free boundary problems in fluid dynamics to show well-posedness (see [3,8] for the vortex patch, [11] for water waves, [30,18,17] for the SQG sharp-front, [12,16] for the Muskat problem and [19,9] for the Peskin problem). Furthermore, it has been applied to prove singularity formation for the water waves, the SQG sharp-front and the Muskat problem [6,4,29,18,5].…”

Well-Posedness for the Dispersive Hunter–Saxton Equation

“…In the case of SQG patches, Gancedo-Nguyen-Patel proved in [14] that under a suitable parametrization, the contour dynamics evolution is locally well-posed in H s (T) when s > 2. The gSQG case with α ∈ (0, 2) and α = 1, was also considered by Gancedo-Patel [15], where they in particular showed local well-posedness in H 2 for α ∈ (0, 1) and H 3 for α ∈ (1, 2).…”