An Improved Regularity Criterion and Absence of Splash-like Singularities for g-SQG Patches

TL;DR

This paper proves that splash-like singularities cannot occur for regular patch solutions of the g-SQG equation with certain parameters, and provides an improved regularity criterion ensuring global regularity under bounded boundary norms.

Contribution

It establishes the non-occurrence of splash-like singularities for g-SQG patches with   and improves the global regularity criterion for solutions in this regime.

Findings

Splash-like singularities are excluded for   g-SQG patches.

Finite-time singularities are prevented if the  norms of patch boundaries stay bounded.

Results extend to all   for excluding boundary segment touches.

Abstract

We prove that splash-like singularities cannot occur for sufficiently regular patch solutions to the generalized surface quasi-geostrophic equation on the plane or half-plane with parameter $\alpha\le \frac 14$. This includes potential touches of more than two patch boundary segments in the same location, an eventuality that has not been excluded previously and presents nontrivial complications (in fact, if we do a priori exclude it, then our results extend to all $\alpha\in(0,1)$). As a corollary, we obtain an improved global regularity criterion for $H^3$ patch solutions when $\alpha\le\frac 14$, namely that finite time singularities cannot occur while the $H^3$ norms of patch boundaries remain bounded.