Local regularity for the modified SQG patch equation

TL;DR

This paper investigates the local regularity of patch solutions for the modified SQG equations, covering a range of parameters between 0 and 1/2, and demonstrates conditions for regularity and singularity formation.

Contribution

It establishes local-in-time regularity results for the modified SQG patch equations across different domains and parameter ranges, extending understanding of their dynamics.

Findings

Local regularity for all  in the whole plane.

Regular initial data can lead to finite-time singularities on the half-plane.

Results connect the behavior of modified SQG to classical Euler and SQG equations.

Abstract

We study the patch dynamics on the whole plane and on the half-plane for a family of active scalars called modified SQG equations. These involve a parameter $\alpha$ which appears in the power of the kernel in their Biot-Savart laws and describes the degree of regularity of the equation. The values $\alpha=0$ and $\alpha=\frac 12$ correspond to the 2D Euler and SQG equations, respectively. We establish here local-in-time regularity for these models, for all $\alpha\in(0,\frac 12)$ on the whole plane and for all small $\alpha>0$ on the half-plane. We use the latter result in [16], where we show existence of regular initial data on the half-plane which lead to a finite time singularity.