On evolution of corner-like gSQG patches

TL;DR

This paper investigates how corner-like patches evolve in generalized SQG equations, revealing that corners bend instantly depending on parameters, and demonstrating the emergence of non-convex patches from convex ones.

Contribution

It provides new insights into the instantaneous bending of corners and the transition from convex to non-convex patches in generalized SQG dynamics.

Findings

Corners bend instantaneously depending on angle and kernel order.

Existence of smooth convex patches that become immediately non-convex.

Corners can bend either upward or downward depending on parameters.

Abstract

We study the evolution of corner-like patch solutions to the generalized SQG equations. Depending on the angle size and order of the velocity kernel, the corner instantaneously bents either downward or upward. In particular, we obtain the existence of strictly convex and smooth patch solutions which become immediately non-convex.