On nonexistence of splash singularities for the $\alpha$-SQG patches

TL;DR

This paper establishes new criteria preventing splash singularities in the $\alpha$-SQG patch solutions, refining previous results by linking curvature growth to collision avoidance and providing bounds on patch separation.

Contribution

It introduces refined conditions on curvature growth and patch separation that prevent splash singularities in $\alpha$-SQG patches, advancing understanding of singularity formation.

Findings

Derived new criteria for the absence of splash singularities.

Provided exponential lower bounds on patch separation.

Refined previous curvature growth conditions.

Abstract

In this paper, we consider patch solutions to the $\alpha$-SQG equation and derive new criteria for the absence of splash singularity where different patches or parts of the same patch collide in finite time. Our criterion refines a result due to Gancedo and Strain \cite{GS}, providing a condition on the growth of curvature of the patch necessary for the splash and an exponential in time lower bound on the distance between patches with bounded curvature.