A Note on Singularity Formation for a Nonlocal Transport Equation

TL;DR

This paper investigates the $ ext{alpha}$-patch model, demonstrating that solutions develop finite-time singularities and characterizing their profiles at the singularity.

Contribution

It provides the first finite-time singularity formation results for the $ ext{alpha}$-patch model and describes the solution behavior at singularity.

Findings

Solutions form singularities in finite time

Characterization of solution profile at singularity

Finite-time blow-up established

Abstract

The $\alpha$-patch model is used to study aspects of fluid equations. We show that solutions of this model form singularities in finite time and give a characterization of the solution profile at the singular time.