Blowup with vorticity control for a 2D model of the Boussinesq equations

TL;DR

This paper introduces a simplified 2D model inspired by the Boussinesq equations that demonstrates controlled finite-time blowup, providing insights into singularity formation mechanisms in hyperbolic flows.

Contribution

It presents a new 2D model with nonlocal flux and vorticity control that exhibits stable finite-time blowup, advancing understanding of singularity development in fluid dynamics.

Findings

Proves finite-time blowup for a class of initial data.

Establishes bounds on vorticity up to blowup time.

Provides a framework for studying singularity formation mechanisms.

Abstract

We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a simplified vorticity stretching term and Biot-Savart law and provide insight into the underlying intrinsic mechanisms of singularity formation. We prove stable, controlled finite time blowup involving upper and lower bounds on the vorticity up to the time of blowup for a wide class of initial data.