Finite-time singularity via multi-layer degenerate pendula for the 2D Boussinesq equation with uniform $C^{1,\sqrt{\frac{4}{3}}-1-\epsilon}\cap L^2$ force

TL;DR

This paper demonstrates finite-time singularity formation in the 2D Boussinesq equation with specific regularity forces, using a novel mechanism involving multi-layer degenerate pendula and hysteresis effects.

Contribution

It introduces a new blow-up mechanism for the 2D Boussinesq equation involving multi-layer degenerate pendula and hysteresis, with forces maintaining regularity at singularity.

Findings

Finite-time singularity solutions with compact support.

Forces preserve regularity at blow-up time.

Hysteresis mechanism causes vorticity blow-up.

Abstract

We establish the existence of compactly supported solutions of the inviscid incompressible 2D Boussinesq equation with $C^{1,\sqrt{\frac{4}{3}}-1-\varepsilon}\cap L^{2}$ force that develop a singularity in finite time. Importantly, the force preserves this regularity at the blow-up time. Moreover, the forces in the vorticity and density equations have compact support. The mechanism behind the blow-up is an accumulated hysteresis effect on the vorticity caused by an infinite chain of "degenerate" pendula and flickering density.