From Instability to Singularity Formation in Incompressible Fluids

TL;DR

This paper proves finite-time singularity formation in certain incompressible fluid models by leveraging classical instability effects, overcoming previous regularity obstructions, and extending results to 3D Euler systems.

Contribution

It introduces a novel method exploiting second-order instability effects to demonstrate singularities in incompressible fluids, including 2D Boussinesq and 3D Euler systems.

Findings

Finite-time singularity formation in 2D Boussinesq solutions.

Extension of singularity results to 3D Euler system.

Overcoming previous regularity obstructions in singularity proofs.

Abstract

We establish finite-time singularity formation for $C^{1,\alpha}$ solutions to the Boussinesq system that are compactly supported on $\mathbb{R}^2$ and infinitely smooth except in the radial direction at the origin. The solutions are smooth in the angular variable at the blow-up point, which was a fundamental obstruction in previous works. This is done by exploiting a second-order effect, related to the classical Rayleigh--B\'enard instability, that overcomes the regularizing effect of transport. A similar result is established for the 3d Euler system based on the Taylor--Couette instability.