Analysis of PDEs (Mathematics)
Finite time singularities to the 3D incompressible Euler equations for solutions in $C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,α}\cap L^2$
Córdoba, Martínez-Zoroa, and Zheng unveil a new blow-up mechanism showing how smooth 3D Euler flows can develop finite-time singularities without self-similar scaling.
Imagine a perfectly frictionless fluid, like an idealized version of water or air with no viscosity. The Euler equations are the rules that describe how this fluid moves and swirls. A big open question is: if you start with a perfectly smooth, well-behaved swirling pattern, could the fluid ever develop an infinitely sharp, jagged feature in a finite amount of time - essentially 'exploding' mathematically at a single point? This paper constructs a clever example where this happens. Instead of using the usual trick (zooming in with a self-similar 'magnifying glass' pattern), the authors build their solution like an onion with infinitely many layers of spinning fluid regions, each separated by calm, non-spinning fluid. By carefully tuning how each layer interacts with the ones around it, they show the whole structure collapses into a singularity at a single point (the origin) at one specific moment in time, even though the fluid was smooth everywhere else just before that moment.
- 01
Introduces a novel blow-up mechanism for the 1D De Gregorio model and the 3D incompressible Euler equations that does not rely on self-similar coordinates
- 02
Constructs singular solutions from infinitely many vorticity regions separated by vortex-free regions
- 03
Produces solutions of the 3D incompressible Euler equations on R3×[-T,0] with velocity in C∞(R3∖{0})∩C1,α∩L2 for t∈(-T,0)
- 04
Shows that the constructed velocity field fails to be C1 at the singular time t=0, demonstrating finite-time singularity formation
We introduce a novel mechanism that reveals finite time singularities within the 1D De Gregorio model and the 3D incompressible Euler equations. Remarkably, we do not construct our blow up using self-similar coordinates, but build it from infinitely many regions with vorticity, separated by vortex-free regions in between. It yields solutions of the 3D incompressible Euler equations in $\mathbb{R}^3\times [-T,0]$ such that the velocity is in the space $C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,α}\cap L^2$ for times $t\in (-T,0)$ and is not $C^1$ at time 0.