Finite-time Singularity Formation for Strong Solutions to the $3D$ Euler equations, II

TL;DR

This paper establishes local well-posedness, analyzes vorticity decomposition, and proves finite-time singularity formation for certain solutions of the 3D Euler equations, highlighting the role of vorticity growth mechanisms.

Contribution

It introduces a new approach to demonstrate finite-time singularities in the 3D Euler equations via vorticity decomposition and critical space analysis.

Findings

Finite-time singularity formation for 3D Euler solutions with swirl.

Global regularity for 0-swirl solutions in critical spaces.

Vorticity decomposition remains valid over time, influencing singularity development.

Abstract

This work is a companion to [EJE1] and its purpose is threefold: first, we will establish local well-posedness for the axi-symmetric $3D$ Euler equation in the domains $\{(x_1,x_2,x_3) \in \mathbb{R}^3 : x_3^2 \le \mathfrak{c}(x_1^2 + x_2^2) \}$ for $\mathfrak{c}$ sufficiently small in a scale of critical spaces. Second, we will prove that if the vorticity at $t=0$ can be decomposed into a scale-invariant part and a smoother part vanishing at $x=0$, then this decomposition remains valid for $t>0$ so long as the solution exists. This will then immediately imply singularity formation for finite-energy solutions in those critical spaces using that there are scale-invariant solutions which break down in finite time (this was proved in the companion paper [EJE1]). Third, we establish global regularity in the same domain and in the same scale of critical spaces for $0-$swirl solutions to the system. This implies that the finite-time singularity is not coming from the domain or the critical regularity of the data (though, these are important for the present construction), they are coming from a genuine vorticity growth mechanism due to the presence of the swirl. This work and the companion work seem to be the first case when this mechanism has been effectively used to establish finite-time singularity formation for the actual $3D$ Euler system.