Self-focusing of helicity drives finite-time singularities in inviscid flows

TL;DR

This paper investigates finite-time singularities in inviscid Euler flows, proposing a self-similar model driven by helicity focusing, and classifies singularities as point-like or line-like with potential implications for Navier-Stokes.

Contribution

It introduces a semi-analytical self-similar framework for Euler equations, highlighting helicity's role in singularity formation and classifying singularities based on their geometric nature.

Findings

Helicity drives the finite-time blow-up via self-focusing.

Flow separates into a shrinking tubular region and an outer zero-helicity region.

Point-like singularity recovers Leray scaling with exponent 1/2.

Abstract

This paper deals with the longstanding quest of the possible existence of finite-time singularities in the equations governing the dynamics of inviscid fluids, namely, Euler equations. Here, two contributions are brought for the case of perfect fluids with finite initial energy. First, a self-similar velocity field inspired by Leray Ansatz is proposed which allows for a separation of variables that transforms the original partial differential Euler equations to a nonlinear system of ordinary differential equations. This system can be solved semi-analytically and allows a continuum set of solutions parametrised by a self-similar exponent, $\nu$. Second, we use the conservation laws of Euler equations to select the possible finite-time singular solutions and the related self-similar exponents. We find that the helicity is the driving mechanism of the blow-up through a self-focusing mechanism. The flow near the singularity separates into two phases. A first phase is within a tubular region that shrinks as a power-law $(t_c-t)^\nu$, with $t_c$ the blow-up time, where the helicity is focused. This region is separated by a sharp interface from an outer region where the vorticity, and thus helicity, is identically zero. We found that the finite-time singularity may be either point-like or line-like depending on the dynamics of the tubular region along its axis of symmetry. Incidentally for a point-like singularity we recover the Leray scaling $\nu=1/2$ paving the way to a generalisation of this approach for the Navier-Stokes equations. Finally, we conjecture that if the helicity vanishes initially, no finite-time singularity would be possible, since in this case the singularity occurs at infinite time from the initial condition.