Chaotic blowup in the 3D incompressible Euler equations on a logarithmic lattice

TL;DR

This paper introduces a new logarithmic lattice model for the 3D Euler equations, providing clear evidence of finite-time blowup through chaotic attractors, challenging current DNS methods and advancing understanding of fluid singularities.

Contribution

The authors propose a novel logarithmic lattice model that captures the blowup behavior of 3D Euler equations, clarifying the controversy and highlighting limitations of current DNS techniques.

Findings

Evidence of finite-time blowup via chaotic attractors

The model spans six decades of spatial scales

Current DNS resolutions are inadequate for blowup analysis

Abstract

The dispute on whether the three-dimensional (3D) incompressible Euler equations develop an infinitely large vorticity in a finite time (blowup) keeps increasing due to ambiguous results from state-of-the-art direct numerical simulations (DNS), while the available simplified models fail to explain the intrinsic complexity and variety of observed structures. Here, we propose a new model formally identical to the Euler equations, by imitating the calculus on a 3D logarithmic lattice. This model clarifies the present controversy at the scales of existing DNS and provides the unambiguous evidence of the following transition to the blowup, explained as a chaotic attractor in a renormalized system. The chaotic attractor spans over the anomalously large six-decade interval of spatial scales. For the original Euler system, our results suggest that the existing DNS strategies at the resolution accessible now (and presumably rather long into the future) are unsuitable, by far, for the blowup analysis, and establish new fundamental requirements for the approach to this long-standing problem.