Preprint B100/2019
Fluid Dynamics on Logarithmic Lattices and Singularities of Euler Flow
Ciro Sobrinho Campolina Martins
Keywords: Euler equations | blowup | chaos | incompressible flow

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 technique, which considers the Fluid Dynamics equations restricted to a logarithmic lattice in Fourier space with specially designed calculus and algebraic operations, giving rise to simplified models structurally identical to the original ones. The application of this technique to the 3D Euler flow clarifies the present controversy at the scales of existing DNS and provides unambiguous evidence of the following transition to the blowup, explained as a chaotic attractor in a renormalized system. The chaotic attractor has an anomalous multiscale structure, suggesting that the existing DNS strategies at the resolution accessible now (and presumably rather long into the future) may be unsuitable for the analysis of blowup.