Finite-time blow-up in an elementary model of the 3D Navier-Stokes equations

TL;DR

This paper proves finite-time blow-up in a simplified shell model of 3D Navier-Stokes equations with smooth initial data, highlighting potential singularity formation in fluid dynamics models.

Contribution

It introduces a realistic shell model demonstrating finite-time singularity, contrasting with previous models that either regularize or rely on artificial interactions.

Findings

Finite-time blow-up demonstrated in the shell model.

Singularity formation occurs just above the energy level in the inviscid case.

Discussion on embedding the model's behavior into full Euler and Navier-Stokes equations.

Abstract

We demonstrate finite-time blow-up in a simple, realistic shell model of the 3D Navier-Stokes equations, equipped with "smooth" (i.e., rapidly decaying in frequency) initial data and forcing. Previously studied models either exhibit a turbulent cascade that regularizes the three-dimensional viscous dynamics, or rely on highly artificial interactions not transparently realized in the true Euler nonlinearity. We also treat the inviscid, unforced case and obtain singularity formation just above the energy level. We conclude with a discussion of the prospects for embedding the behavior of the dyadic model into the full Euler and Navier-Stokes equations.