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

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.