Fluids and Analysis Seminar

Global regularity for the three-dimensional incompressible Navier--Stokes equations is an open problem in mathematical fluid dynamics. A natural approach to isolating the key mechanisms is to study dyadic shell models — infinite systems of ODEs that share the quadratic nonlinearity, energy conservation, and scaling of the full equations. Tao formalized this connection by showing that such models can be realized as averaged Navier--Stokes equations at the partial differential equations level; he then constructed a solution to an averaged Navier--Stokes partial differential equation that blows up in finite time, showing that these structural features alone do not preclude singularity formation.

In the inviscid setting, among all dyadic models satisfying these structural conditions, two basic building blocks emerge: the Katz--Pavlović model, in which the nonlinearity forcefully pushes energy toward ever-finer scales, blows up in finite time for any nontrivial data; and the Obukhov model (originally proposed in the study of atmospheric turbulence), where the cascade mechanism is regulated, admits global smooth solutions. The passage to the viscous problem is nontrivial: viscosity breaks the exact energy conservation underlying the inviscid argument. Viscosity interacts non-trivially with the cascade to reinforce the underlying regularizing structure. We prove that the viscous Obukhov model is globally regular for all sufficiently smooth initial data.