Math Graduate Student Seminar

Dyadic shell models are infinite systems of ODEs that retain the quadratic nonlinearity, energy identity, and scaling of the 3D incompressible Navier-Stokes equations. Such models can be realized as averaged Navier-Stokes equations at the PDE level, and some of them blow up in finite time despite satisfying the energy identity, showing that these structural features do not preclude singularity formation. Among dyadic models with these features, the Obukhov model is distinguished by its regulated cascade mechanism: the inviscid version is globally regular. However, the inviscid proof relies on exact energy conservation, which viscosity destroys. We prove local well-posedness at the critical index and global regularity of the viscous Obukhov model for all initial data in the critical Sobolev space and above, and we explain why recent work shows this result is sharp. The argument turns on a rescaling that reveals a viscous activation threshold for each shell. We also obtain growth rates for these Sobolev norms.