Applied Mathematics Colloquium: Dr Zachary Bradshaw

Title: Sparseness and regularity of Navier-Stokes flows

Abstract: In a series of papers Grujic and collaborators have developed a novel regularity theory for the 3D Navier-Stokes equations based on the sparseness of superlevel sets of the vorticity. In this talk, we give a simple proof that sufficiently sparse Navier-Stokes solutions do not develop singularities. This provides an alternative to the original approach of Grujic, which is based on analyticity and the harmonic measure maximum principle. We additionally explore Grujic et. al.'s approach to bridging the scaling gap by demonstrating that, at the level of specific functions possessing different sparseness properties, the bridging inclusions identified in the work of Bradshaw, Farhat and Grujic as well as Grujic and Xu do not rule out any additional singularities compared to membership in the energy class. This represents joint work with Dallas Albritton.