Turbulence, Regularity, and Geometry in Solutions to the Navier-Stokes and Magnetohydrodynamic Equations

Inertial transport in the Navier--Stokes and magnetohydrodynamic equations is shown to concentrate enstrophy towards smaller scales under physically motivated and numerically supported assumptions. This is possible with an assumption of vorticity coherence wherever the velocity has large gradients in combination with interpreting enstrophy concentration in physical space, using averaged fluxes through spherical shells. Concentration of enstrophy is consistent with the dynamically generated vortex filaments or current sheets seen in numerical simulations of turbulent fluids. Complementing this, a Besov space regularity criterion is proven by relating the analytic condition of scaling behavior of the amplitude of high frequency components with the geometric property of sparseness of a super-level set. Together these results demonstrate deep connections between geometric aspects of velocity fields, regularity of solutions of deterministic fluid equations, and turbulence.