On Turbulent Dynamics and Related Theoretical Topics Associated with Diffusive Incompressible Fluid Models

This dissertation aims to establish rigorous results concerning the behavior of diffusive or fractionally diffusive fluid regimes and identify connections to and implications for theoretical topics relating to the possible irregularity of the corresponding mathematical models. The particular models of interest are the three dimensional Navier-Stokes equations, the magnetohydrodynamical equations, and the fractionally diffusive surface quasi-geostrophic equations.

Two issues concerning turbulent transport are examined. The first relates to the characterization and existence of turbulent cascades across inertial ranges in viscous (or fractionally diffusive) fluid systems and amounts to providing rigorous support for physically and numerically motivated descriptions of turbulent media. The second aims to establish sharp lower bounds for the dissipative length scales as these play an important role in turbulent transport dynamics and have applications to conditional regularity criteria involving anisotropic diffusion of the transverse length scales associated with coherent vortex structures.

The regularity problems associated with these models are challenging and generally remain open. An important issue in this area relates to the fact that the considered models are supercritical. Frequently, to obtain regularity criteria in such a setting, conditions must be included to render this supercritical behavior critical or, better yet, subcritical. Several results along these lines are included which are physically motivated by the anitropic filamentary description of turbulent fluids.