2D Boussinesq equations with logarithmically super-critical conditions

Abstract

This thesis focuses on the regularity problem of two generalized two dimensional Boussinesq equations. The first model contains the critical level of diffusion and a double logarithmically super-critical velocity. The second model contains logarithmically super-critical dissipation. The proof takes the advantage of the two equivalent definitions of the dissipative operator. We also extend the Besov spaces to better suit the new operator. In Chapter 5, we give a small data regularity result for super-critical Surface Quasi-Geostrophic equations. This is achieved by generalize the definition of Only Small Shock first introduced in [21]. The proof also use the modulus of continuity approach in [53]. The last chapter deal with an axisymmetric Navier-Stokes model by Hou and Li in n-dimensional setting. The local and global regularity result is achieved by requiring a strong enough fractional Laplacian dissipation.