Determining the long time behavior of many partial differential equations modeling fluids has been a challenge for many years. In particular,
for many of these equations, the question of whether solutions exist for all time or form singularities is still open. The structure of the nonlinearity
and non-locality in these equations makes their analysis difficult using classical methods. In recent years, many models have been
proposed to study fluid equations. In this thesis, we will review some new result in regards to these models as well as give insight into the relation between these models and the true equations.
First, we analyze a one-dimensional model for the two-dimensional surface quasi-geostrophic equation and vortex sheets. The model
gained prominence due to the work of Cordoba, Cordoba, and Fontelos and is often referred to as the CCF model. We
will show that solutions are globally regular in the presence of logarithmically supercritcal dissipation and that solutions eventually
gain regularity in the presence of supercritical dissipation. Finally, by analyzing a dyadic model of the equation, we will gain insight into
how certain possible singularities in the CCF model can be supressed by dissipation.
For the second part of this thesis, we study some one-dimensional model equations for the Euler equations. These models are
influenced by the recent numerical simulations of Tom Hou and Guo Luo. They observed possible singularity formation for the
three-dimensional Euler equation at the boundary of a cylindrical domain under certain symmetry assumptions. Under these assumptions,
a singularity was observed numerically and the solution was observe to have hyperbolic structure near the singularity. Hou and Luo
proposed a one-dimensional model system to study singularity formation theoritically. We will study a family of one-dimensional models
generalizing their model. The results in chapter 2 are the results of joint work with A. Kiselev, V. Hoang, M. Radosz, and X. Xu.