Regularity of Solution Maps of the Generalized Surface Quasi-Geostrophic Equations

In Chapter 1, we present some preliminaries. In Chapter 2 we drive generalized SQG equations as Euler-Arnold equations. Chapter 3, we establish analyticity of the exponential map associated to the generalized SQG equations. In Chapter 4, we establish that the Eulerian data-to-solution map for the generalized SQG equations fails to be uniformly continuous in Sobolev spaces.