Singular Sets of Riemannian Exponential Maps in Hydrodynamics

We prove that the singular sets for the Lagrangian solution maps of the two-dimensional inviscid Euler and generalized surface quasi-geostrophic equations are Gaussian null sets. To achieve this we carry out a spectral analysis of an operator related to the coadjoint representation of the algebra of divergence-free vector fields on the fluid domain. In particular, we establish sharp results on the Schatten-von Neumann class to which this operator belongs. Furthermore, its failure to be compact is directly connected to the absence of Fredholm properties of the corresponding Lagrangian solution maps. We show that, for the three-dimensional Euler equations and the standard surface quasi-geostrophic equation, this failure is in fact essential.