The Geometry of the Euler and the Navier-Stokes Equations

The first chapter of this thesis is devoted to the study of the Euler equations of hydrodynamics as geodesic equations on the infinite-dimensional group of volume-preserving diffeomorphisms of a compact two-dimensional manifold M. We prove that the set of conjugate vectors for the L2 exponential map contains an open and dense subset, called regular conjugate vectors, which forms a smooth, codimension one submanifold of the tangent space. We then proceed to obtain several normal forms for this map in a neighborhood of a regular conjugate vector. These results apply, more generally, for infinite-dimensional Riemannian manifolds with Fredholm exponential maps. In particular, they are true for any finite-dimensional Riemannian manifold, and some are new even in this context.

In the second chapter we study the Cauchy problem for the Navier-Stokes equations on a hyperbolic manifold of dimension three. Our main result is that the problem is ill-posed, in the sense that there are smooth initial data in the Leray-Hopf class for which there are non-unique Leray-Hopf solutions.