Analysts intersted in PDEs and in Hamiltonian dynamical systems have developed techniques for the phase space analysis of many model nonlinear Hamiltonian evolution equations. In this talk I will describe some applications of these ideas to problems in fluid dynamics, and in particular a version of an infinite dimensional KAM theory. The main application concerns the interaction of two near-parallel vortex filaments in three dimensions. As well, I will speculate about further applications of phase space analysis of Hamiltonian PDEs to other nonlinear systems of fluid dynamics, in the form of nonlinear evolution problems of physical significance.