  We consider a wide class of approximate models of evolution of singular
distributions of vorticity in three dimensional incompressible fluids and we
show that they have global smooth solutions. The proof exploits the existence
of suitable Hamiltonian functions. The approximate models we analyze
(essentially discrete and continuous vortex filaments and vortex loops) are
related to some problem of classical physics concerning turbulence and also to
the numerical approximation of flows with very high Reynolds number. Finally,
we interpret our results as a basis to theoretical validation of numerical
methods used in state-of-the-art computations of turbulent flows.
