On the global evolution of vortex filaments, blobs, and small loops in 3D ideal flows

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 Hamilto- nian functions. The approximate models we analyze (essentially discrete and continuous vortex filaments and vortex loops) are related to some problem of classical physics con- cerning turbulence and also to the numerical approximation of flows with very high Reynolds number. Finally, we apply our strategy to discrete models for filaments used in numerical methods.