Superfluids are remarkable because they lack mechanisms of viscous dissipations, and because vorticity is concentrated in thin vortex lines - a property which arises from the existence and uniqueness of a macroscopic wave function. In this talk I shall review recent experiments and numerical simulations which reveal analogies and differences between the flow of ordinary fluids and the flow of superfluids. In particular, I shall describe conditions under which, in a homogeneous isotropic turbulent superfluid away from boundaries, the distribution of kinetic energy over the length scales is similar to the classical Kolmogorov distribution, and new insight into the properties of superfluid flow near boundaries.[-]

We consider the 1-D Schrödinger system with point vortex-type interactions that was derived by R. Klein, A. Majda and K. Damodaran and by V. Zakharov to modelize the dynamics of N nearly parallel vortex filaments in a 3-D incompressible fluid. We first prove a global in time result and display several classes of solutions. Then we consider the problem of collisions. In particular we establish rigorously the existence of a pair of almost parallel vortex filaments, with opposite circulation, colliding at some point in finite time. These results are joint works with E. Faou and E. Miot.[-]