Vortex dynamics in perfect fluids

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Abstract
I review the current status of a problem, relevant to both plasma physics and ordinary fluid mechanics, namely the long-time behaviour of solutions of the perfect fluid equations. In two space dimensions, thanks in particular to the work of D. Montgomery, the situation is now quite clear, since one expects the formation at long times of large vortices in a background of potential flow. In three dimensions, the situation is blurred, although its understanding is a central issue for fully developped turbulence. I present some new estimates for a possible scenario of self-similar blow up of solutions of 3D Euler. That turns out to be a rather subtle question, if one tries to stay consistent with the conservation of circulation and of energy.