A VLASOV DESCRIPTION OF THE EULER EQUATION

A form of the Euler equation using an impulse formulation is presented . This form is based on a representation of the divergence-free projec tion operator in terms of a continuous distribution of vortex dipoles which have a finite self-induced velocity. A generalization of the Eul er equation is presented as a kinetic equation similar to the Vlasov-P oisson equation. An interesting feature of this generalization of the Euler equation is that it has nontrivial solutions in one space dimens ion. The stability of the spatially homogeneous solution is also studi ed. Distribution functions with a single maximum are found to be linea rly stable, whereas those with two maxima can be unstable and the init ial value problem ill-posed. Weak solutions Of this kinetic equation a re found using a water-bag model and a simple model of inviscid 1D tur bulence is developed.