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 projection operator in terms of a continuous distribution of vortex dipoles which have a finite self-induced velocity. A generalization of the Euler equation is presented as a kinetic equation similar to the Vlasov-Poisson equation. An interesting feature of this generalization of the Euler equation is that it has nontrivial solutions in one space dimension. The stability of the spatially homogeneous solution is also studied. Distribution functions with a single maximum are found to be linearly stable, whereas those with two maxima can be unstable and the initial value problem ill-posed. Weak solutions Of this kinetic equation are found using a water-bag model and a simple model of inviscid 1D turbulence is developed.