2 00 6 Nonunique solution of the Cauchy problem for vortical flow of ideal barotropic fluid ?

The Cuachy problem for the 3D vortical flow of ideal barotropic fluid is considered. It is shown that the solution of the Cauchy problem is unique, if one considers seven dynamic equations for seven dependent variables: the density ρ, the velocity v and Lagrangian variables ξ = {ξ 1 , ξ 2 , ξ 3 }, labeling the fluid particles. If one considers only the closed (Euler) system of four equations for four dependent variables ρ, v, the solution is not unique. The fact is that the Euler system describes both the fluid motion at fixing labeling and the evolution of the fluid labeling, whose evolution is described by the Lin constraints (equations for variables ξ). If one ignores the Lin constraints at the solution of the Euler system, (or one considers the Lin constraints on the basis of the solution of the Euler system), nonunique solution of the Cauchy problem is obtained.