Statement of problem on vortical inviscid flow of barotropic and   incompressible fluids
  The question what information is necessary for determination of a unique solution of hydrodynamic equations for ideal fluid is investigated. Arbitrary inviscid flows of the barotropic fluid and of incompressible fluid are considered. After integrating hydrodynamic equations, all information on the fluid flow is concentrated in dynamic equations in the form of indefinite functions, whereas the initial and boundary conditions contain information on the fluid particle labeling. It is shown that for determination of the stationary flow of the incompressible fluid the vorticity on any stream line must be given. Giving the velocity on the boundary, one does not determine the vorticity, in general. If there are closed stream lines, the vorticity cannot be given on them via boundary conditions. This circumstance explains existence of different stationary vortical flows under the same boundary conditions. 
Yuri A. Rylov