Goldshtik’s problem of pasting of vortical currents of an ideal liquid in the axially symmetric case

We consider that axially symmetric model of vortical flows of an ideal incompressible liquid with discontinuous nonlinear vorticity. The proposed model is a generalization of the Lavrentev’s scheme planar separated flows of an ideal fluid for the axially symmetric case. In terms of the the flow function we solve the Dirichlet problem for the inhomogeneous elliptic Euler-Poisson-Darboux equation with discontinuous nonlinearity is relative to the decision in the right part of the equation. This problem is a generalization of the well-known problem of Goldshtik of pasting planar vortical and potential flows of an ideal liquid on the axially symmetric case. The existence of the so-called trivial solution, which corresponds to the potential flows in the whole domain is shown. On a model example (flow in the ball) we establish the existence of two non-trivial solutions. For the general case of the problem we prove the existence of a nontrivial solution, indicating the existence of this class of axially symmetric vortical flows of an ideal liquid. In the model it is assumed that the stationary flow of an ideal liquid is a limiting flow of a viscous with viscosity tends to zero.