Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3

In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R 3 with corners π/ n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S 1 of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space W p l (Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners\documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm rot }\upsilon {\rm = }\omega {\rm, div }\upsilon = 0,\upsilon \cdot \bar n||_{\partial \Omega } = 6 $$\end{document} where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary,(2) the existence of solutions of the following evolution problem for given velocity vector\documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} \omega _t + \upsilon ^\kappa \omega _x \kappa - \omega ^\kappa \upsilon _x \kappa = F \equiv {\rm rot }f \\ \omega |_{t = 0} = \omega _0,\omega |_{s1} = \eta \\ \end{array} $$\end{document} (3) the method of successive approximations, using solvability of problems (1) and (2).