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

摘要

AbstractIn this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R3with corners π/n, n= 2, 3… We consider the case where the tangent components of the vorticity vector are given on the partS1of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev spaceWpl(Ω), for arbitrary naturallandp>1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with cornersdocumentclass{article}pagestyle{empty}begin{document}$$ {rm rot }upsilon {rm = }omega {rm, div }upsilon = 0,upsilon cdot bar n||_{partial Omega } = 6 $$end{document}wherev– velocity vector, ω – vorticity vector andnis an unit outward vector normal to the boundary,(2) the existence of solutions of the following evolution problem for given velocity vectordocumentclass{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