Solvability of initial-boundary value problems for Euler's equations for flows of an ideal incompressible nonhomogeneous fluid and an ideal barotropic fluid bounded by free surfaces

The Nishida version of the abstract nonlinear Cauchy–Kovalevskaya theorem is used to obtain the results indicated in the title. In this connection, one has to construct special scales of Banach spaces and to estimate in them the solutions of elliptic equations