On classical solutions of the nonstationary Navier-Stokes equations in two and three dimensions

Proofs are presented of estimates based on Galerkin approximations for the existence of solutions to the nonstationary Navier-Stokes equation. Conditions are defined for solution of initial boundary value problems in three dimensions. Eigenfunctions of the Stokes problem in a bounded domain are obtained as a basis for a system of Galerkin approximations. Sequences of identities are found and integrated for an energy estimate, valid in any number of dimensions, and suitable for all Galerkin approximations. Classical solutions in two and three dimensions are analytically explored, and an estimate is found for arbitrary two-dimensional domains.