Classical solutions of the Navier-Stokes equations

The simplest, most elementary proofs of the existence of solutions of the Navier-Stokes equations are given via Galerkin approximation. The core of such proofs lies in obtaining estimates for the approximations from which one can infer their convergence (or at least the convergence of a subsequence of the approximations) as well as some degree of regularity of the resulting solution. The first to use this approach was Hopf [ 5 ], who based an existence theorem for the initial boundary value problem on an energy estimate for Galerkin approximations. However, based on this single estimate, Hopf's theorem provides very little regularity of the solution, in fact, insufficient regularity to prove the solution's uniqueness if the domain is three-dimensional. To remedy this situation, Kiselev and Ladyzhenskaya [ 7 ] introduced a second estimate for the approximations which yields enough further regularity for a uniqueness theorem. As is well known, this second estimate holds only locally in time unless the data are small or the domain is two-dimensional, a circumstance which has stimulated much speculation over the question of "unique solvability in the large". On the other hand, even during the time interval for which it holds, the estimate of Kiselev and Ladyzhenskaya provides far less than the full classical regularity of the solution.