An intrinsic expansion approach to the Galerkin approximations for the Navier-Stokes equations

Abstract

We study the Galerkin approximation of the three-dimensional Navier-Stokes equations. In particular, we examine the convergence of these solutions in a sequence of finite dimensional spaces as the dimension goes to infinity. For any sequence of steady state or, respectively, time dependent Galerkin solutions that converges to a solution of the Navier-Stokes equations, we obtain a subsequence with an intrinsic asymptotic expansion in appropriate nested function spaces. Consequently, an induced asymptotic expansion is obtained in a more standard spatial Sobolev or, respectively, spatiotemporal Sobolev-Lebesgue space. In the case of steady states, we establish certain relations among leading terms of this expansion.