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

TL;DR

This paper investigates the convergence of Galerkin approximations for the 3D Navier-Stokes equations, establishing an intrinsic asymptotic expansion in nested function spaces and analyzing relations among leading terms for steady states.

Contribution

It introduces an intrinsic asymptotic expansion framework for Galerkin solutions of Navier-Stokes equations, linking finite-dimensional approximations to the full solution.

Findings

Existence of subsequences with asymptotic expansions

Induced expansions in Sobolev spaces

Relations among leading terms for steady states

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.