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

Luan Hoang, Michael S. Jolly

TL;DR

The paper develops an intrinsic asymptotic-expansion framework for Galerkin approximations to the three-dimensional Navier–Stokes equations with a fixed Grashof number. It proves the existence of unitary or degenerate intrinsic expansions for both stationary and time-dependent Galerkin solutions in nested Sobolev-type spaces and derives explicit relations between leading expansion terms and the NSE nonlinearities. For stationary problems, the authors present two complementary methods to characterize the nonlinear-term expansions and obtain linear relations among leading vectors; for time-dependent problems, they establish a time-resolved intrinsic expansion using a compact embedding lemma that extends Aubin–Lions concepts to fractional time derivatives. These results provide a detailed asymptotic structure of Galerkin approximations, connecting the dominant terms to the NSE operators and potentially informing numerical analysis and the understanding of convergence behavior in 3D flows.

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.

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

TL;DR

The paper develops an intrinsic asymptotic-expansion framework for Galerkin approximations to the three-dimensional Navier–Stokes equations with a fixed Grashof number. It proves the existence of unitary or degenerate intrinsic expansions for both stationary and time-dependent Galerkin solutions in nested Sobolev-type spaces and derives explicit relations between leading expansion terms and the NSE nonlinearities. For stationary problems, the authors present two complementary methods to characterize the nonlinear-term expansions and obtain linear relations among leading vectors; for time-dependent problems, they establish a time-resolved intrinsic expansion using a compact embedding lemma that extends Aubin–Lions concepts to fractional time derivatives. These results provide a detailed asymptotic structure of Galerkin approximations, connecting the dominant terms to the NSE operators and potentially informing numerical analysis and the understanding of convergence behavior in 3D flows.

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.
Paper Structure (15 sections, 12 theorems, 175 equations, 3 figures)

This paper contains 15 sections, 12 theorems, 175 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Steady state case
  4. Time dependent solutions
  5. Navier--Stokes preliminaries
  6. Background on intrinsic asymptotic expansions
  7. Stationary solutions
  8. Existence of the intrinsic expansions
  9. Properties of the expansions
  10. First method: Direct expansion for the nonlinear terms
  11. Second method: Expansion by substitution for the nonlinear terms
  12. Time-dependent solutions
  13. A compact embedding lemma
  14. Application to the NSE
  15. Computational example - by Chengzhang Fu

Key Result

Theorem 2.1

Let $s=1$ if $f\in V$, or $s\in [0,1)$ if $f\in H$. There exists a subsequence of $(v_n)_{n=1}^\infty$ of $(u_n)_{n=1}^\infty$ with a unitary or degenerate expansion, in the sense of Definition defsimnorm, of the form where $\mathcal{N}=\emptyset$ or $\mathcal{N}=\{1,2,\ldots N\}$ for some integer $N\ge 1$ or $\mathcal{N}=\mathbb N$, $w_k\in D(A^s)$ and $\Gamma_{k,n}>0$ for all $k\in \mathcal{N}$

Figures (3)

  • Figure 1: Physical-space plots of the scalar field $\omega$ (left) and the forcing term $g$ (right)
  • Figure 2: Left: convergence plots for $\Gamma_{1,n}$, $\rho_{1,n}$, and $G_{n}$. Right: the corresponding quotient plots.
  • Figure 3: Left: the product $\|v_n -v\|_{D(A)} \cdot \lambda_{n+1}^{\alpha_*}$ for various values of $\alpha_*$. Right: convergence $\mathcal{E}_n \to 0$.

Theorems & Definitions (31)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3: HJ1
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.6: HJ1
  • ...and 21 more