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On Galerkin approximations of the Navier-Stokes equations in the limit of large Grashof numbers

Ciprian Foias, Luan Hoang, Michael S. Jolly

Abstract

We examine how stationary solutions to Galerkin approximations of the Navier--Stokes equations behave in the limit as the Grashof number $G$ tends to $\infty$. An appropriate scaling is used to place the Grashof number as a new coefficient of the nonlinear term, while the body force is fixed. A new type of asymptotic expansion, as $G\to\infty$, for a family of solutions is introduced. Relations among the terms in the expansion are obtained by following a procedure that compares and totally orders positive sequences generated by the expansion. The same methodology applies to the case of perturbed body forces and similar results are obtained. We demonstrate with a class of forces and solutions that have convergent asymptotic expansions in $G$. All the results hold in both two and three dimensions, as well as for both no-slip and periodic boundary conditions.

On Galerkin approximations of the Navier-Stokes equations in the limit of large Grashof numbers

Abstract

We examine how stationary solutions to Galerkin approximations of the Navier--Stokes equations behave in the limit as the Grashof number tends to . An appropriate scaling is used to place the Grashof number as a new coefficient of the nonlinear term, while the body force is fixed. A new type of asymptotic expansion, as , for a family of solutions is introduced. Relations among the terms in the expansion are obtained by following a procedure that compares and totally orders positive sequences generated by the expansion. The same methodology applies to the case of perturbed body forces and similar results are obtained. We demonstrate with a class of forces and solutions that have convergent asymptotic expansions in . All the results hold in both two and three dimensions, as well as for both no-slip and periodic boundary conditions.
Paper Structure (14 sections, 17 theorems, 173 equations, 3 figures)

This paper contains 14 sections, 17 theorems, 173 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Implications for the NSE
  5. General expansion results
  6. A numerical demonstration for the 3D NSE
  7. On the uniqueness of strict unitary expansions
  8. Examples
  9. Proofs of main results
  10. Proof of Proposition \ref{['Sprop']}
  11. Proof of Theorem \ref{['thm1']}
  12. Proof of Proposition \ref{['v0eg']}
  13. Proof of Proposition \ref{['vAinvg']}
  14. Expanded force case

Key Result

Lemma 3.2

If $X$ is a countable subset of $\mathcal{X}$, then it has a totally comparable subsequential set.

Figures (3)

  • Figure 1: Bifurcation diagram.
  • Figure 2: Sequences in \ref{['seqarray']}.
  • Figure 3: Sequence ratios.

Theorems & Definitions (46)

  • Definition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Definition 3.4
  • Theorem 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Definition 4.1
  • Lemma 4.2: Expansion lemma
  • proof
  • ...and 36 more