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Asymptotic expansions with subordinate variables for solutions of the Navier-Stokes equations

Luan Hoang

Abstract

We study the three-dimensional Navier-Stokes equations in a periodic domain with the force decaying in time. Although the force has a certain coherent decay, as time tends to infinity, it can be too complicated for the previous theory of asymptotic expansions to be applicable. To deal with this issue, we systematically develop a new theory of asymptotic expansions containing the so-called subordinate variables which can be defined recursively. We apply it to obtain an asymptotic expansion for any Leray-Hopf weak solutions. The expansion, in fact, is constructed explicitly and the impact of the subordinate variables can be clearly specified. The complexifications of the Gevrey-Sobolev spaces, and of the Stokes and bilinear operators of the Navier-Stokes equations are utilized to facilitate such a construction.

Asymptotic expansions with subordinate variables for solutions of the Navier-Stokes equations

Abstract

We study the three-dimensional Navier-Stokes equations in a periodic domain with the force decaying in time. Although the force has a certain coherent decay, as time tends to infinity, it can be too complicated for the previous theory of asymptotic expansions to be applicable. To deal with this issue, we systematically develop a new theory of asymptotic expansions containing the so-called subordinate variables which can be defined recursively. We apply it to obtain an asymptotic expansion for any Leray-Hopf weak solutions. The expansion, in fact, is constructed explicitly and the impact of the subordinate variables can be clearly specified. The complexifications of the Gevrey-Sobolev spaces, and of the Stokes and bilinear operators of the Navier-Stokes equations are utilized to facilitate such a construction.
Paper Structure (21 sections, 16 theorems, 269 equations)

This paper contains 21 sections, 16 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. The functional setting
  4. The real Gevrey--Sobolev spaces
  5. Complexified spaces and mappings
  6. General complexification
  7. Complexification in the context of the NSE
  8. Functions with subordinate variables and asymptotic expansions
  9. Functions with subordinate variables
  10. Substitutions for the subordinate variables
  11. Asymptotic expansions
  12. Main results
  13. Statements with complexified spaces
  14. Statements with real spaces
  15. Scenarios and examples
  16. ...and 6 more sections

Key Result

Lemma 2.2

If $L:X\to Y$ is a bounded linear mapping between two real inner product spaces, then $L_\mathbb C$ is also a bounded linear mapping, and Here, $\|\cdot\|_{\mathcal{B}(\cdot,\cdot)}$ denotes the norm of a bounded linear mapping.

Theorems & Definitions (47)

  • Definition 2.1
  • Lemma 2.2: H7
  • Definition 2.3
  • Lemma 2.4: H6
  • Lemma 3.1: CaH3
  • proof
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • ...and 37 more