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

TL;DR

This paper develops a new asymptotic expansion theory with subordinate variables for solutions of the 3D Navier-Stokes equations under decaying forcing, enabling explicit construction and analysis of weak solutions' long-term behavior.

Contribution

It introduces a recursive asymptotic expansion framework with subordinate variables, extending previous theories to more complex decaying forces in Navier-Stokes solutions.

Findings

Explicit asymptotic expansions for Leray-Hopf solutions

Impact of subordinate variables clearly specified

Utilizes complexified Gevrey-Sobolev spaces for construction

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.