Asymptotic expansions in a general system of decaying functions for solutions of the Navier-Stokes equations

TL;DR

This paper develops a framework for asymptotic expansions of solutions to the 3D Navier-Stokes equations under decaying forces, extending previous results to more general decay types including logarithmic and iterated logarithmic decay.

Contribution

It introduces a general system of decaying functions and proves that solutions admit unique asymptotic expansions determined by the force, covering new decay behaviors.

Findings

Solutions have asymptotic expansions matching the force's decay in Gevrey-Sobolev spaces.

The expansions are unique and solution-independent.

Includes new results for logarithmic and iterated logarithmic decay cases.

Abstract

We study the long-time dynamics of the Navier-Stokes equations in the three-dimensional periodic domains with a body force decaying in time. We introduce appropriate systems of decaying functions and corresponding asymptotic expansions in those systems. We prove that if the force has a large-time asymptotic expansion in Gevrey-Sobolev spaces in such a general system, then any Leray-Hopf weak solution admits an asymptotic expansion of the same type. This expansion is uniquely determined by the force, and independent of the solutions. Various applications of the abstract results are provided which particularly include the previously obtained expansions for the solutions in case of power decay, as well as the new expansions in case of the logarithmic and iterated logarithmic decay.