Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forces

TL;DR

This paper investigates the long-time asymptotic behavior of solutions to the 3D Navier-Stokes equations with non-potential forces, extending previous potential force results by analyzing how force structure affects solution decay and expansion.

Contribution

It establishes asymptotic expansions for solutions under non-potential forces, broadening understanding beyond potential force cases with detailed regularity and decay analysis.

Findings

Asymptotic expansions are derived for solutions with non-potential forces.

The structure of the force influences the solution's asymptotic behavior.

Results extend previous potential force analyses to more general force cases.

Abstract

We study the long-time behavior of spatially periodic solutions of the Navier-Stokes equations in the three-dimensional space. The body force is assumed to possess an asymptotic expansion or, resp., finite asymptotic approximation, in either Sobolev or Gevrey spaces, as time tends to infinity, in terms of polynomial and decaying exponential functions of time. We establish an asymptotic expansion, or resp., finite asymptotic approximation, of the same type for the Leray-Hopf weak solutions. This extends the previous results, obtained in the case of potential forces, to the non-potential force case, where the body force may have different levels of regularity and asymptotic approximation. In fact, our analysis identifies precisely how the structure of the force influences the asymptotic behavior of the solutions.