Asymptotic Expansions in Time for Rotating Incompressible Viscous Fluids

TL;DR

This paper derives asymptotic expansions for three-dimensional rotating incompressible viscous fluids governed by Navier-Stokes equations, using Gevrey space techniques and Poincaré wave analysis, applicable to all non-zero rotation speeds.

Contribution

It introduces a novel method combining Poincaré wave decomposition and Gevrey norms to analyze long-time behavior of rotating Navier-Stokes solutions.

Findings

Asymptotic expansions valid in all Gevrey spaces for weak solutions.

Results hold for all non-zero rotation speeds and solution averages.

Identification of infinite-dimensional invariant manifolds.

Abstract

We study the three-dimensional Navier--Stokes equations of rotating incompressible viscous fluids with periodic boundary conditions. The asymptotic expansions, as time goes to infinity, are derived in all Gevrey spaces for any Leray-Hopf weak solutions in terms of oscillating, exponentially decaying functions. The results are established for all non-zero rotation speeds, and for both cases with and without the zero spatial average of the solutions. Our method makes use of the Poincar\'e waves to rewrite the equations, and then implements the Gevrey norm techniques to deal with the resulting time-dependent bi-linear form. Special solutions are also found which form infinite dimensional invariant linear manifolds.