The Navier-Stokes equations with body forces decaying coherently in time

TL;DR

This paper investigates the long-time behavior of solutions to the 3D Navier-Stokes equations with time-decaying body forces, establishing that solutions have a unique asymptotic expansion dictated by the force's asymptotics.

Contribution

It proves that all Leray-Hopf weak solutions have a unique asymptotic expansion determined by the body force, extending understanding of long-term dynamics under complex decay conditions.

Findings

Solutions admit an asymptotic expansion independent of initial conditions

The expansion is uniquely determined by the asymptotic behavior of the body force

The proof employs complexified Gevrey-Sobolev spaces and advanced functional analysis

Abstract

The long-time behavior of solutions of the three-dimensional Navier--Stokes equations in a periodic domain is studied. The time-dependent body force decays, as time $t$ tends to infinity, in a coherent manner. In fact, it is assumed to have a general and complicated asymptotic expansion which involves complex powers of $e^t$, $t$, $\ln t$, or other iterated logarithmic functions of $t$. We prove that all Leray-Hopf weak solutions admit an asymptotic expansion which is independent of the solutions and is uniquely determined by the asymptotic expansion of the body force. The proof makes use of the complexifications of the Gevrey-Sobolev spaces together with those of the Stokes operator and the bilinear form of the Navier-Stokes equations.