Stability of time-dependent Navier-Stokes flow and algebraic energy decay

TL;DR

This paper investigates the algebraic decay rates of energy disturbances in time-dependent Navier-Stokes flows, extending stability results for small flows in weak-$L^n$ spaces to long-term decay behavior.

Contribution

It establishes algebraic energy decay rates for disturbances in time-dependent Navier-Stokes flows under smallness conditions, building on prior stability results.

Findings

Proves algebraic decay rates of energy disturbances over time

Extends stability analysis to flows in weak-$L^n$ spaces

Provides conditions for long-term decay in viscous fluid flows

Abstract

Let $V$ be a given time-dependent Navier-Stokes flow of an incompressible viscous fluid in the whole space ($n=3,4$). Assume such $V$ to be small in $L^\infty(0,\infty; L^{n,\infty})$, where $L^{n,\infty}$ denotes the weak-$L^n$ space. The energy stability of this basic flow $V$ with respect to any initial disturbance in $L^2_\sigma$ has been established by Karch, Pilarczyk and Schonbek. In this paper we study, under reasonable conditions, the algebraic rates of energy decay of disturbances as $t\to\infty$.