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

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 $t\to\infty$.