Energy Decay for a Weak Solution of the Navier–Stokes Equation with Slowly Varying External Forces

We consider the Navier–Stokes system with slowly decaying external forces[formula]We show that the energy norm of a weak solution has non-uniform decay,‖u(t)‖2→0 (t→∞),under suitable conditions on the data f and a which make the energy of solution bounded in time. Also, we show the exact rate of the decay (uniform decay) of the energy,‖u(t)‖2⩽C(1+t)−ε,for external forces with a given explicit decay rate.