Asymptotic behavior of weak solutions to the damped Navier-Stokes equations

Whether or not weak solutions of classical Navier-Stokes equations decay to zero in L ² as time tends to infinity was posed by Leray in his pioneering paper [12,13] and has been well solved (see [15] for instance). This paper examines the three dimensional Navier-Stokes equations with damping term |u| ^β−1 u and proves that the weak solutions decay to zero in L ² as time tends to infinity for any β≥1, by developing local-in-space estimate for weak solutions.