The asymptotic behavior solutions to damped Navier-Stokes equations in unbounded domain

This paper is concerned with time decay rates of the weak solutions to the 3D Navier-Stokes equations with nonlinear damping |u| p-1 u in R 3 + . With the aid of the spectral decomposition of the Stokes operator and L p — L q estimates, it shows that the weak solution decay in L 2 norm like ct −3/2r + 3/4 with the initial velocity u 0 ∊ L 2 σ ∩ L r for 1 R 3 + |x 3 u 0 (x)| r , and then we have the decay in L 2 norm like ct −3/2r + 3/4 .