Temporal decay for the generalized Navier–Stokes equations

In this paper, we study the asymptotic behavior of solutions to the generalized incompressible Navier–Stokes equations ∂tu+(−Δ)αu+u⋅∇u+∇π=0,∂tu+(−Δ)αu+u⋅∇u+∇π=0,divu=0. We show that some weighted negative Besov norms of solutions are preserved along time evolution, and we obtain the optimal time decay rates of the higher-order spatial derivatives of solutions both in the subcritical case α∈(12,1] and the critical case α=12 by using the Fourier splitting approach and the interpolation techniques.