On the weighted decay for solutions of the Navier–Stokes system

We consider the question of to what extent the temporal decay ‖u(⋅,t)‖L2=O(t−γ0)‖u(⋅,t)‖L2=O(t−γ0) for solutions uu of the Navier–Stokes equation influences the decay of weighted norms. We prove that ‖|x|au(⋅,t)‖L2=O(t−γ0+a/2)‖|x|au(⋅,t)‖L2=O(t−γ0+a/2) holds under the condition 0≤a