Temporal decay rates for weak solutions of the Navier-Stokes Equations with supercritical fractional dissipation

In this paper, we establish temporal decay for a weak solution $u(x,t)$ (with initial data u₀) of the Navier-Stokes equations with supercritical fractional dissipation α ∈ (0,5/4) in L²(R³) and Ḣˢ(R³) (s≤0). More precisely, we prove that u satisfies the following upper bound: \|u(t)\|₂²≤ C(1+t)-3-2p/2α, ∀ t>0. This estimate leads us to show the next inequality: \|u(t)\|_Ḣ-δ²≤ C(1+t)-3-2δ-2p/2α, ∀ t>0. These results are obtained by applying standard Fourier Analysis and they hold for α∈(0,5/4), p∈[-1,3/2), δ∈ [0, 3-2p/2) and u₀∈ L²(R³)∩ Yᵖ(R³) (and also u₀∈ L¹(R³) for $p=-1$ and a certain finite set of values of α).