On space-time derivative estimates for the fractional Navier-Stokes equations

In this paper, we are concerned with space-time derivative estimates of solutions to the fractional Navier-Stokes equations. It is shown that $Λ^{nα}u^{(m)}_{t} \in L^{\frac{2(6α-5)}{4mα+2nα+4 α-5}}~~~~~~~(0,T; L^{2}(\mathbb{R}^{3}))$ and $ Λ^{n }u^{(m)}_{t} \in L^{\frac{2(6α-5)}{4mα+2n +4 α-5}}~~