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

Abstract

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}}~~~~~(0,T; L^{2}(\mathbb{R}^{3}))$. This generalizes a priori bounds for the classical Navier-Stokes system by Duff in [7, Acta Math. 164, 1990] and Boutros and Gibbon's spatial derivative estimates in [1, Nonlinearity 37, 2024]. In addition, we derive that $ u \in L^{\frac{q}{q-3}}~~(0,T;L^{q} (\mathbb{R}^{3}))$ with $ 6\leq q\leq\infty $ and $Λ^{k}u \in L^{\frac{q}{ q(k+1)-3}}~~~(0,T;L^{q} (\mathbb{R}^{3})) $ with $k\geq1, 2\leq q\leq\infty $ in the standard Navier-Stokes equations.