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 ${\mathrm{\Lambda }}^{n\alpha }{u}_t^{(m)}\in L^{\frac{2(6\alpha -5)}{4m\alpha +2n\alpha +4\alpha -5}}(0,T;L^2({\mathbb{R}}^3))$ and ${\mathrm{\Lambda }}^n{u}_t^{(m)}\in L^{\frac{2(6\alpha -5)}{4m\alpha +2n+4\alpha -5}}(0,T;L^2({\mathbb{R}}^3))$. 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 $6\le q\le \mathrm{\infty }$ and ${\mathrm{\Lambda }}^{k}u\in L^{\frac{q}{q(k+1)-3}}(0,T;L^q({\mathbb{R}}^3))$ with $k\ge 1,2\le q\le \mathrm{\infty }$ in the standard Navier-Stokes equations.