  In this paper, we present the result of maximum regularity of the mild
solution of the fractional Cauchy problem. As our main result, we investigate
the uniqueness of mild solutions for time-fractional Navier-Stokes equations in
class $C\left([0,\infty);L^{N}\left( \mathbb{R}^{N}\right)^{N}\right)$ by means
of the estimates $L^{p}-L^{q}$ of Giga-Shor inequality and the Gronwall
inequality.
