  On a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \geq 3$, we
continue the study of Shen and of Kunstmann and Weis of the Stokes operator on
$\mathrm{L}^p_{\sigma} (\Omega)$. We employ their results in order to determine
the domain of the square root of the Stokes operator as the space
$\mathrm{W}^{1 , p}_{0 , \sigma} (\Omega)$ for $\lvert \frac{1}{p} -
\frac{1}{2} \rvert < \frac{1}{d} + \varepsilon$ and some $\varepsilon > 0$.
This characterization provides gradient estimates as well as
$\mathrm{L}^p$-$\mathrm{L}^q$-mapping properties of the corresponding
semigroup. In the three-dimensional case this provides a means to show the
existence of solutions to the Navier--Stokes equations in the critical space
$\mathrm{L}^{\infty} (0 , \infty ; \mathrm{L}^3_{\sigma} (\Omega))$ whenever
the initial velocity is small in the $\mathrm{L}^3$-norm. Finally, we present a
different approach to the $\mathrm{L}^p$-theory of the Navier--Stokes equations
by employing the maximal regularity proven by Kunstmann and Weis.
