On the ${\mathrm{L}}^p$-theory of the Navier--Stokes equations on three-dimensional bounded Lipschitz domains

On a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$, $d\ge 3$, we continue the study of Shen and of Kunstmann and Weis of the Stokes operator on ${\mathrm{L}}_{\sigma }^p(\mathrm{\Omega })$. We employ their results in order to determine the domain of the square root of the Stokes operator as the space ${\mathrm{W}}_{0,\sigma }^{1,p}(\mathrm{\Omega })$ for $\lvert \frac{1}{p} - \frac{1}{2} \rvert < \frac{1}{d} + \varepsilon$andsome$\varepsilon > 0$. 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 ${\mathrm{L}}^{\mathrm{\infty }}(0,\mathrm{\infty };{\mathrm{L}}_{\sigma }^3(\mathrm{\Omega }))$ 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.