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

Abstract

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.