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

Abstract

On a bounded Lipschitz domain $\Omega \subset \mathbb {R}^d$, $d \ge 3$, we continue the study of Shen (Arch Ration Mech Anal 205(2):395–424, 2012) and of Kunstmann and Weis (J Evol Equ 387–409, 2016) 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 $|\frac{1}{p} - \frac{1}{2} |< \frac{1}{2d} + \varepsilon$ 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 (J Evol Equ 387–409, 2016).