On the $$\mathrm {L}^p$$ 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).