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 and of Kunstmann and Weis of the Stokes operator on ${\mathrm{L}}_{\sigma }^p(\Omega )$. We employ their results in order to determine the domain of the square root of the Stokes operator as the space ${\mathrm{W}}_{0,\sigma }^{1,p}(\Omega )$ for $|\frac{1}{p}-\frac{1}{2}|<\frac{1}{d}+\epsilon$ and some $\epsilon >0$. 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 ${\mathrm{L}}^{\infty }(0,\infty ;{\mathrm{L}}_{\sigma }^3(\Omega ))$ 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.

arXiv comment19 pages, replaced by a shortened version, some typo's were corrected