The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to $\mathscr H^\infty$-calculus to Fractional Domains

Abstract

We consider the Stokes operator $A$ on smooth exterior domains $\Omega$ of $\mathbb{R}^n$ in homogeneous Sobolev spaces $\widehat H^{\kappa,q}_w(\Omega)$ with radially symmetric Muckenhoupt weights $w\in \mathscr A_q$. A fundamental property is the existence of a bounded $\mathscr H^\infty$-calculus of the Stokes operator on weighted nonhomogeneous and homogeneous $L^q$ Sobolev spaces. This property implies the existence of uniformly bounded purely imaginary powers $A^{it}$, $t\in\mathbb{R}$, and the characterization of domains of fractional powers $A^\theta$ equipped with nonhomogeneous ($\| u\|_{L^q_w} + \|A^\theta u\|_{L^q_w}$) as well as homogeneous norm ($\|A^\theta u\|_{L^q_w}$) as complex interpolation spaces. The final aim is the identification with homogeneous spaces $\widehat{\mathcal D}((-\Delta_{q,w})^\theta) = [L^q_{w},\widehat{\mathcal D}(-\Delta_{q,w})]_\theta$ intersected by a space of solenoidal vector fields. Moreover, we obtain weighted variational inequalities for weak solutions of the Stokes equations, weighted $L^q$-$L^r$ decay estimates of the Stokes semigroup and $L^p$-maximal regularity on $L^q_{\sigma,w}(\Omega)$.