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

Abstract

We consider the Stokes operator $A$ on smooth exterior domains $\Omega$ of ${\mathbb{R}}^n$ in homogeneous Sobolev spaces ${\hat{H}}_w^{\kappa ,q}(\Omega )$ with radially symmetric Muckenhoupt weights $w\in {\mathcal{A}}_q$. A fundamental property is the existence of a bounded ${\mathcal{H}}^{\infty }$-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 ($\Vert u{\Vert }_{L_w^q}+\Vert A^{\theta }u{\Vert }_{L_w^q}$) as well as homogeneous norm ($\Vert A^{\theta }u{\Vert }_{L_w^q}$) as complex interpolation spaces. The final aim is the identification with homogeneous spaces $\hat{\mathcal{D}}((-{\Delta }_{q,w}{)}^{\theta })=[L_w^q,\hat{\mathcal{D}}(-{\Delta }_{q,w}){]}_{\theta }$ 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 $L_{\sigma ,w}^q(\Omega )$.