MiS Preprint Repository

We consider a generalization of the Stokes resolvent equation, where the constant viscosity is replaced by a general given positive function. Such a system arises in many situations as linearized system, when the viscosity of an incompressible, viscous fluid depends on some other quantities. We prove that an associated Stokes-like operator generates an analytic semi-group and admits a bounded $H_{\mathrm{\infty }}$-calculus, which implies the maximal $L^q$-regularity of the corresponding parabolic evolution equation. The analysis is done for a large class of unbounded domains with $W_r^{2-\frac{1}{r}}$-boundary for some $r>d$ with $r\ge q$. In particular, the existence of an $L^q$-Helmholtz projection is assumed.