On Stokes operators with variable viscosity in bounded and unbounded domains

Abstract

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 ∞ -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 $${W^{2-\frac1r}_r}$$ -boundary for some r > d with r ≥ q, q′. In particular, the existence of an L q -Helmholtz projection is assumed.